Document Zbl 1468.62338

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Time Ser. Anal.</a> <a href="/?q=in%3A457401" title="Articles in this Issue">42, No. 1, 63-84 (2021)</a>. </div> <div class="abstract">Summary: We propose a new sequential monitoring scheme for changes in the parameters of a multivariate time series. In contrast to procedures proposed in the literature which compare an estimator from the training sample with an estimator calculated from the remaining data, we suggest to divide the sample at each time point after the training sample. Estimators from the sample before and after all separation points are then continuously compared calculating a maximum of norms of their differences. For open-end scenarios our approach yields an asymptotic level $\alpha$ procedure, which is consistent under the alternative of a change in the parameter. By means of a simulation study it is demonstrated that the new method outperforms the commonly used procedures with respect to power and the feasibility of our approach is illustrated by analyzing two data examples.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7364921">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62M10" title="MSC2020">62M10</a> </td> <td class="space"> Time series, auto-correlation, regression, etc. in statistics (GARCH) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62H15" title="MSC2020">62H15</a> </td> <td class="space"> Hypothesis testing in multivariate analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62L12" title="MSC2020">62L12</a> </td> <td class="space"> Sequential estimation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G10" title="MSC2020">62G10</a> </td> <td class="space"> Nonparametric hypothesis testing </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Achange+point+analysis">change point analysis</a>; <a href="/?q=ut%3Aopen-end+procedures">open-end procedures</a>; <a href="/?q=ut%3Asequential+monitoring">sequential monitoring</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1468.62338" data-ciurl="/ci/07364921" data-biburl="/bibtex/07364921.bib" data-amsurl="/amsrefs/07364921.bib" data-xmlurl="/xml/07364921.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07364921.pdf" title="Zbl 1468.62338 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1111/jtsa.12555" aria-label="DOI for “A new approach for open-end sequential change point monitoring”" title="10.1111/jtsa.12555">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1906.03225"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Anatolyev, S.; Kosenok, G., Sequential testing with uniformly distributed size, Journal of Time Series Econometrics, 10, 2018 · <a href="/1499.62280" class="nowrap">Zbl 1499.62280</a></td> </tr><tr> <td>[2]</td> <td class="space">Andrews, D., Heteroskedasticity and autocorrelation consistent covariance matrix estimation, Econometrica, 59, 817-858, 1991 · <a href="/0732.62052" class="nowrap">Zbl 0732.62052</a></td> </tr><tr> <td>[3]</td> <td class="space">Aue, A.; Dienes, C.; Fremdt, S.; Steinebach, J., Reaction times of monitoring schemes for ARMA time series, Bernoulli, 21, 1238-1259, 2015 · <a href="/1388.62246" class="nowrap">Zbl 1388.62246</a></td> </tr><tr> <td>[4]</td> <td class="space">Aue, A.; Hörmann, S.; Horváth, L.; Hušková, M.; Steinebach, JG, Sequential testing for the stability of high‐frequency portfolio betas, Econometric Theory, 28, 804-837, 2012 · <a href="/1245.91089" class="nowrap">Zbl 1245.91089</a></td> </tr><tr> <td>[5]</td> <td class="space">Aue, A.; Hörmann, S.; Horváth, L.; Reimherr, M., Break detection in the covariance structure of multivariate time series models, The Annals of Statistics, 37, 4046-4087, 2009 · <a href="/1191.62143" class="nowrap">Zbl 1191.62143</a></td> </tr><tr> <td>[6]</td> <td class="space">Aue, A.; Horváth, L., Delay time in sequential detection of change, Statistics & Probability Letters, 67, 221-231, 2004 · <a href="/1059.62085" class="nowrap">Zbl 1059.62085</a></td> </tr><tr> <td>[7]</td> <td class="space">Aue, A.; Horváth, L., Structural breaks in time series, Journal of Time Series Analysis, 34, 1-16, 2013 · <a href="/1274.62553" class="nowrap">Zbl 1274.62553</a></td> </tr><tr> <td>[8]</td> <td class="space">Aue, A.; Horváth, L.; Hušková, M.; Kokoszka, P., Change‐point monitoring in linear models, The Econometrics Journal, 9, 373-403, 2006 · <a href="/1106.62067" class="nowrap">Zbl 1106.62067</a></td> </tr><tr> <td>[9]</td> <td class="space">Aue, A.; Horváth, L.; Reimherr, ML, Delay times of sequential procedures for multiple time series regression models, Journal of Econometrics, 149, 174-190, 2009 · <a href="/1429.62380" class="nowrap">Zbl 1429.62380</a></td> </tr><tr> <td>[10]</td> <td class="space">Aue, A.; Hörmann, S.; Horváth, L.; Hušková, M., Dependent functional linear models with applications to monitoring structural change, Statistica Sinica, 24, 1043-1073, 2014 · <a href="/1534.62070" class="nowrap">Zbl 1534.62070</a></td> </tr><tr> <td>[11]</td> <td class="space">Berkes, I.; Gombay, E.; Horváth, L.; Kokoszka, P., Sequential change‐point detection in GARCH(p, q) models, Econometric Theory, 20, 1140-1167, 2004 · <a href="/1069.62058" class="nowrap">Zbl 1069.62058</a></td> </tr><tr> <td>[12]</td> <td class="space">Borodin, A.; Salminen, P., Handbook of Brownian motion – facts and formulae, Probability and Its Applications, 1996, Birkhäuser: Basel · <a href="/0859.60001" class="nowrap">Zbl 0859.60001</a></td> </tr><tr> <td>[13]</td> <td class="space">Chen, Z.; Tian, Z., Modified procedures for change point monitoring in linear models, Mathematics and Computers in Simulation, 81, 62-75, 2010 · <a href="/1201.62073" class="nowrap">Zbl 1201.62073</a></td> </tr><tr> <td>[14]</td> <td class="space">Chu, C‐SJ; Stinchcombe, M.; White, H., Monitoring structural change, Econometrica, 64, 1045-1065, 1996 · <a href="/0856.90027" class="nowrap">Zbl 0856.90027</a></td> </tr><tr> <td>[15]</td> <td class="space">Dette, H.; Gösmann, J., A likelihood ratio approach to sequential change point detection for a general class of parameters, Journal of the American Statistical Association, 2019 · <a href="/1441.62212" class="nowrap">Zbl 1441.62212</a></td> </tr><tr> <td>[16]</td> <td class="space">Eberlein, E., On strong invariance principles under dependence assumptions, The Annals of Probability, 14, 260-270, 1986 · <a href="/0589.60031" class="nowrap">Zbl 0589.60031</a></td> </tr><tr> <td>[17]</td> <td class="space">Feller, W., The asymptotic distribution of the range of sums of independent random variables, The Annals of Mathematical Statistics, 22, 427-432, 1951 · <a href="/0043.34201" class="nowrap">Zbl 0043.34201</a></td> </tr><tr> <td>[18]</td> <td class="space">Fremdt, S., Asymptotic distribution of the delay time in Page’s sequential procedure, Journal of Statistical Planning and Inference, 145, 74-91, 2014 · <a href="/1432.62273" class="nowrap">Zbl 1432.62273</a></td> </tr><tr> <td>[19]</td> <td class="space">Fremdt, S., Page’s sequential procedure for change‐point detection in time series regression, Statistics, 49, 128-155, