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The proposed estimators share a structural similarity with the adjusted projection estimators and the difference estimators but they adopt a different inferential approach and informative setup. The pseudo-calibration estimators can be employed when the target variable is observed in the probability sample and, in the non-probability sample, it is observed correctly, observed with error, or predicted. This paper also introduces an original application of the jackknife-type method for variance estimation. A simulation study shows that the proposed estimators are robust and efficient compared to the regression data integration estimators that use the same informative setup. 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The proposed estimators share a structural similarity with the adjusted projection estimators and the difference estimators but they adopt a different inferential approach and informative setup. The pseudo-calibration estimators can be employed when the target variable is observed in the probability sample and, in the non-probability sample, it is observed correctly, observed with error, or predicted. This paper also introduces an original application of the jackknife-type method for variance estimation. A simulation study shows that the proposed estimators are robust and efficient compared to the regression data integration estimators that use the same informative setup. 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The proposed estimators share a structural similarity with the adjusted projection estimators and the difference estimators but they adopt a different inferential approach and informative setup. The pseudo-calibration estimators can be employed when the target variable is observed in the probability sample and, in the non-probability sample, it is observed correctly, observed with error, or predicted. This paper also introduces an original application of the jackknife-type method for variance estimation. A simulation study shows that the proposed estimators are robust and efficient compared to the regression data integration estimators that use the same informative setup. 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Sciences","Humanities","Law"],"image":["https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig1_HTML.png","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig2_HTML.png","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig3_HTML.png","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig4_HTML.png"],"isPartOf":{"name":"Statistical Methods & Applications","issn":["1613-981X","1618-2510"],"volumeNumber":"33","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer Berlin Heidelberg","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Natalia Golini","url":"http://orcid.org/0000-0003-4457-5781","affiliation":[{"name":"University of Turin","address":{"name":"Department of Economics and Statistics “Cognetti de Martiis”, University of Turin, Turin, Italy","@type":"PostalAddress"},"@type":"Organization"}],"email":"natalia.golini@unito.it","@type":"Person"},{"name":"Paolo Righi","affiliation":[{"name":"Italian National Statistical Institute (Istat)","address":{"name":"Italian National Statistical Institute (Istat), Rome, Italy","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" >  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>   <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" 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c-article-identifiers--cite-list"> <li class="c-article-identifiers__item"> <span data-test="journal-volume">Volume 33</span>, pages 555–580, (<span data-test="article-publication-year">2024</span>) </li> <li class="c-article-identifiers__item c-article-identifiers__item--cite"> <a href="#citeas" data-track="click" data-track-action="cite this article" data-track-category="article body" data-track-label="link">Cite this article</a> </li> </ul> <div class="app-article-masthead__buttons" data-test="download-article-link-wrapper" data-track-context="masthead"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s10260-023-00740-y.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" 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Integrating probability and big non-probability samples data to produce Official Statistics </div> <div data-test="inCoD" data-track-context="sticky banner"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s10260-023-00740-y.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> </div> </div> </div> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Natalia-Golini-Aff1" data-author-popup="auth-Natalia-Golini-Aff1" data-author-search="Golini, Natalia" data-corresp-id="c1">Natalia Golini<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0003-4457-5781"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0003-4457-5781</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup> & </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Paolo-Righi-Aff2" data-author-popup="auth-Paolo-Righi-Aff2" data-author-search="Righi, Paolo">Paolo Righi</a><sup class="u-js-hide"><a href="#Aff2">2</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>1644 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1007/s10260-023-00740-y/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>This paper introduces the pseudo-calibration estimators, a novel method that integrates a non-probability sample of big size with a probability sample, assuming both samples contain relevant information for estimating the population parameter. The proposed estimators share a structural similarity with the adjusted projection estimators and the difference estimators but they adopt a different inferential approach and informative setup. The pseudo-calibration estimators can be employed when the target variable is observed in the probability sample and, in the non-probability sample, it is observed correctly, observed with error, or predicted. This paper also introduces an original application of the jackknife-type method for variance estimation. A simulation study shows that the proposed estimators are robust and efficient compared to the regression data integration estimators that use the same informative setup. Finally, a further evaluation using real data is carried out.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1080/15598608.2013.856359?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1080/15598608.2013.856359">Improved Family of Estimators of Population Variance in Simple Random Sampling </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">01 June 2015</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs13571-024-00346-8/MediaObjects/13571_2024_346_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s13571-024-00346-8?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/s13571-024-00346-8">Semiparametric Model-Assisted Approach to Probabilistic Sampling of Finite Populations With High Right-Skew and Kurtosis </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">08 November 2024</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs10260-017-0380-4/MediaObjects/10260_2017_380_Fig1_HTML.gif" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s10260-017-0380-4?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/s10260-017-0380-4">Small area estimation based on M-quantile models in presence of outliers in auxiliary variables </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">22 March 2017</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1743586580, embedded_user: 'null' } }); </script> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=10260" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>In recent years, new data sources have emerged as a result of increased interactions with digital technologies by both citizens and business units, along with the growing capability of these technologies to generate digital trails. These sources, known as Big Data (BD) sources, encompass extensive amounts of digital information, including web surveys, search queries, website visits, social media activity, online purchases, self-reported administrative data sets, and other online interactions. BD sources typically comprise numerous records, often containing unstructured information, and are primarily generated for non-statistical purposes. They represent non-probability samples of the reference population. In many cases, they do not accurately represent the population of interest. Consequently, using them, for instance, to compute a simple mean of the observed values can lead to biased population mean estimates and erroneous conclusions, despite the large sample size (Bethlehem <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2010" title="Bethlehem J (2010) Selection bias in web surveys. Int Stat Rev 78(2):161–188" href="/article/10.1007/s10260-023-00740-y#ref-CR2" id="ref-link-section-d84953921e387">2010</a>; Vehovar et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2016" title="Vehovar V, Toepoel V, Steinmetz S (2016) Non-probability sampling, vol 1. The Sage handbook of survey methods" href="/article/10.1007/s10260-023-00740-y#ref-CR43" id="ref-link-section-d84953921e390">2016</a>; Meng <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2018" title="Meng XL (2018) Statistical paradises and paradoxes in big data (i) law of large populations, big data paradox, and the 2016 us presidential election. Ann Appl Stat 12(2):685–726" href="/article/10.1007/s10260-023-00740-y#ref-CR28" id="ref-link-section-d84953921e393">2018</a>). Notwithstanding these limitations, BD sources offer quick, easy, and cost-effective alternatives for obtaining data. They are becoming increasingly relevant in research and, notably, they present challenging sources of information for producing Official Statistics.</p><p>The use of BD sources is leading to a paradigm shift for National Statistical Institutes (NSIs), transitioning from planned statistics achieved through a designed process to data-oriented or data-driven statistics. Traditionally, NSIs rely on a designed process for collecting statistical data. This involves identifying the target population and its records, defining the target variables, planning the sampling design, and using efficient estimators. In the data-driven approach, the primary focus is on choosing the estimator that is most suitable for the task based on the observed variables. The process involves using a specific data collection tool, usually a digital device, on a sub-population selected through an unknown sampling technique. Horrigan (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2013" title="Horrigan MW (2013) Big data: A perspective from the BLS. AMSTAT News January:25–27" href="/article/10.1007/s10260-023-00740-y#ref-CR16" id="ref-link-section-d84953921e399">2013</a>) emphasizes the importance of creating transparent methodological documentation (metadata) describing how BD are used to construct any type of estimate. Citro (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2014" title="Citro CF (2014) From multiple modes for surveys to multiple data sources for estimates. Surv Methodol 40(2):137–162" href="/article/10.1007/s10260-023-00740-y#ref-CR8" id="ref-link-section-d84953921e402">2014</a>), Tam and Clarke (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2015a" title="Tam SM, Clarke F (2015) Big data, official statistics and some initiatives by the Australian Bureau of Statistics. Int Stat Rev 83(3):436–448" href="/article/10.1007/s10260-023-00740-y#ref-CR37" id="ref-link-section-d84953921e405">2015a</a>), Pfeffermann (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2015" title="Pfeffermann D (2015) Methodological issues and challenges in the production of official statistics: 24th annual Morris Hansen lecture. J Surv Stat Methodol 3(4):425–483" href="/article/10.1007/s10260-023-00740-y#ref-CR29" id="ref-link-section-d84953921e408">2015</a>) address the methodological uses and challenges of BD sources in the production of Official Statistics. Many reports have developed suitable statistical frameworks (among others: EUROSTAT <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2018" title="EUROSTAT (2018) Report describing the quality aspects of big data for official statistics. In: Work Package 8 Quality Deliverable 8.2. ESSnet Big Data" href="/article/10.1007/s10260-023-00740-y#ref-CR13" id="ref-link-section-d84953921e411">2018</a>; Japec et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2015" title="Japec L, Kreuter F, Berg M et al (2015) Big data in survey research: AAPOR task force report. Public Opin Q 79(4):839–880. 
 https://doi.org/10.1093/poq/nfv039
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR17" id="ref-link-section-d84953921e415">2015</a>) and quality frameworks (UNECE Big Data Quality Task Team <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2014" title="UNECE Big Data Quality Task Team (2014) A suggested framework for the quality of big data. Deliverables of the UNECE Big Data Quality Task Team" href="/article/10.1007/s10260-023-00740-y#ref-CR39" id="ref-link-section-d84953921e418">2014</a>; United Nations <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2019" title="United Nations (2019) United Nations National Quality Assurance Frameworks Manual for Official Statistics. United Nations publication" href="/article/10.1007/s10260-023-00740-y#ref-CR40" id="ref-link-section-d84953921e421">2019</a>; EUROSTAT <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="EUROSTAT (2020) Deliverable k3: Revised version of the quality guidelines for the acquisition and usage of big data. In: Workpackage K Methodology and quality. ESSnet Big Data II" href="/article/10.1007/s10260-023-00740-y#ref-CR14" id="ref-link-section-d84953921e424">2020</a>) that outline the fundamental principles and guidelines for using BD sources in producing Official Statistics. Several papers focusing on the accuracy and reliability of BD sources emphasize the growing need to determine the conditions under which BD sources can provide valid inferences. In this regard, many authors agree with the necessity of using methods combining data from big non-probability and probability samples to not severely sacrifice the quality of the estimates (Beaumont <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Beaumont JF (2020) Are probability surveys bound to disappear for the production of official statistics. Surv Methodol 46(1):1–28" href="/article/10.1007/s10260-023-00740-y#ref-CR1" id="ref-link-section-d84953921e427">2020</a>). Valliant (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Valliant R (2020) Comparing alternatives for estimation from nonprobability samples. J Surv Stat Methodol 8(2):231–263" href="/article/10.1007/s10260-023-00740-y#ref-CR41" id="ref-link-section-d84953921e430">2020</a>) and Rao (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Rao J (2021) On making valid inferences by integrating data from surveys and other sources. Sankhya B 83:242–272" href="/article/10.1007/s10260-023-00740-y#ref-CR30" id="ref-link-section-d84953921e434">2021</a>) provide insightful reviews of these methods. Kim (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2022" title="Kim JK (2022) A gentle introduction to data integration in survey sampling. Surv Stat 85:19–29" href="/article/10.1007/s10260-023-00740-y#ref-CR18" id="ref-link-section-d84953921e437">2022</a>) offers an extensive review of data integration techniques for combining a probability sample with a non-probability sample when the study variable is only observed in the non-probability sample. Most methods assume that the variable of interest is available only in the non-probability sample, while other auxiliary variables are present in both samples.</p><p>In this work, we assume that the target variable is observed in the probability sample, while in the big non-probability sample, it is (a) observed correctly, (b) observed with error, or (c) predicted using covariates collected in the big non-probability sample. A real case study inspiring our research is the 2018 European Community survey data on ICT usage and e-commerce in enterprises, conducted annually by Istat. The ICT probability survey sample data can be combined with the internet data scraped from the enterprises’ websites belonging to the ICT target population (big non-probability sample data). The target variables related to e-commerce functionalities, social media links, and presence of job advertisements can be observed, according to assumption (a), or predicted, according to assumption (c), using text-mining techniques on the scraped website data (Righi et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2019" title="Righi P, Bianchi G, Nurra A et al (2019) Integration of survey data and big data for finite population inference in official statistics: statistical challenges and practical applications. Stat Appl XVII(2):135–158" href="/article/10.1007/s10260-023-00740-y#ref-CR31" id="ref-link-section-d84953921e443">2019</a>). By integrating this additional information with the ICT survey, one can significantly improve the accuracy of the estimates. Another real case illustrating the type of BD we consider in this paper is given in Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2015" title="Tam SM (2015) A statistical framework for analysing big data. Surv Stat 72:36–51" href="/article/10.1007/s10260-023-00740-y#ref-CR36" id="ref-link-section-d84953921e446">2015</a>) and Tam and Clarke (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2015b" title="Tam SM, Clarke F (2015) Big data, statistical inference and official statistics—methodology research papers. Australian Bureau of Statistics, Canberra" href="/article/10.1007/s10260-023-00740-y#ref-CR38" id="ref-link-section-d84953921e449">2015b</a>). In these papers, the use of remote sensing for agricultural statistics using geo-localized satellite imagery and other satellite data (e.g., moisture, temperature) is investigated. After transforming the images into structured data (for instance, the reflectance data from frequency bands), the target variables (land use, crop type, crop yield) are predicted by supervised machine learning classification techniques. A probability sample of geo-localized areas collecting ground truth data is used as a training set. Another example is given by Rueda et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2023" title="Rueda MDM, Pasadas-del-Amo S, Rodríguez BC et al (2023) Enhancing estimation methods for integrating probability and nonprobability survey samples with machine-learning techniques. An application to a survey on the impact of the COVID-19 pandemic in Spain. Biom J 65(2):2200035" href="/article/10.1007/s10260-023-00740-y#ref-CR34" id="ref-link-section-d84953921e452">2023</a>) where an application of data integration techniques using a similar informative setup is provided. They consider a probability survey on the impact of the COVID-19 pandemic in Spain combined with a non-probability web-based survey. Both samples share the same questionnaire and measures.</p><p>In this paper, we introduce a novel class of estimators called pseudo-calibration (PC) estimators. They are based on big non-probability sample data, integrated with probability survey sample data and administrative or statistical registers. We also propose a variance estimation method based on the Delete-a-Group Jackknife technique (Kott <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2001" title="Kott PS (2001) Delete-a-group jackknife. J Off Stat 17(4):521–526" href="/article/10.1007/s10260-023-00740-y#ref-CR22" id="ref-link-section-d84953921e458">2001</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2006a" title="Kott PS (2006) Delete-a-group variance estimation for the general regression estimator under Poisson sampling. J Off Stat 22(4):759–767" href="/article/10.1007/s10260-023-00740-y#ref-CR23" id="ref-link-section-d84953921e461">2006a</a>). Specifically, we formalize the PC estimators initially introduced in an Istat technical report<sup><a href="#Fn1"><span class="u-visually-hidden">Footnote </span>1</a></sup> and employed in Righi et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2019" title="Righi P, Bianchi G, Nurra A et al (2019) Integration of survey data and big data for finite population inference in official statistics: statistical challenges and practical applications. Stat Appl XVII(2):135–158" href="/article/10.1007/s10260-023-00740-y#ref-CR31" id="ref-link-section-d84953921e475">2019</a>). The PC estimators are developed within a model-based framework, although an automatic calibration procedure, typical of model-assisted estimators, is carried out. We highlight that the proposed estimators have a similar structure to the <i>adjusted projection estimator</i> (Kim and Rao <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2011" title="Kim JK, Rao JNK (2011) Combining data from two independent surveys: a model-assisted approach. Biometrika 99(1):85–100" href="/article/10.1007/s10260-023-00740-y#ref-CR19" id="ref-link-section-d84953921e482">2011</a>) and the <i>difference estimators</i> (Breidt and Opsomer <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2017" title="Breidt FJ, Opsomer JD (2017) Model-assisted survey estimation with modern prediction techniques. Stat Sci 32:190–205" href="/article/10.1007/s10260-023-00740-y#ref-CR4" id="ref-link-section-d84953921e488">2017</a>), but a different inferential approach and informative setup. Furthermore, we show the analogies of the proposed estimators with the <i>doubly robust estimators</i> (Chen et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e494">2020</a>). Yet, we compare the proposed estimators with the <i>data integration estimators</i> proposed by Kim (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2022" title="Kim JK (2022) A gentle introduction to data integration in survey sampling. Surv Stat 85:19–29" href="/article/10.1007/s10260-023-00740-y#ref-CR18" id="ref-link-section-d84953921e501">2022</a>), developed in the same informative setup. The data integration estimators utilize both a probability and non-probability sample from the reference population. The target variable is observed in both samples, but there is a possibility of inaccurate measurement in one of the samples. The PC and data integration estimators employ calibration techniques, which are well-established methods used by National Statistical Institutes (NSIs), making them suitable for producing Official Statistics. However, the calibration methods differ significantly between these two classes of estimators. Precisely, the PC estimators aim to compute the weights of units in the non-probability sample; the data integration estimators seek to compute the weights of the probability sample units according to a model-assisted approach. With few exceptions, the two classes of estimators produce different estimates of the target parameter.</p><p>The paper is structured as follows. Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec2">2</a> introduces the basic notation and the informative context. A brief introduction of the data integration estimators (Kim and Tam <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e511">2021</a>) is in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec4">3</a>. Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec5">4</a> illustrates the novel class of PC estimators, and Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec10">5</a> shows the jackknife-type variance estimator. Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec11">6</a> presents the results of a Monte Carlo simulation on the performance of PC estimators, the comparison with the data integration estimators, and the accuracy of the jackknife-type variance estimator. Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec14">7</a> shows an application of the two classes of estimators on the motivating real survey data and BD source introduced above. Finally, some concluding remarks are in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec17">8</a>.</p><p>This paper is an extended version of the paper presented at the 51st Scientific Meeting of the Italian Statistical Society on June 2022 (Righi et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2022" title="Righi P, Golini N, Bianchi G (2022) Big data and official statistics: some evidence. In: Balzanella A, Bini M, Cavicchia C et al (eds) Book of short the papers: 51st scientific meeting of the Italian statistical society. Pearson, Hoboken, pp 723–734" href="/article/10.1007/s10260-023-00740-y#ref-CR32" id="ref-link-section-d84953921e536">2022</a>).</p></div></div></section><section data-title="Informative context"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Informative context</h2><div class="c-article-section__content" id="Sec2-content"><p>We estimate the parameters of the finite target population using a big non-probability sample, where the values of the target variable may either be observed correctly, observed with error, or predicted. To ensure valid inferences, we assume the following: (i) there exists a reference survey, with a probability sample drawn from the target population, where the target variable is observed correctly; (ii) it is possible to identify which units in the probability sample also belong to the non-probability sample; (iii) in the big non-probability sample, a set of auxiliary variables related to the target variable are available.</p><p>Assumptions (i) and (ii) are necessary for implementing the proposed PC estimators when the values of the target variable are observed with error or predicted in the large non-probability sample. Assumption (iii) underlines that the non-probability sample can serve as a source to gather informative covariates for predicting the target variable.</p><h3 class="c-article__sub-heading" id="Sec3"><span class="c-article-section__title-number">2.1 </span>Notation and basic setup</h3><p>Let <span class="mathjax-tex">$\mathcal {U} = \{1,\ldots ,N\}$</span> denote the target population of size <i>N</i>, let <span class="mathjax-tex">$Y = \Sigma _{i=1}^N y_i$</span> be the target parameter and <span class="mathjax-tex">$y_i$</span> the observed value of the variable <span class="mathjax-tex">$\mathcal {Y}$</span> for the unit <i>i</i>. We have two independent samples from the finite population <span class="mathjax-tex">$\mathcal {U}$</span>: a probability sample <span class="mathjax-tex">$\mathcal {S}_A$</span> of size <span class="mathjax-tex">$n_A$</span> and a big non-probability sample <span class="mathjax-tex">$\mathcal {S}_B$</span> of size <span class="mathjax-tex">$n_B$</span>. For each unit <span class="mathjax-tex">$i \in \mathcal {S}_A$</span>, we observe the values of a vector of auxiliary variables <span class="mathjax-tex">$\textbf{x}_i$</span> and the target variable <span class="mathjax-tex">$y_i$</span>. Within a design-based framework, <span class="mathjax-tex">$\hat{Y}_{HT,A} = \Sigma _{i \in A} d_i^{A} y_i$</span> stands as the design-unbiased Horvitz-Thompson estimator of <i>Y</i>, where <span class="mathjax-tex">$d_i^A = 1/\pi ^{A}_{i}$</span> denotes the sampling weight and <span class="mathjax-tex">$\pi ^{A}_{i} = Pr(i \in \mathcal {S}_A)$</span> is the first-order inclusion probability in <span class="mathjax-tex">$\mathcal {S}_A$</span>.</p><p>In the big non-probability sample <span class="mathjax-tex">$\mathcal {S}_B$</span>, the target variable <span class="mathjax-tex">$\mathcal {Y}$</span> can be observed correctly, with error, or predicted using a parametric or non-parametric model. In the first case, <span class="mathjax-tex">$y_i$</span> represents the observed value of <span class="mathjax-tex">$\mathcal {Y}$</span> for the unit <i>i</i>. In the latter two cases, the value of <span class="mathjax-tex">$\mathcal {Y}$</span> is denoted as <span class="mathjax-tex">$\tilde{y}_i$</span>. We use the notation <span class="mathjax-tex">$y^{*}_{i}$</span> to indicate either <span class="mathjax-tex">$y_i$</span> or <span class="mathjax-tex">$\tilde{y}_i$</span>.</p><p>We observe a vector of auxiliary variables <span class="mathjax-tex">$\textbf{x}_{i}$</span> for each unit <span class="mathjax-tex">$i \in \mathcal {U}$</span> and an additional vector of auxiliary variables <span class="mathjax-tex">$\textbf{x}_{i,B}$</span> for each unit <span class="mathjax-tex">$i \in \mathcal {S}_B$</span>. When the target variable cannot be observed in <span class="mathjax-tex">$\mathcal {S}_B$</span>, the vector <span class="mathjax-tex">$\textbf{x}_{i,B}$</span> contains good predictors for it.</p><p>The probability of a unit being included in the big non-probability sample, say <span class="mathjax-tex">$\pi _{i}^{B} = Pr(i \in \mathcal {S}_B)$</span>, is unknown. This probability is referred to as the propensity score. Let <span class="mathjax-tex">$\delta _i = I(i \in \mathcal {S}_B)$</span> be the indicator variable such that, <span class="mathjax-tex">$\delta _i = 1$</span> if <span class="mathjax-tex">$i \in \mathcal {S}_B$</span> and <span class="mathjax-tex">$\delta _i = 0$</span> if <span class="mathjax-tex">$i \notin \mathcal {S}_B$</span> <span class="mathjax-tex">$(i = 1, \ldots , N)$</span>. The propensity scores are given by <span class="mathjax-tex">$\pi _i^B=E_{p}(\delta _i\mid \textbf{x}_{i},y_i)= Pr(\delta _i = 1 \mid {\textbf {x}}_i, y_i)$</span>, where <i>p</i> refers to the model for generating <span class="mathjax-tex">$S_B$</span>.</p><p>Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab1">1</a> displays the data set available for the two samples and their representativeness.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-1"><figure><figcaption class="c-article-table__figcaption"><b id="Tab1" data-test="table-caption">Table 1 Data available for <span class="mathjax-tex">$\mathcal {S}_A$</span> and <span class="mathjax-tex">$\mathcal {S}_B$</span></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s10260-023-00740-y/tables/1" aria-label="Full size table 1"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>As in Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e2172">2021</a>) and Chen et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e2175">2020</a>), we assume that units belonging to <span class="mathjax-tex">$\mathcal {S}_A$</span> can be recognized in <span class="mathjax-tex">$\mathcal {S}_B$</span>. Therefore, it is possible to specify <span class="mathjax-tex">$\delta _i$</span> for each unit <span class="mathjax-tex">$i\in \mathcal {S}_A$</span>.</p></div></div></section><section data-title="Data integration estimators"><div class="c-article-section" id="Sec4-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec4"><span class="c-article-section__title-number">3 </span>Data integration estimators</h2><div class="c-article-section__content" id="Sec4-content"><p>The data integration (DI) estimators, developed by Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e2286">2021</a>), provide a versatile tool for properly utilizing big non-probability samples in finite population inference. The big non-probability sample (BD source) is treated as a finite population of incomplete or inaccurate observations that can be used as auxiliary information. Thus, a calibration estimator can be directly used to adjust sampling weights for each <span class="mathjax-tex">$i\in \mathcal {S}_A$</span>, to reproduce certain known population totals for both the target population <span class="mathjax-tex">$\mathcal {U}$</span> and a non-probability sample <span class="mathjax-tex">$\mathcal {S}_B$</span>. In Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e2360">2021</a>), the authors point out that if the fraction of the non-probability sample present in the finite population is not substantial, the efficiency gain achieved by the DI estimators is limited. Additionally, it is worth highlighting that making design-based inference is advantageous for NSIs, as they typically use this approach to produce Official Statistics.</p><p>The general form of the class of DI estimators is the Regression DI (RegDI) estimator which is defined as</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{RegDI} = \sum _{i \in \mathcal {S}_A} w^{A}_{i} y_i, \end{aligned}$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>where <span class="mathjax-tex">$\{w^{A}_{i}: i\in \mathcal {S}_A\}$</span> is the vector of calibrated weights. These weights are determined by solving the following optimization problem</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\left\{ \begin{array}{ll} \min \sum _{i \in \mathcal {S}_A} Q(d^{A}_{i}, w^{A}_{i})/q_i \\ \sum _{i \in \mathcal {S}_A} w^{A}_{i} \textbf{x}_{i} = \textbf{X} \end{array}\right. }, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>where <span class="mathjax-tex">$d^{A}_{i}$</span> represents the base sampling weight, <span class="mathjax-tex">$q_i$</span> is a known positive weight independent of <span class="mathjax-tex">$d_i^A$</span> and <span class="mathjax-tex">$\textbf{X} = \sum _{i \in \mathcal {U}} \textbf{x}_i$</span> is a vector of totals, including the totals of <span class="mathjax-tex">$\delta _i$</span> and <span class="mathjax-tex">$\delta _i y^{*}_{i}$</span>. These totals are assumed to be known or possibly to be estimated by a large and accurate survey (e.g., Dever and Valliant <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2010" title="Dever J, Valliant R (2010) A comparison of variance estimators for post-stratification to estimated control totals. Surv Methodol 36:45–56" href="/article/10.1007/s10260-023-00740-y#ref-CR9" id="ref-link-section-d84953921e2853">2010</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2016" title="Dever J, Valliant R (2016) General regression estimation adjusted for undercoverage and estimated control totals. J Surv Stat Methodol 4:289–318" href="/article/10.1007/s10260-023-00740-y#ref-CR10" id="ref-link-section-d84953921e2856">2016</a>). The function <span class="mathjax-tex">$Q(\cdot )$</span> is a distance function that can be defined, for example, as</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} Q(d^{A}_{i}, w^{A}_{i}; q_i) = \sum _{i \in \mathcal {S}_A} \frac{d^{A}_{i}}{q_i} \left( \frac{w^{A}_{i}}{d^A_i} - 1 \right) ^2. \end{aligned}$$</span></div><div class="c-article-equation__number"> (3) </div></div><p>It is important to note that, in practice, uniform weighting (<span class="mathjax-tex">$q_i=1$</span>) is commonly used, although sometimes different weights are employed (Deville and Särndal <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1992" title="Deville JC, Särndal CE (1992) Calibration estimators in survey sampling. J Am Stat Assoc 87:367–382" href="/article/10.1007/s10260-023-00740-y#ref-CR11" id="ref-link-section-d84953921e3067">1992</a>).</p><p>By specifying the terms of the RegDI estimator, one can derive various DI estimators. Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e3073">2021</a>) gives insight into the specific estimators. Furthermore, if we use alternative distance functions, we can obtain the calibration data integration estimators according to the definition by Deville and Särndal (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1992" title="Deville JC, Särndal CE (1992) Calibration estimators in survey sampling. J Am Stat Assoc 87:367–382" href="/article/10.1007/s10260-023-00740-y#ref-CR11" id="ref-link-section-d84953921e3076">1992</a>). Regression and calibration estimators are useful statistical tools for enhancing the precision of the sampling estimates and are commonly used to deal with unit non-response and the frame list under-coverage (Kott <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2006b" title="Kott PS (2006) Using calibration weighting to adjust for nonresponse and coverage errors. Surv Methodol 32(2):133" href="/article/10.1007/s10260-023-00740-y#ref-CR24" id="ref-link-section-d84953921e3079">2006b</a>; Särndal and Lundström <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2005" title="Särndal CE, Lundström S (2005) Estimation in surveys with nonresponse. John Wiley & Sons, Hoboken" href="/article/10.1007/s10260-023-00740-y#ref-CR35" id="ref-link-section-d84953921e3082">2005</a>).</p> <h3 class="c-article__sub-heading" id="FPar1">Remark 1</h3> <p>A special case of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ1">1</a>) can be obtained considering the distance function (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ3">3</a>) and setting <span class="mathjax-tex">$\textbf{x}_i=\delta _i y_i^*$</span> and <span class="mathjax-tex">$q_i=\delta _i y_i^*$</span>. We can define the RegDI estimator as</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{RegDI}=\hat{Y}_{HT,A}+\frac{\hat{Y}_{HT,A}}{\hat{Y}^{*(B)}_{HT,A}} \left( Y^{*(B)}-\hat{Y}^{*(B)}_{HT,A}\right) =\frac{\hat{Y}_{HT,A}}{\hat{Y}^{*(B)}_{HT,A}} Y^{*(B)}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (4) </div></div><p>where <span class="mathjax-tex">$\hat{Y}_{HT,A} = \sum _{i \in \mathcal {S}_A} d^{A}_{i} y_i$</span>, <span class="mathjax-tex">$\hat{Y}^{*(B)}_{HT,A}=\sum _{i \in \mathcal {S}_A} d^{A}_{i} \delta _i y^*_i$</span> and <span class="mathjax-tex">$Y^{*(B)}=\sum _{i \in \mathcal {S}_B} y^*_i$</span>.</p> <h3 class="c-article__sub-heading" id="FPar2">Remark 2</h3> <p>The RegDI estimators utilize <span class="mathjax-tex">$y_i^*$</span> as auxiliary information in a design-based approach. They exhibit greater efficiency compared to the Horvitz-Thompson estimator when <span class="mathjax-tex">$y_i^*$</span> is correlated with the variable <span class="mathjax-tex">$y_i$</span> observed in <span class="mathjax-tex">$\mathcal {S}_A$</span>, with maximum efficiency achieved when <span class="mathjax-tex">$y_i^* = y_i$</span>. It is worth noting that in large-scale multi-purpose surveys, more than one target variable may be observed or predicted in <span class="mathjax-tex">$\mathcal {S}_B$</span>. Consequently, the RegDI estimators could face excess calibration constraints, potentially making the calibration process unfeasible. Sampling errors may be notably large when these constraints are satisfied.</p> </div></div></section><section data-title="Pseudo-calibration estimators"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5"><span class="c-article-section__title-number">4 </span>Pseudo-calibration estimators</h2><div class="c-article-section__content" id="Sec5-content"><h3 class="c-article__sub-heading" id="Sec6"><span class="c-article-section__title-number">4.1 </span>Model-based estimators</h3><p>In this section, we consider the case (a), i.e., where the target variable is observed in both samples.</p><p>Unlike the design-based approach, the model-based approach utilizes data from a big non-probability sample and directly estimates the finite population parameter. This is achieved by summing the observed target variable for <span class="mathjax-tex">$i\in \mathcal {S}_B$</span> and the target variable predicted for <span class="mathjax-tex">$i \notin \mathcal {S}_B$</span>. In this case, inference can be made within a model-based framework. Prediction methods rely on defining a super-population model that generates the target variable <span class="mathjax-tex">$\mathcal {Y}$</span> (Valliant et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2000" title="Valliant R, Dorfman AH, Royall RM (eds) (2000) Finite population sampling and inference: a prediction approach. Wiley Series in Survey Methodology" href="/article/10.1007/s10260-023-00740-y#ref-CR42" id="ref-link-section-d84953921e3978">2000</a>). Let’s suppose that the finite population <span class="mathjax-tex">$(\textbf{x}_i, y_i)$</span>, for all <span class="mathjax-tex">$i\in \mathcal {U}$</span>, can be viewed as a random sample from the model <span class="mathjax-tex">$y_i = \mu (\textbf{x}_i) + \epsilon _i$</span>, where <span class="mathjax-tex">$\mu (\cdot )$</span> can take a parametric or an unspecified non-parametric form, and <span class="mathjax-tex">$\epsilon _i$</span> is an independent variable with zero mean and variance <span class="mathjax-tex">$V(\epsilon _i)=v(\textbf{x}_i) \sigma ^2$</span>, with the form of the variance function <span class="mathjax-tex">$v(\cdot )$</span> being known. This outcome model describes the dependence of the target variable on a vector of auxiliary variables <span class="mathjax-tex">$\textbf{x}$</span>. We can use this relationship to predict the values of the units not belonging to <span class="mathjax-tex">$\mathcal {S}_B$</span>, provided that the <span class="mathjax-tex">$\textbf{x}$</span> values are known for all <span class="mathjax-tex">$i\in \mathcal {U}$</span>. In practice, we utilize the dataset of the pooled sample <span class="mathjax-tex">$\{(\textbf{x}_{i}, y_i), i\in \mathcal {S}_B\cup \mathcal {S}_A\}$</span> to construct the outcome model and make predictions. Given a parametric outcome model, <span class="mathjax-tex">$y_i=\mu (\textbf{x}_i; \varvec{\beta }) + \epsilon_i$</span>, and a consistent estimator <span class="mathjax-tex">$\varvec{\hat{\beta }}$</span> of the parameters <span class="mathjax-tex">$\varvec{\beta }$</span>, we obtain the predictions as <span class="mathjax-tex">${{\hat{y}} }_i=\mu (\textbf{x}_i; \varvec{\hat{\beta }})$</span>. The estimator for the population total <i>Y</i> is defined as <span class="mathjax-tex">$\hat{Y}_m=\sum _{i\in \mathcal {S}_B}y_i+\sum _{i\notin \mathcal {S}_B} {\hat{y}}_i$</span>. If the assumption <span class="mathjax-tex">$E_{\mu }(y_i\mid \textbf{x}_i, \delta _i)=E_{\mu }(y_i\mid \textbf{x}_i)$</span> holds, we obtain unbiased model-based estimates. Finally, the model-based estimator can be defined as a weighted sum of the observed values, <span class="mathjax-tex">$\hat{Y}_m=\sum _{i\in \mathcal {S}_B} \omega _i y_i$</span>, where <span class="mathjax-tex">$\omega _i$</span> are the appropriate weights representing the units not belonging to <span class="mathjax-tex">$\mathcal {S}_B$</span> (Valliant et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2000" title="Valliant R, Dorfman AH, Royall RM (eds) (2000) Finite population sampling and inference: a prediction approach. Wiley Series in Survey Methodology" href="/article/10.1007/s10260-023-00740-y#ref-CR42" id="ref-link-section-d84953921e4894">2000</a>).</p><p>Another class of model-based estimators involves estimating the propensity scores. Since the selection mechanism of <span class="mathjax-tex">$\mathcal {S}_B$</span> is unknown, <span class="mathjax-tex">$\pi _i^B$</span> is estimated by a propensity score model exploting the dataset <span class="mathjax-tex">$\{(\delta _i,\delta _{i}y_{i},\textbf{x}_i), i\in \mathcal {V} \}$</span>, where <span class="mathjax-tex">$\mathcal {V}$</span> is either <span class="mathjax-tex">$\mathcal {U}$</span> or <span class="mathjax-tex">$\mathcal {S}_B$</span>. For example, let the propensity score model be parametric, <span class="mathjax-tex">$\pi _{i}^{B} = \pi (\textbf{x}_i,y_i, \varvec{\theta })$</span>, and let <span class="mathjax-tex">$\hat{\varvec{\theta }}$</span> be a consistent estimator of <span class="mathjax-tex">$\varvec{\theta }$</span>. The estimate of <span class="mathjax-tex">$\pi _{i}^{B}$</span> is then <span class="mathjax-tex">$\hat{\pi _{i}}^{B} = \pi (\textbf{x}_i,y_i, \hat{\varvec{\theta }})$</span>. Once estimated <span class="mathjax-tex">$\pi _i^B$</span>, the model-based estimator is given by <span class="mathjax-tex">$\hat{Y}_{\pi } = \sum _{i \in \mathcal {S}_B} y_i / \hat{\pi }_{i}^{B}$</span>. In practice, <span class="mathjax-tex">$\varvec{\theta }$</span> cannot be estimated when the model depends on the <span class="mathjax-tex">$y_i$</span> values since they are not observed for <span class="mathjax-tex">$i\notin \mathcal {S}_B$</span>. Given the assumptions</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">A.1:</span> <p>the selection indicator <span class="mathjax-tex">$\delta _i$</span> and the target variable <span class="mathjax-tex">$y_i$</span> are independent given the vector of covariates <span class="mathjax-tex">$\textbf{x}_i$</span>;</p> </li> <li> <span class="u-custom-list-number">A.2:</span> <p><span class="mathjax-tex">$\pi _i^B>0$</span> for all <span class="mathjax-tex">$i\in \mathcal {U}$</span>;</p> </li> <li> <span class="u-custom-list-number">A.3:</span> <p>the variables <span class="mathjax-tex">$\delta _i$</span> and <span class="mathjax-tex">$\delta _j$</span> are independent given <span class="mathjax-tex">$\textbf{x}_i$</span> and <span class="mathjax-tex">$\textbf{x}_j$</span> for <span class="mathjax-tex">$i\ne j$</span> with <span class="mathjax-tex">$i,j\in \mathcal {U},$</span></p> </li> </ol><br><p>then, by Chen et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e5795">2020</a>), <span class="mathjax-tex">$\pi _{i}^{B} = Pr(\delta _i = 1 \mid \textbf{x}_{i}, y_i) = Pr(\delta _i = 1 \mid \textbf{x}_i)$</span>. This model corresponds to the Missing At Random mechanism (MAR) as defined by Rubin (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1976" title="Rubin DB (1976) Inference and missing data. Biometrika 63:581–590" href="/article/10.1007/s10260-023-00740-y#ref-CR33" id="ref-link-section-d84953921e5912">1976</a>) and Little and Rubin (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2019" title="Little RJA, Rubin DB (2019) Statistical analysis with missing data, 3rd edn. Wiley, Hoboken" href="/article/10.1007/s10260-023-00740-y#ref-CR26" id="ref-link-section-d84953921e5915">2019</a>). The MAR model parameters can be estimated using the dataset <span class="mathjax-tex">$\{(\delta _i,\textbf{x}_i), i\in \mathcal {V} \}$</span>. For example, one may opt for a logistic propensity score model and employ a maximum likelihood consistent estimator when <span class="mathjax-tex">$\mathcal {V}\equiv \mathcal {U}$</span>. However, if <span class="mathjax-tex">$\mathcal {V}\equiv \mathcal {S}_B$</span>, the (log)likelihood function cannot be completely computed. The method relies on the reference survey sample, collecting the <span class="mathjax-tex">$\textbf{x}$</span> values for <span class="mathjax-tex">$i\in \mathcal {S}_A$</span>. Afterwards, a pseudo-likelihood function can be defined, and the maximum pseudo-likelihood estimates of <span class="mathjax-tex">$\varvec{\theta }$</span> can be computed (for details, refer to formula (4) in Chen et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e6105">2020</a>)). Given the propensity score estimates, the inverse probability weighted estimator can be estimated as <span class="mathjax-tex">$\hat{Y}_{IPW}=\sum _{i \in \mathcal {S}_B} y_i/\hat{\pi }_{i}^{B}$</span> being <span class="mathjax-tex">$\hat{\pi _{i}}^{B}=\pi ({\textbf {x}}_i, \hat{\varvec{\theta }})$</span> (Kott <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1994" title="Kott PS (1994) A note on handling nonresponse in sample surveys. J Am Stat Assoc 89(426):693–696" href="/article/10.1007/s10260-023-00740-y#ref-CR21" id="ref-link-section-d84953921e6266">1994</a>; Elliot and Valliant <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2017" title="Elliot M, Valliant R (2017) Inference for nonprobability samples. Stat Sci 32:249–264" href="/article/10.1007/s10260-023-00740-y#ref-CR12" id="ref-link-section-d84953921e6269">2017</a>). Chen et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e6272">2020</a>) show that, assuming the logistic regression model for the propensity scores, under the regularity conditions A1–A3 and other reasonable conditions (C1-C6 specified in the supplementary materials), then <span class="mathjax-tex">$\hat{Y}_{IPW}-Y =O_p(n_B^{-1/2})$</span>.</p><h3 class="c-article__sub-heading" id="Sec7"><span class="c-article-section__title-number">4.2 </span>Pseudo-calibration estimators when the target variable is observed in <span class="mathjax-tex">$\mathcal {S}_B$</span> </h3><p>In the case (a), we derive the PC estimators from the inverse probability weighted estimator. In this case, the maximum pseudo-likelihood estimator is replaced by a consistent estimator based on unbiased estimating functions. Consider the following class of estimating equations</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {U}} \delta _i \varvec{h}(\textbf{x}_i,\varvec{\theta }) - \sum _{i \in \mathcal {U}} \pi (\textbf{x}_i,\varvec{\theta }) \varvec{h}(\textbf{x}_i,\varvec{\theta }) = \varvec{0}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (5) </div></div><p>where <span class="mathjax-tex">$\varvec{h}(\textbf{x}_i,\varvec{\theta })$</span> is a predefined smooth function of <span class="mathjax-tex">$\varvec{\theta }$</span> that ensures the system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ5">5</a>) has a unique solution.</p><p>When <span class="mathjax-tex">$\textbf{x}_i$</span> is known for each <span class="mathjax-tex">$i\in \mathcal {U}$</span> and <span class="mathjax-tex">$\varvec{h}(\textbf{x}_i,\varvec{\theta })=\pi (\textbf{x}_i,\varvec{\theta })^{-1}\textbf{x}_i$</span>, the system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ5">5</a>) becomes the conventional calibration equations</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_B} w_{i}^{B} \textbf{x}_i = \sum _{i \in \mathcal {U}} \textbf{x}_i, \end{aligned}$$</span></div><div class="c-article-equation__number"> (6) </div></div><p>where <span class="mathjax-tex">$w_{i}^{B}=1/\pi (\textbf{x}_i,\varvec{\theta })$</span>.</p><p>When <span class="mathjax-tex">$\textbf{x}_i$</span> can only be observed for the units belonging to <span class="mathjax-tex">$\mathcal {S}_A$</span>, Chen et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e6993">2020</a>) propose to replace <span class="mathjax-tex">$\sum _{i \in \mathcal {U}} \pi (\textbf{x}_i,\varvec{\theta }) \varvec{h}(\textbf{x}_i,\varvec{\theta })$</span> with <span class="mathjax-tex">$\sum _{i \in \mathcal {S}_A} d^{A}_{i} \pi (\textbf{x}_i,\varvec{\theta }) \varvec{h}(\textbf{x}_i,\varvec{\theta })$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ5">5</a>), obtaining the class of estimating equations,</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_B} \varvec{h}(\textbf{x}_i,\varvec{\theta }) - \sum _{i \in \mathcal {S}_A} d_{i}^{A} \pi (\textbf{x}_i,\varvec{\theta }) \varvec{h}(\textbf{x}_i,\varvec{\theta }) = \varvec{0}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (7) </div></div><p>When <span class="mathjax-tex">$\varvec{h}(\textbf{x}_i,\varvec{\theta }) = \textbf{x}_i$</span>, the system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ7">7</a>) simplifies to</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_B} w_{i}^{B} \textbf{x}_i = \sum _{i \in \mathcal {S}_A} d_{i}^{A} \textbf{x}_i, \end{aligned}$$</span></div><div class="c-article-equation__number"> (8) </div></div><p>where the calibration is based on the estimated totals from the reference survey.</p><p>We can obtain the solution to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ6">6</a>) or (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ8">8</a>) through the standard calibration process (Deville and Särndal <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1992" title="Deville JC, Särndal CE (1992) Calibration estimators in survey sampling. J Am Stat Assoc 87:367–382" href="/article/10.1007/s10260-023-00740-y#ref-CR11" id="ref-link-section-d84953921e7572">1992</a>). The weights <span class="mathjax-tex">$w^{B}_{i}$</span> are determined by solving the optimization problem</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\left\{ \begin{array}{ll} \min \sum _{i \in \mathcal {S}_B} Q(d^{B}_{i}, w^{B}_{i}; q_i) \\ \sum _{i \in \mathcal {S}_B} w^{B}_{i} \textbf{x}_{i} = \textbf{X}^* \end{array}\right. }, \end{aligned}$$</span></div><div class="c-article-equation__number"> (9) </div></div><p>where <span class="mathjax-tex">$Q(\cdot )$</span> is a convex distance function, which may take the same form as illustrated in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ3">3</a>), replacing <span class="mathjax-tex">$d_{i}^{A}$</span> and <span class="mathjax-tex">$w_{i}^{A}$</span> by <span class="mathjax-tex">$d_{i}^{B}$</span> and <span class="mathjax-tex">$w_{i}^{B}$</span>, respectively. Additionally, the summation is indexed for <span class="mathjax-tex">$i\in \mathcal {S}_B$</span>. Here, <span class="mathjax-tex">$d^{B}_{i}$</span> represent the base sampling weights, <span class="mathjax-tex">$w^{B}_{i}$</span> are the calibration weights, and <span class="mathjax-tex">$\textbf{X}^*$</span> is a vector of known totals, denoted as <span class="mathjax-tex">$\textbf{X}$</span>, or estimated totals, denoted as <span class="mathjax-tex">${\varvec{\hat{X}}}$</span>, derived from an accurate reference survey (Dever and Valliant <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2010" title="Dever J, Valliant R (2010) A comparison of variance estimators for post-stratification to estimated control totals. Surv Methodol 36:45–56" href="/article/10.1007/s10260-023-00740-y#ref-CR9" id="ref-link-section-d84953921e8062">2010</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2016" title="Dever J, Valliant R (2016) General regression estimation adjusted for undercoverage and estimated control totals. J Surv Stat Methodol 4:289–318" href="/article/10.1007/s10260-023-00740-y#ref-CR10" id="ref-link-section-d84953921e8065">2016</a>).</p><p>At first glance, the PC estimators may appear to be a slight variation of the DI indicators proposed by Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e8072">2021</a>). However, there are key distinctions: the PC estimators operate on the propensity score of the non-probability sample, whereas the DI estimators work on the weights of the probability sample; the inference for the PC estimators is based on the outcome and a propensity score model, while the inference for the DI estimators is based on a model-assisted approach; the calibration constraints of the PC estimators do not use the target variable(s), while the calibration constraints of the DI estimators are strictly target variable-dependent. Since <span class="mathjax-tex">$\mathcal {S}_B$</span> is not a probability sample, the propensity scores <span class="mathjax-tex">$\pi _{i}^{B}$</span> and the base sampling weights <span class="mathjax-tex">$d^{B}_{i}=1/\pi _{i}^{B}$</span> for the units in <span class="mathjax-tex">$\mathcal {S}_B$</span> are unknown. Nevertheless, we make two alternative assumptions. The first is that we plan a census, but the frame list of <span class="mathjax-tex">$\mathcal {S}_B$</span> under-covers the target population <span class="mathjax-tex">$\mathcal {U}$</span>. Then, <span class="mathjax-tex">$\pi _{i}^{B}=1$</span> for all <span class="mathjax-tex">$i \in \mathcal {S}_B$</span> and the base sampling weights <span class="mathjax-tex">$d^{B}_{i}$</span> are adjusted to account for the under-coverage bias through a calibration estimator (Little and Rubin <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2019" title="Little RJA, Rubin DB (2019) Statistical analysis with missing data, 3rd edn. Wiley, Hoboken" href="/article/10.1007/s10260-023-00740-y#ref-CR26" id="ref-link-section-d84953921e8324">2019</a>; Kott <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2006b" title="Kott PS (2006) Using calibration weighting to adjust for nonresponse and coverage errors. Surv Methodol 32(2):133" href="/article/10.1007/s10260-023-00740-y#ref-CR24" id="ref-link-section-d84953921e8328">2006b</a>). The second assumption is that in the absence of information about the process generating <span class="mathjax-tex">$\mathcal {S}_B$</span>, the maximum likelihood estimate of <span class="mathjax-tex">$\pi _{i}^{B}$</span> is <span class="mathjax-tex">$n_B/N$</span> for all <span class="mathjax-tex">$i \in \mathcal {S}_B$</span>, and <span class="mathjax-tex">$d^{B}_{i} = d^{B}$</span> with <span class="mathjax-tex">$d^B=N/n_B$</span>. After observing the sample and the auxiliary variables within it, we improve the estimates of <span class="mathjax-tex">$\pi _{i}^{B}$</span>. In the space of possible final weight vectors, we look for the vector closest to the initial value of <span class="mathjax-tex">$d^{B}$</span>, reducing the variability of <span class="mathjax-tex">$w_{i}^{B}$</span> as much as possible. Furthermore, Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10260-023-00740-y#FPar7">7</a> shows that by setting <span class="mathjax-tex">$d^B=N/n_B$</span> for all <span class="mathjax-tex">$i \in \mathcal {S}_B$</span>, the RegDI estimator in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ4">4</a>) can be expressed as a special case of PC estimator. Eventually, the two proposed guesses provide the same vector of calibrated weights and, in general, we achieve the same solution when using <span class="mathjax-tex">$d^{B}_{i} = d^{B}$</span> regardless of the value of <span class="mathjax-tex">$d^{B}$</span>.</p><p>The general expression of PC estimators is given by</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{PC} = \sum _{i \in \mathcal {S}_B} w_{i}^B y^{}_{i}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (10) </div></div><p>The PC estimators ensure that the weighted distribution of the non-probability sample across auxiliary variables aligns with the distribution of those variables in the target population. They offer a simple and direct implementation and utilize well-established and widely used statistical calibration tools in NSIs.</p> <h3 class="c-article__sub-heading" id="FPar3">Remark 3</h3> <p>(Justification of the optimization problem). We solve the calibration equations, <span class="mathjax-tex">$\sum _{i \in \mathcal {S}_B} w^{B}_{i} \textbf{x}_{i} = \textbf{X}^*$</span>, by setting up the optimization problem (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ9">9</a>). In the special case of <span class="mathjax-tex">$d_i^B=1$</span>, we encounter a frame list under-coverage problem for <span class="mathjax-tex">$\mathcal {S}_B$</span>. Solving the optimization problem given in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ9">9</a>) is a commonly used strategy to address this issue. The basic idea is to limit the variability of <span class="mathjax-tex">$w_i^B$</span> and, through the choice of specific distance functions, prevent the occurrence of very large or negative values of <span class="mathjax-tex">$w_i^B$</span>. With <span class="mathjax-tex">$d_i^B=N/n_B$</span>, the optimization starts from the simple propensity score mean model, which does not incorporate any auxiliary variables. Then, we enhance the model by introducing explanatory auxiliary variables, aiming to find the model closest to the parsimonious mean model, obtaining the final weights.</p> <h3 class="c-article__sub-heading" id="FPar4">Remark 4</h3> <p>The pseudo-calibrated estimate converges to the true value of the target parameter as <span class="mathjax-tex">$n_B\rightarrow N$</span>, but its accuracy is sensitive to potential failures of the propensity score model, such as the violation of the MAR assumption, especially when dealing with small sample sizes. This is particularly notable in the case of sub-population estimates. A possible approach to enhancing the robustness of the PC estimators is to integrate a prediction model for the target variable and use a doubly robust estimator. When <span class="mathjax-tex">$\textbf{x}_i$</span> is known for each <span class="mathjax-tex">$i\in \mathcal {U}$</span>, we have</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{DR1} = \sum _{i \in \mathcal {S}_B} w_{i}^B (y^{}_{i}-\hat{{y}}_i)+\sum _{i \in \mathcal {U}} \hat{{y}}_i. \end{aligned}$$</span></div><div class="c-article-equation__number"> (11) </div></div><p>When <span class="mathjax-tex">$\textbf{x}_i$</span> is known for <span class="mathjax-tex">$i\in \mathcal {S}_B$</span> or <span class="mathjax-tex">$i\in \mathcal {S}_A$</span>, we have</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{DR2} = \sum _{i \in \mathcal {S}_B} w_{i}^B (y^{}_{i}-{\hat{y}}_i)+\sum _{i \in \mathcal {S}_A} d_i^A {\hat{y}}_i. \end{aligned}$$</span></div><div class="c-article-equation__number"> (12) </div></div><p> Chen et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e9535">2020</a>) show the theoretical properties of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ11">11</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ12">12</a>).</p> <h3 class="c-article__sub-heading" id="FPar5">Remark 5</h3> <p>We can test the MAR assumption of the propensity score model (MAR model) using the dataset <span class="mathjax-tex">$\{(\delta _i,y_i,\textbf{x}_i,), i\in \mathcal {S}_A \}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar6">Remark 6</h3> <p>The <span class="mathjax-tex">$\hat{Y}_{PC}$</span> estimator, with <span class="mathjax-tex">$\textbf{x}_i=x_i$</span> (where <span class="mathjax-tex">$x_i$</span> is a scalar), a distance function in the form given in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ3">3</a>), and <span class="mathjax-tex">$q_i=x_i$</span>, can be expressed as</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{PC}=\sum _{i\in {S}_B} y_i d_i^B\left( \frac{X}{\hat{X}_B}\right) , \end{aligned}$$</span></div><div class="c-article-equation__number"> (13) </div></div><p>where <span class="mathjax-tex">$X= \sum _{i \in \mathcal {U}} x_i$</span> and <span class="mathjax-tex">$\hat{X}_B=\sum _{i \in \mathcal {S}_B} x_i d_i^B$</span>. <span class="mathjax-tex">$\hat{Y}_{PC}$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ13">13</a>) is the PC ratio estimator. Note we only use the <span class="mathjax-tex">$x_i$</span> values for <span class="mathjax-tex">$i\in \mathcal {S}_B$</span>. Moreover, we can obtain a second version of the PC ratio estimator by replacing <i>X</i> with <span class="mathjax-tex">$\hat{X}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar7">Remark 7</h3> <p>In the case (a), the RegDI estimator in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ4">4</a>) can be reformulated as</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{RegDI} = \frac{N}{n_B}Y^{(B)}\left( \frac{n_B\hat{Y}_{HT,A}}{N\hat{Y}^{(B)}_{HT,A}}\right) =\sum _{i\in \mathcal {S}_B}y_i\frac{N}{n_B}\left( \frac{\hat{Y}_{HT,A}}{\hat{Y}_{HT,AB}}\right) , \end{aligned}$$</span></div><div class="c-article-equation__number"> (14) </div></div><p>where <span class="mathjax-tex">$Y^{(B)}=\sum _{i \in \mathcal {S}_B} y_i$</span>, <span class="mathjax-tex">$\hat{Y}_{HT,A} = \sum _{i \in \mathcal {S}_A} d^{A}_{i} y_i$</span>, <span class="mathjax-tex">$\hat{Y}^{(B)}_{HT,A}=\sum _{i \in \mathcal {S}_A} d^{A}_{i} \delta _i y_i$</span> and <span class="mathjax-tex">$\hat{Y}_{HT,AB}=\sum _{i\in \mathcal {S}_A \cap \mathcal {S}_B} y_i d_i^A \frac{N}{n_B}$</span>. The RegDI estimator in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ14">14</a>) is equivalent to the PC ratio estimator in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ13">13</a>) when <span class="mathjax-tex">$d_i^B=N/n_B$</span>. In this case, the PC ratio estimator incorporates <span class="mathjax-tex">$\mathcal {Y}$</span> as an auxiliary variable and performs calibration based on the unknown total population <span class="mathjax-tex">$X \equiv Y$</span>. Then, <i>Y</i> is replaced with <span class="mathjax-tex">$\hat{Y}_{HT,A}$</span>. Similarly, <span class="mathjax-tex">$\hat{X}_B$</span> is replaced with <span class="mathjax-tex">$\hat{Y}_{HT,AB}$</span>.</p> <h3 class="c-article__sub-heading" id="Sec8"><span class="c-article-section__title-number">4.3 </span>Pseudo-calibration estimators when the target variable is not observed in <span class="mathjax-tex">$\mathcal {S}_B$</span> </h3><p>When the target variable is not observed in <span class="mathjax-tex">$\mathcal {S}_B$</span>, we could be in the cases (b) or (c). In the case (b), we observe the target variable with error, i.e., <span class="mathjax-tex">$\tilde{y}_i$</span> is generated by a measurement error model as <span class="mathjax-tex">$\tilde{y}_i=e(y_i)+\epsilon _i$</span>, where <span class="mathjax-tex">$e(\cdot )$</span> is a <i>method</i> for estimating <span class="mathjax-tex">$\tilde{y}_i$</span>, <span class="mathjax-tex">$\epsilon _i$</span> such that are independent error terms with zero mean and variance <span class="mathjax-tex">$V(\epsilon _i)=v({y}_i) \sigma ^2$</span>. In the case (c) we predict its values according to a prediction model as <span class="mathjax-tex">$\tilde{y}_i = m({\textbf {z}}_i)+\epsilon _i$</span>, where <span class="mathjax-tex">$m(\cdot )$</span> is a <i>method</i> for predicting <span class="mathjax-tex">$\tilde{y}_i$</span>, <span class="mathjax-tex">${\textbf {z}}_i'=({\textbf {x}}'_{i,B},{\textbf {x}}')$</span> such that <span class="mathjax-tex">$\epsilon _i$</span> are independent error terms with zero mean and variance <span class="mathjax-tex">$V(\epsilon _i)=v(\textbf{z}_i)\sigma ^2$</span>. In both the cases, we can use the probability survey sample data, where the target variable is observed, to build the methods. Concerning the prediction methods, <span class="mathjax-tex">$m(\cdot )$</span> can belong to a very broad class of supervised prediction methods, encompassing both parametric and non-parametric methods, as well as machine learning techniques such as kernel methods, regression-tree (Hastie et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2001" title="Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Springer New York Inc., New York" href="/article/10.1007/s10260-023-00740-y#ref-CR15" id="ref-link-section-d84953921e11503">2001</a>) and random forest (Breiman <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2001" title="Breiman L (2001) Random forests. Mach Learn 45:5–32" href="/article/10.1007/s10260-023-00740-y#ref-CR5" id="ref-link-section-d84953921e11507">2001</a>). Non-parametric methods can be useful with high-dimensional and unstructured data, a scenario often encountered in big non-probability sources.</p><p>A real example of the case (c) is estimating the number of websites offering specific services, such as e-commerce. In this case, we can employ a web-scraping technique to collect text documents from the websites, perform text analysis, and then predict the presence of functionalities and services on the website using supervised machine learning techniques (Righi et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2019" title="Righi P, Bianchi G, Nurra A et al (2019) Integration of survey data and big data for finite population inference in official statistics: statistical challenges and practical applications. Stat Appl XVII(2):135–158" href="/article/10.1007/s10260-023-00740-y#ref-CR31" id="ref-link-section-d84953921e11513">2019</a>). Supervised machine learning methods learn from a labelled training set, which consists of predictors (<span class="mathjax-tex">$\textbf{z}_i$</span>) and their corresponding target values (<span class="mathjax-tex">${y}_i$</span>). After training, the method can be employed to make predictions on new, unseen data (<span class="mathjax-tex">$\tilde{y}_i$</span>). We assume to observe the target variable in <span class="mathjax-tex">$\mathcal {S}_A$</span>, where <span class="mathjax-tex">$\mathcal {S}_A\subset \mathcal {S}_B$</span> or <span class="mathjax-tex">$\mathcal {S}_A\cap \mathcal {S}_B \ne \emptyset$</span> (see Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec9">4.3.1</a>). We train <span class="mathjax-tex">$m(\cdot )$</span> on the dataset <span class="mathjax-tex">$\{(y_i, {\textbf {z}}_i): i \in \mathcal {S}_A \cap \mathcal {S}_B\}$</span> to obtain <span class="mathjax-tex">$\hat{m}(\cdot )$</span>. Then, we make deterministic predictions with <span class="mathjax-tex">$\bar{\tilde{y}}_{i} = \hat{m}({\textbf {z}}_i)$</span> or random predictions with <span class="mathjax-tex">$\hat{\tilde{y}}_{i} = \bar{\tilde{y}}_{i} + \hat{\epsilon }_i$</span>, where <span class="mathjax-tex">$\hat{\epsilon }_i$</span> represents the estimated random error terms. Plugging the deterministic predictions in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ10">10</a>), we obtain the projection pseudo-calibration estimator, <span class="mathjax-tex">$\hat{Y}_{PC}^P$</span>, similar to the <i>projection estimator</i> proposed by Kim and Rao (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2011" title="Kim JK, Rao JNK (2011) Combining data from two independent surveys: a model-assisted approach. Biometrika 99(1):85–100" href="/article/10.1007/s10260-023-00740-y#ref-CR19" id="ref-link-section-d84953921e12052">2011</a>). Plugging the random predictions in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ10">10</a>), we obtain</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{PC}^P = \sum _{i \in \mathcal {S}_B} w_{i}^B \hat{\tilde{y}}_{i}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (15) </div></div><p>If the prediction method is misspecified or fails to capture the true relationship between the predictors and the target variable, then the estimates produced by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ15">15</a>) are biased. In cases where <span class="mathjax-tex">$\mathcal {S}_A \subset \mathcal {S}_B$</span>, we introduce a correction term, defining the difference pseudo-calibration estimator in the case (c) as</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}^{D}_{PC} = \sum _{i \in \mathcal {S}_B} w_{i}^B \hat{\tilde{y}}_{i} + \sum _{i \in \mathcal {S}_A} d^{A}_{i} (y_{i} - \hat{\tilde{y}}_{i}). \end{aligned}$$</span></div><div class="c-article-equation__number"> (16) </div></div><p>In the case (b), we can define an estimator similar to <span class="mathjax-tex">$\hat{Y}^{D}_{PC}$</span> by replacing <span class="mathjax-tex">$\hat{\tilde{y}}_{i}$</span> with <span class="mathjax-tex">$\tilde{y}_i$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ16">16</a>).</p> <h3 class="c-article__sub-heading" id="FPar8">Remark 8</h3> <p>The estimator <span class="mathjax-tex">$\hat{Y}^{D}_{PC}$</span> shares similarities with the estimator the <span class="mathjax-tex">$\hat{Y}_{DR2}$</span> described in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ12">12</a>). Notice how <span class="mathjax-tex">$\hat{Y}^{D}_{PC}$</span> reverses the roles of probability and non-probability samples compared to the <span class="mathjax-tex">$\hat{Y}_{DR2}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar9">Remark 9</h3> <p>The estimator <span class="mathjax-tex">$\hat{Y}^{D}_{PC}$</span> shares a similar structure with the adjusted <i>projection estimator</i> proposed by Kim and Rao (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2011" title="Kim JK, Rao JNK (2011) Combining data from two independent surveys: a model-assisted approach. Biometrika 99(1):85–100" href="/article/10.1007/s10260-023-00740-y#ref-CR19" id="ref-link-section-d84953921e12672">2011</a>) and the <i>difference estimator</i> developed by Breidt and Opsomer (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2017" title="Breidt FJ, Opsomer JD (2017) Model-assisted survey estimation with modern prediction techniques. Stat Sci 32:190–205" href="/article/10.1007/s10260-023-00740-y#ref-CR4" id="ref-link-section-d84953921e12678">2017</a>), both defined in the model-assisted framework.</p> <h3 class="c-article__sub-heading" id="FPar10">Remark 10</h3> <p>The asymptotic properties of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ16">16</a>) when <span class="mathjax-tex">$\hat{m}(\cdot )$</span> is used instead of <span class="mathjax-tex">$m(\cdot )$</span> are outlined in Breidt and Opsomer (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2017" title="Breidt FJ, Opsomer JD (2017) Model-assisted survey estimation with modern prediction techniques. Stat Sci 32:190–205" href="/article/10.1007/s10260-023-00740-y#ref-CR4" id="ref-link-section-d84953921e12758">2017</a>). They provide conditions under which the differences <span class="mathjax-tex">$(\tilde{y}_i - \hat{\tilde{y}}_i)$</span> can be considered negligible for many parametric and non-parametric methods. This theory is developed in the model-assisted framework, which is the context of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ16">16</a>) when considering the big non-probability sample frame list affected by under-coverage. Additionally, Chen et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e12823">2020</a>) offer insights into the asymptotic properties in the model-based framework, particularly when the propensity score model is the logistic model, and the outcome model is parametric.</p> <h3 class="c-article__sub-heading" id="FPar11">Remark 11</h3> <p>Given <span class="mathjax-tex">$\hat{Y}_{RegDI}$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ4">4</a>), where <span class="mathjax-tex">$y^{*}_{i} =\hat{\tilde{y}}_{i}$</span>, and assuming <span class="mathjax-tex">$m(\cdot )$</span> such that <span class="mathjax-tex">$E_m(\hat{\tilde{y}}_{i}) = y_i$</span>, then,</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} E_{m}(\hat{Y}_{RegDI})= Y^{(B)}\frac{\hat{Y}_{HT,A}}{\hat{Y}^{(B)}_{HT,A}}+\text {Term of minor order}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (17) </div></div><p>When we consider <span class="mathjax-tex">$\hat{Y}_{PC}^{D}$</span>, with the first term defined as in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ13">13</a>), and <span class="mathjax-tex">$w_i^B$</span> being independent of <span class="mathjax-tex">$y_i$</span>, we have that <span class="mathjax-tex">$E_{m}(\hat{Y}_{RegDI})\approx E_m(\hat{Y}^{D}_{PC})$</span>.</p> <h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec9"><span class="c-article-section__title-number">4.3.1 </span>Pseudo-calibration estimators when <span class="mathjax-tex">$\mathcal {S}_A\cap \mathcal {S}_B \ne \mathcal {S}_A$</span> </h4><p>In some cases, it may happen that <span class="mathjax-tex">$\mathcal {S}_A \not \subset \mathcal {S}_B$</span> and <span class="mathjax-tex">$\mathcal {S}_A\cap \mathcal {S}_B \ne \emptyset$</span>, meaning that for certain units in <span class="mathjax-tex">$\mathcal {S}_A$</span>, we cannot observe <span class="mathjax-tex">${\textbf {x}}_B$</span>. This condition generally arises when we plan <span class="mathjax-tex">$\mathcal {S}_A$</span> independently from <span class="mathjax-tex">$\mathcal {S}_B$</span>, but other practical reasons may also lead to this situation. For instance, in the previous example of business statistics regarding the services and functionalities of enterprise websites, the condition <span class="mathjax-tex">$\mathcal {S}_A \not \subset \mathcal {S}_B$</span> and <span class="mathjax-tex">$\mathcal {S}_A\cap \mathcal {S}_B \ne \emptyset$</span> arises when we select enterprises in <span class="mathjax-tex">$\mathcal {S}_A$</span> that implement anti-scraping techniques to block automatic scraping procedures on their websites. Consequently, these enterprises cannot belong to <span class="mathjax-tex">$\mathcal {S}_B$</span>. We handle this situation as a non-response problem and replace <span class="mathjax-tex">$d^{A}_{i}$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ16">16</a>) with <span class="mathjax-tex">$f^{A}_{i}$</span>, which are the final adjusted weights, since <span class="mathjax-tex">$\mathcal {S}_A$</span> is not fully included in <span class="mathjax-tex">$\mathcal {S}_B$</span>. The difference pseudo-calibration estimator (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ16">16</a>) can be rewritten in this case as</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}^{D}_{PC} = \sum _{i \in \mathcal {S}_B} w_{i}^B \hat{\tilde{y}}_{i} + \sum _{i \in \mathcal {S}_A\cap \mathcal {S}_B} f^{A}_{i} (y_{i} - \hat{\tilde{y}}_{i}). \end{aligned}$$</span></div><div class="c-article-equation__number"> (18) </div></div></div></div></section><section data-title="Variance estimation"><div class="c-article-section" id="Sec10-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec10"><span class="c-article-section__title-number">5 </span>Variance estimation</h2><div class="c-article-section__content" id="Sec10-content"><p>We estimate the variance of PC estimators using a jackknife-type method based on an adjusted version of the Delete-a-Group Jackknife (DAGJK) method (Kott <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2001" title="Kott PS (2001) Delete-a-group jackknife. J Off Stat 17(4):521–526" href="/article/10.1007/s10260-023-00740-y#ref-CR22" id="ref-link-section-d84953921e13975">2001</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2006a" title="Kott PS (2006) Delete-a-group variance estimation for the general regression estimator under Poisson sampling. J Off Stat 22(4):759–767" href="/article/10.1007/s10260-023-00740-y#ref-CR23" id="ref-link-section-d84953921e13978">2006a</a>), which is suitable for handling huge sample sizes. The DAGJK method offers computational advantages over the traditional Jackknife technique. It is well-suited for complex sampling strategies involving stratified design, several sampling phases, adjustment for non-response, calibration and composite estimation (Kott <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2001" title="Kott PS (2001) Delete-a-group jackknife. J Off Stat 17(4):521–526" href="/article/10.1007/s10260-023-00740-y#ref-CR22" id="ref-link-section-d84953921e13981">2001</a>). The variance estimation is asymptotically unbiased when the target parameter is a smooth function of the stratum means. However, it is not possible to guarantee that the DAGJK quantile variance estimation is unbiased.</p><p>The DAGJK method defines <i>G</i> random replication groups drawn from the parent sample, i.e., <span class="mathjax-tex">$\mathcal {S}_B$</span> and <span class="mathjax-tex">$\mathcal {S}_A$</span>. Then, <i>G</i> estimation processes are carried out using the sampled data, excluding the units of one replication group. For the <span class="mathjax-tex">$g-$</span>th <span class="mathjax-tex">$(g=1, \ldots , G)$</span> replicated estimate, the method computes a weight for each unit, <span class="mathjax-tex">$w_i^{B(g)}$</span> and <span class="mathjax-tex">$d_i^{A(g)}$</span> based respectively on the <span class="mathjax-tex">$w_i^B$</span> and <span class="mathjax-tex">$d_i^A$</span> weights adjusted by the exclusion of the units in the group <i>g</i>. When <span class="mathjax-tex">$y_i^*=y_i$</span>, the DAGJK variance estimation is</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} v(\hat{Y}_{PC})=\frac{G-1}{G}\sum _{g=1}^G ( \hat{Y}_{PC}^{(g)}-\hat{Y}_{PC})^2 , \end{aligned}$$</span></div><div class="c-article-equation__number"> (19) </div></div><p>with <span class="mathjax-tex">$\hat{Y}_{PC}^{(g)}=\sum _{i\in \mathcal {S}_B}w_i^{B(g)}y_i$</span>, being <span class="mathjax-tex">$w_i^{B(g)}=0$</span> when the unit <i>i</i> belongs to group <i>g</i>.</p><p>Let <span class="mathjax-tex">$y_i^*=\bar{\tilde{y}}_i$</span> be a deterministic prediction. In this case, we add a variability correction term into (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ19">19</a>) and rewrite the DAGJK variance estimator as</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} v(\hat{Y}_{PC}^{D})&= \frac{G-1}{G}\Bigl \{ \sum _{g=1}^G \biggr ( \sum _{i\in \mathcal {S}_B}w_i^{B(g)}\bar{\tilde{y}}_i-\hat{Y}_{PC}^{P} \biggr )^2 \nonumber \\&+ \sum _{g=1}^G \Bigl [\sum _{i \in \mathcal {S}_A} d^{A(g)}_{i} (y_{i} - \bar{\tilde{y}}_i)-\sum _{i \in \mathcal {S}_A} d^{A}_{i} (y_{i} - \bar{\tilde{y}}_i)\Bigr ]^2 \Bigr \} , \end{aligned}$$</span></div><div class="c-article-equation__number"> (20) </div></div><p>being <span class="mathjax-tex">$w_i^{B(g)}=0$</span> and <span class="mathjax-tex">$d^{A(g)}_{i}=0$</span> when the unit <i>i</i> belongs to group <i>g</i>. Finally, let <span class="mathjax-tex">$y_i^*=\hat{\tilde{y}}_i+\hat{\epsilon }_i$</span> be a random prediction, where <span class="mathjax-tex">$\hat{\epsilon }_i$</span> is the estimated error term obtained from the dataset <span class="mathjax-tex">$\{(y_i,{\textbf {z}}_i): i \in \mathcal {S}_A\cap \mathcal {S}_B \}$</span>. In the case of observations with errors in <span class="mathjax-tex">$\mathcal {S}_B$</span>, we use the dataset <span class="mathjax-tex">$\{(y_i,\tilde{y}_i): i \in \mathcal {S}_A\cap \mathcal {S}_B \}$</span> to estimate the measurement error model and the relative error term. In both cases, the DAGJK method involves a random generation of <span class="mathjax-tex">$y_i^*$</span> values in each group, replacing the <span class="mathjax-tex">$\bar{\tilde{y}}_i$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ20">20</a>).</p><p>We evaluate (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ19">19</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ20">20</a>) in the simulation study presented in the next section.</p></div></div></section><section data-title="Simulation study"><div class="c-article-section" id="Sec11-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec11"><span class="c-article-section__title-number">6 </span>Simulation study</h2><div class="c-article-section__content" id="Sec11-content"><p>Following the simulation study 1) by Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e15487">2021</a>), we generate a finite population, <span class="mathjax-tex">$\mathcal {U}$</span>, of size <span class="mathjax-tex">$N = 1,000,000$</span>. The response variable, <span class="mathjax-tex">$\mathcal {Y}$</span>, is given by the following model:</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} y_i = 3 + 0.7(x_i - 2) + \eta _i, \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$x_i \sim \mathcal {N}(2,1)$</span>, <span class="mathjax-tex">$\eta _i \sim \mathcal {N}(0,0.51)$</span> and <span class="mathjax-tex">$\eta _i$</span> is independent of <span class="mathjax-tex">$x_i$</span>. Next, we generate a contaminated version of <span class="mathjax-tex">$y_i$</span> as follows:</p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \tilde{y}_{i} = 2 + 0.9(y_i - 3) + \epsilon _i, \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\epsilon _i \sim \mathcal {N}(0,0.5^2)$</span> and <span class="mathjax-tex">$\epsilon _i$</span> is independent of <span class="mathjax-tex">$y_i$</span>.</p><p>Additionally, we generate the auxiliary variable <span class="mathjax-tex">$\varvec{\xi }$</span> such that <span class="mathjax-tex">$cor({\textbf {x}}, \varvec{\xi }) = 0.5$</span>, which is given by:</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \xi _{i} = \frac{cov({\textbf {x}},\varvec{\xi })}{var({\textbf {x}})} x_i + \nu _i , \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\nu _i \sim \mathcal {N}(0,1)$</span> and <span class="mathjax-tex">$\nu _i$</span> is independent of <span class="mathjax-tex">$x_i$</span>. We define <span class="mathjax-tex">$\Xi _{1} = \sum _{i \in U} \xi _{1i}$</span> and <span class="mathjax-tex">$\Xi _{2} = \sum _{i \in U} \xi _{2i}$</span>, where <span class="mathjax-tex">$\xi _{1i} = 1$</span> if <span class="mathjax-tex">$\xi _{i} \le 1$</span> and 0 otherwise, and <span class="mathjax-tex">$\xi _{2i} = 1$</span> if <span class="mathjax-tex">$\xi _{i} > 1$</span> and 0 otherwise. We use these variables for prediction and calibration purposes.</p><p>We select two samples: <span class="mathjax-tex">$\mathcal {S}_A$</span> and <span class="mathjax-tex">$\mathcal {S}_B$</span>, representing the probability and the non-probability samples, respectively. <span class="mathjax-tex">$\mathcal {S}_A$</span> is a simple random sample of size <span class="mathjax-tex">$n_A = 1000$</span>, while <span class="mathjax-tex">$\mathcal {S}_B$</span> is selected by a different probability sampling of size <span class="mathjax-tex">$n_B = 500,000$</span>. The latter is obtained by creating two strata in <span class="mathjax-tex">$\mathcal {U}$</span>: stratum 1 consists of units with <span class="mathjax-tex">$x_i \le 2$</span>, while stratum 2 consists of those with <span class="mathjax-tex">$x_i >2$</span>. We define <span class="mathjax-tex">$X_{1} = \sum _{i \in U} x_{1i}$</span> and <span class="mathjax-tex">$X_{2} = \sum _{i \in U} x_{2i}$</span>, where <span class="mathjax-tex">$x_{1i} = 1$</span> if <span class="mathjax-tex">$x_{i} \le 2$</span> and 0 otherwise, and <span class="mathjax-tex">$x_{2i} = 1$</span> if <span class="mathjax-tex">$x_{i} > 2$</span> and 0 otherwise. Within each stratum, we independently select <span class="mathjax-tex">$n_{B1} = 300,000$</span> and <span class="mathjax-tex">$n_{B2} = 200,000$</span> observations, respectively, through simple random sampling. The target parameter is the finite population mean of <span class="mathjax-tex">$\mathcal {Y}$</span>. This sampling procedure implies that the sample mean of <span class="mathjax-tex">$\mathcal {S}_B$</span> is smaller than the population mean.</p><p>The simulation study examines the first two scenarios proposed in Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e17066">2021</a>): </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>Scenario I: we observe <span class="mathjax-tex">$y_i$</span> in both samples;</p> </li> <li> <span class="u-custom-list-number">2.</span> <p>Scenario II: we observe <span class="mathjax-tex">$y_i$</span> in <span class="mathjax-tex">$\mathcal {S}_A$</span> and <span class="mathjax-tex">$\tilde{y}_i$</span> in <span class="mathjax-tex">$\mathcal {S}_B$</span>.</p> </li> </ol><p>The indicator variable <span class="mathjax-tex">$\delta _i$</span> is observed in both <span class="mathjax-tex">$\mathcal {S}_B$</span> and <span class="mathjax-tex">$\mathcal {S}_A$</span>. Therefore, if <span class="mathjax-tex">$\delta _i = 1$</span> in <span class="mathjax-tex">$\mathcal {S}_A$</span>, we have both <span class="mathjax-tex">$y_i$</span> and <span class="mathjax-tex">$\tilde{y}_{i}$</span>.</p><h3 class="c-article__sub-heading" id="Sec12"><span class="c-article-section__title-number">6.1 </span>Estimators</h3><p>The simulation study considers two benchmark estimators: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>Mean <span class="mathjax-tex">$\mathcal {S}_A$</span> <span class="mathjax-tex">$= \frac{1}{n_A}\sum _{i \in \mathcal {S}_A} y_i$</span>,</p> </li> <li> <span class="u-custom-list-number">2.</span> <p>Mean <span class="mathjax-tex">$\mathcal {S}_B$</span> <span class="mathjax-tex">$= \frac{1}{n_B}\sum _{i \in \mathcal {S}_B} y_i$</span>,</p> </li> </ol><p>and compares two classes of estimators integrating <span class="mathjax-tex">$\mathcal {S}_A$</span> and <span class="mathjax-tex">$\mathcal {S}_B$</span>.</p><p>The first class of estimators considers the RegDI methods proposed by Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e17629">2021</a>): </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>RegDI: regression data integration estimator of the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ1">1</a>) with calibration equation </p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_A} w_{i}^{A} (1, \delta _i, \delta _i y_i) = \sum _{i \in U} (1, \delta _i, \delta _i y_i) = (N, n_B, Y_B^{*}). \end{aligned}$$</span></div></div> </li> <li> <span class="u-custom-list-number">2.</span> <p>RegDI<span class="mathjax-tex">$_{(X_{1},X_{2})}$</span>: regression data integration estimator of the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ1">1</a>) with calibration equation </p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_A} w_{i}^{A} (1, \delta _i, \delta _i y_i, x_{1i}, x_{2,1}) = \sum _{i \in U} (1, \delta _i, \delta _i y_i, x_{1i}, x_{2,1}) = (N, n_B, Y_B^{*}, X_1, X_2). \end{aligned}$$</span></div></div> </li> <li> <span class="u-custom-list-number">3.</span> <p>RegDI<span class="mathjax-tex">$_{(\Xi _{1},\Xi _{2})}$</span>: regression data integration estimator of the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ1">1</a>) with calibration equation </p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_A} w_{i}^{A} (1, \delta _i, \delta _i y_i, \xi _{1i}, \xi _{2,1}) = \sum _{i \in U} (1, \delta _i, \delta _i y_i, \xi _{1i}, \xi _{2,1}) = (N, n_B, Y_B^{*}, \Xi _1, \Xi _2). \end{aligned}$$</span></div></div> </li> </ol><p>The second class of estimators includes the PC estimators. For Scenario I, we consider the following: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>PC<span class="mathjax-tex">$_{(X_1,X_2)}$</span>: pseudo-calibration estimator of the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ10">10</a>) with calibration equation </p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_B} w_{i}^{B} (x_{1i}, x_{2i}) = \sum _{i \in U} (x_{1i}, x_{2,1}) = (X_1, X_2). \end{aligned}$$</span></div></div> </li> <li> <span class="u-custom-list-number">2.</span> <p>PC<span class="mathjax-tex">$_{(\Xi _1,\Xi _2)}$</span>: pseudo-calibration estimator of the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ10">10</a>) with calibration equation </p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_B} w_{i}^{B} (\xi _{1i}, \xi _{2i}) = \sum _{i \in U} (\xi _{1i}, \xi _{2,1}) = (\Xi _1, \Xi _2). \end{aligned}$$</span></div></div> </li> </ol><p>For Scenario II, we consider the following estimators: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>Difference Mean <span class="mathjax-tex">$\mathcal {S}_B$</span>= <span class="mathjax-tex">$\frac{1}{n_B} \sum _{i \in \mathcal {S}_B} \tilde{y}_i + \frac{1}{N} \sum _{i \in \mathcal {S}_A} d_{i}^{A} (y_i - \tilde{y}_i)$</span>: the sample mean of predictions in <span class="mathjax-tex">$\mathcal {S}_B$</span>, corrected by the weighted residuals calculated in <span class="mathjax-tex">$\mathcal {S}_A$</span>.</p> </li> <li> <span class="u-custom-list-number">2.</span> <p>PC<span class="mathjax-tex">$^{D}_{(X_1,X_2)}$</span>: pseudo-calibration estimator of the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ16">16</a>) with calibration equation </p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_B} w_{i}^{B} (x_{1i}, x_{2i}) = \sum _{i \in U} (x_{1i}, x_{2,1}) = (X_1, X_2). \end{aligned}$$</span></div></div> </li> <li> <span class="u-custom-list-number">3.</span> <p>PC<span class="mathjax-tex">$^{D}_{(\Xi _1,\Xi _2)}$</span>: pseudo-calibration estimator of the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ16">16</a>) with calibration equation </p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_B} w_{i}^{B} (\xi _{1i}, \xi _{2i}) = \sum _{i \in U} (\xi _{1i}, \xi _{2,1}) = (\Xi _1, \Xi _2). \end{aligned}$$</span></div></div> </li> </ol><h3 class="c-article__sub-heading" id="Sec13"><span class="c-article-section__title-number">6.2 </span>Results</h3><p>The performance of each estimator is evaluated through the bias (Bias), the standard error (SE), and the root mean squared error (MSE) given by the Monte Carlo process.</p><p>The two classes of estimators employ different inference approaches: a design-based approach for the RegDI estimators, where the <span class="mathjax-tex">$y_i$</span> values are treated as fixed, and a model-based approach for the PC estimators, where the variable <span class="mathjax-tex">$\mathcal {Y}$</span> is considered random. For the RegDI estimators, we generate 1000 Monte Carlo samples for both <span class="mathjax-tex">$\mathcal {S}_A$</span> and <span class="mathjax-tex">$\mathcal {S}_B$</span> from the finite population. We simulate 1000 Monte Carlo populations for the PC estimators and draw a single sample for each population for <span class="mathjax-tex">$\mathcal {S}_A$</span> and <span class="mathjax-tex">$\mathcal {S}_B$</span>.</p><p>Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab2">2</a> shows the simulation study results for the design-based estimators. The results for the first three estimators (i.e., Mean <span class="mathjax-tex">$\mathcal {S}_A$</span>, Mean <span class="mathjax-tex">$\mathcal {S}_B$</span>, RegDI) are identical to the ones presented in Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e19891">2021</a>) (see pg. 394, Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab2">2</a>). As discussed in Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e19898">2021</a>), Mean <span class="mathjax-tex">$\mathcal {S}_A$</span> and the RegDI estimators are unbiased in both scenarios. In contrast, Mean <span class="mathjax-tex">$\mathcal {S}_B$</span> estimator is always biased due to the selection bias in sample <span class="mathjax-tex">$\mathcal {S}_B$</span>. The RegDI estimators have lower RMSE values. In particular, the RegDI<span class="mathjax-tex">$_{(X_{1},X_{2})}$</span> estimator, not considered in Kim and Tam (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e20015">2021</a>), has the lowest standard error. On the other hand, the RegDI<span class="mathjax-tex">$_{(\Xi {1},\Xi _{2})}$</span> estimator, which employs calibration variables not strictly related to the <span class="mathjax-tex">$y_i$</span> values, leads to an inflation in the standard error (SE).</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-2"><figure><figcaption class="c-article-table__figcaption"><b id="Tab2" data-test="table-caption">Table 2 Results of the five estimators for the simulation study based on design-based Monte Carlo simulations of size 1000</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s10260-023-00740-y/tables/2" aria-label="Full size table 2"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab3">3</a> shows the simulation study results of the model-based estimators. The Mean <span class="mathjax-tex">$\mathcal {S}_B$</span> estimator remains seriously biased due to the selection bias in <span class="mathjax-tex">$\mathcal {S}_B$</span>. Focusing on Scenario I, the PC<span class="mathjax-tex">$_{(X_1,X_2)}$</span> estimator shows competitiveness compared to the RegDI estimators, while the PC<span class="mathjax-tex">$_{(\Xi _1,\Xi _2)}$</span> estimator is affected by the use of a slightly wrong propensity score model, implicitly defined by the <span class="mathjax-tex">$\xi _1$</span> and <span class="mathjax-tex">$\xi _2$</span> variables. In Scenario II, as shown in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab3">3</a>, the Mean <span class="mathjax-tex">$\mathcal {S}_B$</span> estimator increases the bias. In this case, indeed, both the outcome model for <span class="mathjax-tex">$\tilde{y}_i$</span> and the propensity score model for <span class="mathjax-tex">$w_i^B$</span> come into play. By utilizing the outcome model, the Difference Mean <span class="mathjax-tex">$\mathcal {S}_B$</span> estimator significantly reduces the bias. Scenario II does not present results for the projection-type estimators, PC<span class="mathjax-tex">$^{P}_{(X_1,X_2)}$</span> and PC<span class="mathjax-tex">$^P_{(\Xi _1,\Xi _2)}$</span>, as they exclusively rely on the propensity score model. The PC<span class="mathjax-tex">$^{P}_{(X_1,X_2)}$</span> and PC<span class="mathjax-tex">$^P_{(\Xi _1,\Xi _2)}$</span> estimators exhibit bias levels closer to the Mean <span class="mathjax-tex">$\mathcal {S}_B$</span> estimator than the Difference Mean <span class="mathjax-tex">$\mathcal {S}_B$</span> estimator. The PC<span class="mathjax-tex">$^{D}_{(X_1,X_2)}$</span> estimator uses both models and remains competitive with the RegDI estimators. The PC<span class="mathjax-tex">$^D_{(\Xi _1,\Xi _2)}$</span> estimator reduces the bias compared to the Difference Mean <span class="mathjax-tex">$\mathcal {S}_B$</span> estimator. However, it is still more biased than the RegDI estimators.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-3"><figure><figcaption class="c-article-table__figcaption"><b id="Tab3" data-test="table-caption">Table 3 Results of the six estimators for the simulation study based on Monte Carlo populations of size 1000</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s10260-023-00740-y/tables/3" aria-label="Full size table 3"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>The second part of the simulation study is devoted to the variance estimator presented in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10260-023-00740-y#Sec10">5</a>. We use the DAGJK method with 100 random groups. Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab4">4</a> shows the standard error estimates of the PC estimators using either <span class="mathjax-tex">$(X_1,X_2)$</span> or <span class="mathjax-tex">$(\Xi _1,\Xi _2)$</span> in both scenarios. The DAGJK values represent the mean values of the DAGJK estimates computed on 1000 samples of the Monte Carlo simulation. In Scenario I, the DAGJK variance estimates are close to the Monte Carlo variances. As expected, in Scenario II we slightly overestimate the DAGJK variance estimates.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-4"><figure><figcaption class="c-article-table__figcaption"><b id="Tab4" data-test="table-caption">Table 4 Standard error estimates of four PC estimators for the simulation study based on 1000 Monte Carlo populations of size and on Delete-a-Group Jackknife estimator using 100 random groups</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s10260-023-00740-y/tables/4" aria-label="Full size table 4"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></section><section data-title="An application to the European community survey data on ICT usage and e-commerce in enterprises"><div class="c-article-section" id="Sec14-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec14"><span class="c-article-section__title-number">7 </span>An application to the European community survey data on ICT usage and e-commerce in enterprises</h2><div class="c-article-section__content" id="Sec14-content"><p>We implement the RegDI and PC estimators using the 2018 European Community Survey data on ICT usage and e-commerce in enterprises. This ICT survey is conducted yearly by Istat and by other member states of the EU. Additionally, we consider internet data scraped from enterprises’ websites that fall within the ICT target population. The primary objective of the ICT survey is to supply users with indicators related to internet connectivity and usage, encompassing aspects such as website usage, social media engagement, and cloud computing. The survey’s target population refers to enterprises with ten or more employees working in the industry and non-financial market services. The population frame is the Italian Business Register (Asia), which was last updated two years before the survey’s reference period. For the 2018 ICT survey, this population comprises 199,914 units. The ICT survey considers a stratified simple random sampling design with strata given by four classes of number of persons employed (0–9; 10–19; 20–249; 250 or more), economic activities (24 Nace groups) and geographical breakdown (21 administrative regions at NUTS 2 level). The strata, including the fourth size class (enterprises with 250 and more persons employed), are taken entirely. The number of units within these strata is 3342. For the 2018 ICT survey, the sample of respondents consists of 22,097 units. The survey posed questions to enterprises, including whether a) the website enables online ordering, reservations, or bookings and b) there are links to social media on the website. We assign specific variable names, WEBORD (<span class="mathjax-tex">$\mathcal {Y}_1$</span>) and WEBSM (<span class="mathjax-tex">$\mathcal {Y}_2$</span>), to these two questions, respectively. The current ICT survey estimator employs a calibration method, which considers the number of enterprises and persons employed based on economic activity, size class, and administrative region, according to a complex combination of these variables. We use Internet data as a big non-probability sample (i.e., a big data source). This process starts with text documents collected through web scraping from the enterprise’s websites. Specifically, we have gathered 93,848 scraped websites representing the units falling in <span class="mathjax-tex">$\mathcal {S}_B$</span>. It is worth mentioning that the total number of websites in the target population is unknown. The ICT survey estimates approximately 134,655.82 enterprises with a relative error of about 1%. The web-scraping step returns information retrieval for the WEBSM variable. That means that we observe the variable with <span class="mathjax-tex">$y_{2i}=1$</span> when the website has a link to social media and with <span class="mathjax-tex">$y_{2i}=0$</span> otherwise. Using the text document of each website, we predict the WEBORD variable using a machine learning technique (Random Forest) as described in Bianchi et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Bianchi G, Bri R, Scalfati F (2020) Identifying e-commerce in enterprises by means of text mining and classification algorithms Hindawi. Math Probl Eng 2018:1–8. 
 https://doi.org/10.1155/2018/7231920
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR3" id="ref-link-section-d84953921e22436">2020</a>) and Bruni and Bianchi (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Bruni R, Bianchi G (2020) Website categorization: a formal approach and robustness analysis in the case of e-commerce detection. Expert Syst Appl 142(113):001" href="/article/10.1007/s10260-023-00740-y#ref-CR6" id="ref-link-section-d84953921e22439">2020</a>). We use a deterministic prediction for the WEBORD, meaning we use the estimated probability that the website incorporates functionalities for online ordering, reservations, or bookings. Further insights into the ICT survey, web scraping, and machine learning procedure can be found in Righi et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2019" title="Righi P, Bianchi G, Nurra A et al (2019) Integration of survey data and big data for finite population inference in official statistics: statistical challenges and practical applications. Stat Appl XVII(2):135–158" href="/article/10.1007/s10260-023-00740-y#ref-CR31" id="ref-link-section-d84953921e22442">2019</a>).</p><h3 class="c-article__sub-heading" id="Sec15"><span class="c-article-section__title-number">7.1 </span>Estimators</h3><p>We compare a simplified version of the estimator used by Istat for the ICT survey, denoted as T0, with four different RegDI estimators (RegDI.1, RegDI.2, RegDI.3, RegDI.4) and three other PC estimators (PC.1, PC.2, PC.3) for the population total.</p><p>T0 is a calibration estimator of the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ1">1</a>) that uses the number of enterprises and employed persons for four enterprise-size classes (0–9; 10–19; 20–249; +249) and for three macro-regions (aggregation of NUTS 2 regions, North, Centre and South) as known totals. We set <span class="mathjax-tex">$\textbf{x}_i=(1 \mathbf {\lambda }_{i}',e_i \mathbf {\lambda }_{i}')'$</span>, where <span class="mathjax-tex">$e_i$</span> is the number of employed persons in unit <i>i</i>, and <span class="mathjax-tex">$\mathbf {\lambda }_i = (\lambda _{i(0-9)},\lambda _{i(10-19)},\lambda _{i(20-249)},\lambda _{i(+249)},\lambda _{i(North)},\lambda _{i(Centre)},\lambda _{i(South)})'$</span>, where the generic element of <span class="mathjax-tex">$\mathbf {\lambda }_i$</span>, <span class="mathjax-tex">$\lambda _{i(d)}$</span>, is equal to 1 if <i>i</i> belongs to one of four enterprise-size classes or one of three macro-regions, and equal to 0 otherwise. For example, if the enterprise <i>i</i> has ten employed persons and is located in southern Italy, then <span class="mathjax-tex">$\mathbf {\lambda }_i = (0,1,0,0,0,0,1)'$</span>. Then, the calibration equation is</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{i \in \mathcal {S}_A} w_{i}^{A}(1 \lambda _{i}',e_i \lambda _{i}') = \sum _{i \in \mathcal {U}}&(N_{0-9},N_{10-19},N_{20-249},N_{+249},N_{Centre},N_{North},N_{South}, \\ {}&E_{0-9},E_{10-19}, E_{20-249},E_{+249},E_{Centre},E_{North},E_{South}), \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$N_{d}$</span> and <span class="mathjax-tex">$E_{d}$</span> are the total number of enterprises and employed persons, respectively, in each enterprise-size class and macro-region. As a result, the estimates of the total population and the seven sub-populations using T0 can be derived as</p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}_{j,T0} = \sum _{i \in \mathcal {S}_A} w^{A}_{i} y_{ji} \quad \text {and} \quad \hat{Y}_{j,T0(d)} = \sum _{i \in \mathcal {S}_A} w^{A}_{i} y_{ji} \lambda _{i(d)} \quad \text {for} \quad j = 1,2. \end{aligned}$$</span></div><div class="c-article-equation__number"> (21) </div></div><p>The calibration variables are <span class="mathjax-tex">$(\textbf{x}_{i}^{'},\delta _i \mathbf {\lambda }_{i}^{'})'$</span>, <span class="mathjax-tex">$(\textbf{x}_{i}^{'},\delta _i \mathbf {\lambda }_{i}^{'},\delta _i y_{1i} \mathbf {\lambda }_{i}^{'})'$</span>, <span class="mathjax-tex">$(\textbf{x}_{i}^{'},\delta _i \mathbf {\lambda }_{i}^{'},\delta _i y_{2i} \mathbf {\lambda }_{i}^{'})'$</span> and <span class="mathjax-tex">$(\textbf{x}_{i}^{'},\delta _i \mathbf {\lambda }_{i}^{'},\delta _i y_{1i} \mathbf {\lambda }_{i}^{'},\delta _i y_{2i} \mathbf {\lambda }_{i}^{'})'$</span> for the RegDI.1, RegDI.2, RegDI.3 and RegDI.4 estimators, respectively. It follows that the estimates of the population total and the seven sub-populations using the RegDI estimators have the same form of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ21">21</a>) but different calibration weights.</p><p>The PC.1 estimator calibrates the weights using the same totals as T0. It corresponds to the estimator (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ10">10</a>) for WEBSM and the estimator (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ15">15</a>) for WEBORD. Additionally, for the WEBORD total, we implement the PC.2 and PC.3 estimators following the formula (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10260-023-00740-y#Equ18">18</a>). In the PC.2 estimator, the sampling calibrated weights are adjusted by the factor <span class="mathjax-tex">$f_{2i}^{A} = \sum _{\mathcal {S}_A} \phi _i / \sum _{\mathcal {S}_A}{\delta _i}$</span>, with <span class="mathjax-tex">$\phi _i = 1$</span> when the enterprise has the website and <span class="mathjax-tex">$\phi _i = 0$</span> otherwise. The PC.3 estimator uses the factor <span class="mathjax-tex">$f_{3i}^{A} = \sum _{\mathcal {S}_A} \phi _i w_{i}^{A} / \sum _{\mathcal {S}_A} \delta _i w_{i}^{A}$</span>, where <span class="mathjax-tex">$w_{i}^{A}$</span> is the calibrated sampling weight of the ICT survey estimator (T0). It follows that the estimates of the population total and the seven sub-populations using the PC.1 estimator can be derived as</p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}^{P}_{1,PC.1}= & {} \sum _{i \in \mathcal {S}_B} w^{B}_{i} \hat{\tilde{y}}_{1i} \quad \text {and} \quad \hat{Y}^{P}_{1,PC.1(d)} = \sum _{i \in \mathcal {S}_B} w^{B}_{i} \hat{\tilde{y}}_{1i} \lambda _{i(d)}, \\ \hat{Y}_{2,PC.1}= & {} \sum _{i \in \mathcal {S}_B} w^{B}_{i} y_{2i} \quad \text {and} \quad \hat{Y}_{2,PC.1(d)} = \sum _{i \in \mathcal {S}_B} w^{B}_{i} y_{2i} \lambda _{i(d)}. \end{aligned}$$</span></div></div><p>The estimates of the population total and the seven sub-populations using the PC.2 and PC.3 estimators can be obtained as</p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \hat{Y}^{D}_{1,PC.l}= & {} \sum _{i \in \mathcal {S}_B} w^{B}_{i} \hat{\tilde{y}}_{1i} + \sum _{i \in \mathcal {S}_A} f^{A}_{li} (y_{1i} - \hat{\tilde{y}}_{1i}) \quad \text {(for} \quad l = 2, 3) \quad \text {and} \\ \hat{Y}^{D}_{1,PC.l(d)}= & {} \sum _{i \in \mathcal {S}_B} w^{B}_{i} \tilde{y}_{1i} \lambda _{i(d)} + \sum _{i \in \mathcal {S}_A} f^{A}_{li} (y_{1i} - \hat{\tilde{y}}_{1i}) \lambda _{i(d)} \quad \text {(for} \quad l = 2, 3). \end{aligned}$$</span></div></div><h3 class="c-article__sub-heading" id="Sec16"><span class="c-article-section__title-number">7.2 </span>Results</h3><p>Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab5">5</a> shows the estimates for the totals at the national level. We can observe that the RegDI.1 estimator does not affect the Coefficient of Variation (CV) of the estimates compared to T0. On the other hand, the RegDI.2 and RegDI.3 estimators reduce the CV for the variable involved in the calibration. Only when we apply the RegDI.4 estimator we observe a substantial decrease in CV for both the WEBORD and WEBSM variables. These results highlight a crucial crossroads in a multi-purpose survey. The choices are twofold: (1) make a massive calibration, risking either the non-attainment of the optimal solution to the optimization problem or the inflation of variance due to excessively small or large final weights; (2) omit certain target variables from the calibration process and risking to compromise in the enhancement of their estimation accuracy. The estimates of WEBSM given by RegDI.3 and RegDI.4 fall outside the 95% Confidence Interval (CI) of T0. Since the RegDI.3 and RegDI.4 estimators are unbiased, the findings suggest that the T0 estimate has such a large error that it considerably underestimates the WEBSM total.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-5"><figure><figcaption class="c-article-table__figcaption"><b id="Tab5" data-test="table-caption">Table 5 Results of the considered estimators at the national level</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s10260-023-00740-y/tables/5" aria-label="Full size table 5"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>The analysis of the PC estimators reveals some important findings. The PC.1 estimates fall outside the 95% CI of T0. While we know the PC.1-WEBORD estimate can be biased since it bypasses the correction term, the PC.1-WEBSM estimate appears different from the corresponding T0 estimate. Nevertheless, the 95% CI of the PC.1-WEBSM estimator overlaps the CI of the RegDI.3 and RegDI.4 estimates. The consistency among these three estimates suggests that the PC.1-WEBSM estimator is unbiased. The CI of PC.1-WEBSM estimator is much narrower compared to the CIs of RegDI. We apply the pseudo-calibration difference estimators, PCE.2 and PCE.3, for the WEBORD total estimate. The PCE.2 and PCE.3 estimates fall within the 95% CI of the T0 estimate, and their 95% CIs overlap the RegDI-WEBORD estimators’ 95% CIs. This suggests that we may have effectively adjusted for the bias in the measurement error of the big data target variable. The CV of the PC.3 estimator is smaller than that of T0 and is roughly equivalent to the CVs of the others RegDI.3 and RegDI.4.</p><p>We compare the estimates of WEBORD and WEBSM totals by size class domains (Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig1">1</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig2">2</a>) and macro-regions domains (Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig3">3</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig4">4</a>).</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Fig. 1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s10260-023-00740-y/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig1_HTML.png?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="340"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>Estimator CIs (95%) by size class for WEBORD total</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s10260-023-00740-y/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Fig. 2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s10260-023-00740-y/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig2_HTML.png?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="438"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Estimator CIs (95%) by size class for WEBSM total</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s10260-023-00740-y/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Fig. 3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s10260-023-00740-y/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig3_HTML.png?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="438"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>Estimator CIs (95%) by macro-regions for WEBORD total</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s10260-023-00740-y/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Fig. 4"><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 4</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s10260-023-00740-y/figures/4" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig4_HTML.png?as=webp"><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs10260-023-00740-y/MediaObjects/10260_2023_740_Fig4_HTML.png" alt="figure 4" loading="lazy" width="685" height="438"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>Estimator CIs (95%) by macro-regions for WEBSM total</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s10260-023-00740-y/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>In Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig1">1</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig2">2</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig3">3</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig4">4</a>, it is evident that the RegDI estimator 95% CIs always overlap the 95% CI of the T0 estimates except for WEBSM estimates in the domain <span class="mathjax-tex">$0-9$</span> size class or North macro-region. We underlined this evidence for the national estimate. While the 95% CIs of the RegDI estimators appear similar in length, they are slightly narrower than the 95% CI of the T0 estimate for some domains (such as the size class <span class="mathjax-tex">$0-9$</span> for WEBORD and WEBSM). The PC estimators give the shortest intervals. As expected, in certain domains, the WEBSM estimates significantly deviate from those of T0 (specifically, in the <span class="mathjax-tex">$0-9$</span> size class, Center and North macro-regions) (see Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig2">2</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig4">4</a>). Regarding the WEBORD totals, the PC.1 estimates fall outside the CIs of T0 and frequently deviate significantly from those generated by the RegDI.1 estimator, which utilizes the same auxiliary variables. This outcome is anticipated because the PC.1 estimator ignores the correction term, and the prediction method (i.e., random forest technique) could fail to accurately capture the true relationship between the predictors and the target variable in specific domains. Consequently, the PC.1 estimator can be biased. The difference estimator adjusts the PC.1-WEBORD estimates that fall within the 95% CIs of the T0 estimate, or at least produces 95% CIs that overlap the 95% CIs of the RegDI.3 and RegDI.4 estimators. Figures <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig1">1</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s10260-023-00740-y#Fig2">2</a> also include the T<span class="mathjax-tex">$_b$</span> estimator, which is a naïve PC estimator defined as <span class="mathjax-tex">$(\hat{N}_W /n_B ) \sum _{\mathcal {S}_B} y_i^{*}$</span>, where <span class="mathjax-tex">$\hat{N}_W$</span> is the survey-based estimate of the number of units with a website. Finally, it is noteworthy that the estimates from PC.2 and PC.3 differ (though not significantly) from those generated by the RegDI.4 estimator, which incorporates all auxiliary variables. We interpret these findings as indicative of intrinsic distinctions arising from the utilization of information within the two classes of estimators.</p><p>Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab6">6</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab7">7</a> investigate the sampling errors of the estimators of the cross-classified domain size class by macro-region (12 domains) by the average CV (%) for WEBORD and WEBSM total estimate, respectively. We categorize the domains into two groups: six domains with a sample size between 344 and 547 units (Group 1) and six domains with a sample size between 1558 and 8299 sample units (Group 2).</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-6"><figure><figcaption class="c-article-table__figcaption"><b id="Tab6" data-test="table-caption">Table 6 Coefficient of variation of the estimators for size classes by the macro-region domain of WEBORD total</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s10260-023-00740-y/tables/6" aria-label="Full size table 6"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-7"><figure><figcaption class="c-article-table__figcaption"><b id="Tab7" data-test="table-caption">Table 7 Coefficient of variation of the estimators for size classes by macro-region domain of WEBSM total</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s10260-023-00740-y/tables/7" aria-label="Full size table 7"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab6">6</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s10260-023-00740-y#Tab7">7</a> show that the PC estimators are more efficient. Among the RegDI estimators, those in Group 2 (large domains) are more efficient than T0. Conversely, the average CVs (%) for the RegDI estimators in Group 1 (small domains) are greater than the CV of T0. We attribute this to the increased number of calibration constraints, resulting in some units having extreme weights, which in turn leads to higher variance estimates. This effect of extreme weights is more pronounced in domains with small sample sizes.</p></div></div></section><section data-title="Discussion"><div class="c-article-section" id="Sec17-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec17"><span class="c-article-section__title-number">8 </span>Discussion</h2><div class="c-article-section__content" id="Sec17-content"><p>The PC estimators integrate data from various sources, including probability and big non-probability samples, administrative records, or statistical registers. They are applicable when the target variable is observed in the probability sample and, in the big non-probability sample, it is (a) observed correctly, (b) observed with error, or (c) predicted using highly correlated auxiliary variables. In the case (a), the PC estimators are inverse probability weighted estimators (Chen et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Chen Y, Li P, Wu C (2020) Doubly robust inference with nonprobability survey samples. J Am Stat Assoc 115(532):2011–2021. 
 https://doi.org/10.1080/01621459.2019.1677241
 
 " href="/article/10.1007/s10260-023-00740-y#ref-CR7" id="ref-link-section-d84953921e26586">2020</a>). In the cases (b) and (c), they represent a novel class of estimators with forms akin to the adjusted projection estimator (Kim and Rao <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2011" title="Kim JK, Rao JNK (2011) Combining data from two independent surveys: a model-assisted approach. Biometrika 99(1):85–100" href="/article/10.1007/s10260-023-00740-y#ref-CR19" id="ref-link-section-d84953921e26589">2011</a>) and difference estimators (Breidt and Opsomer <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2017" title="Breidt FJ, Opsomer JD (2017) Model-assisted survey estimation with modern prediction techniques. Stat Sci 32:190–205" href="/article/10.1007/s10260-023-00740-y#ref-CR4" id="ref-link-section-d84953921e26592">2017</a>) developed within the model-assisted framework. In these cases, the PC estimators reverse the roles of probability and non-probability samples in the estimation process compared to the doubly robust estimators.The paper outlines the RegDI estimators (Kim and Tam <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Kim JK, Tam SM (2021) Data integration by combining big data and survey sample data for finite population inference. Int Stat Rev 89(2):382–401" href="/article/10.1007/s10260-023-00740-y#ref-CR20" id="ref-link-section-d84953921e26595">2021</a>), as they share the same informative context (cases (a) and (b)) required by the PC estimators. Both the PC and RegDI estimators employ calibration techniques, albeit with distinct approaches. The use of calibration tools is standard in the inferential context of data integration estimators, while it is less conventional in the context of PC estimators, although it has been previously suggested in the literature (Lee and Valliant <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2009" title="Lee S, Valliant R (2009) Estimation for volunteer panel web surveys using propensity score adjustment and calibration adjustment. Sociol Methods Res 37(3):319–343" href="/article/10.1007/s10260-023-00740-y#ref-CR25" id="ref-link-section-d84953921e26598">2009</a>). With few exceptions, one of which is shown in the paper, the RegDI and PC estimators yield different point estimates. A jackknife-type variance estimator is introduced for the PC estimators suitable for large-scale datasets. Moreover, a comparative analysis is conducted between the PC and RegDI estimators, both defined within the same informative framework. This analysis leverages a Monte Carlo simulation and an experiment using real data from the ICT enterprise survey and information scraped from enterprise websites. The PC estimators show competitiveness with the RegDI estimators in Monte Carlo simulations, with the jackknife-type variance estimates very close to the Monte Carlo variances. In the experiment with real data, PC estimates are not significantly different from the RegDI estimates, and the confidence intervals are narrower than those of the RegDI.</p></div></div></section> </div> <section data-title="Notes"><div class="c-article-section" id="notes-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="notes">Notes</h2><div class="c-article-section__content" id="notes-content"><ol class="c-article-footnote c-article-footnote--listed"><li class="c-article-footnote--listed__item" id="Fn1" data-counter="1."><div class="c-article-footnote--listed__content"><p><a href="https://www.istat.it/it/files//2020/05/Tech_Report_ICT2018.pdf">https://www.istat.it/it/files//2020/05/Tech_Report_ICT2018.pdf</a></p></div></li></ol></div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ul class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item"><p class="c-article-references__text" id="ref-CR1">Beaumont JF (2020) Are probability surveys bound to disappear for the production of official statistics. 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Integrating probability and big non-probability samples data to produce Official Statistics. <i>Stat Methods Appl</i> <b>33</b>, 555–580 (2024). https://doi.org/10.1007/s10260-023-00740-y</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s10260-023-00740-y?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-12-06">06 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c-article-share-box__additional-info"> Provided by the Springer Nature SharedIt content-sharing initiative </p></div></div><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=Big%20data&facet-discipline="Statistics"" data-track="click" data-track-action="view keyword" data-track-label="link">Big data</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Calibration%20weighting&facet-discipline="Statistics"" data-track="click" data-track-action="view keyword" data-track-label="link">Calibration weighting</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Data%20integration&facet-discipline="Statistics"" data-track="click" data-track-action="view keyword" data-track-label="link">Data integration</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Missing%20at%20random&facet-discipline="Statistics"" data-track="click" data-track-action="view keyword" data-track-label="link">Missing at random</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Model-based%20inference&facet-discipline="Statistics"" data-track="click" data-track-action="view keyword" data-track-label="link">Model-based inference</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Variance%20estimation&facet-discipline="Statistics"" data-track="click" data-track-action="view keyword" data-track-label="link">Variance estimation</a></span></li></ul><div data-component="article-info-list"></div></div></div></div></div></section> </div> </main> <div class="c-article-sidebar u-text-sm u-hide-print l-with-sidebar__sidebar" id="sidebar" data-container-type="reading-companion" data-track-component="reading companion"> <aside aria-label="reading companion"> <div 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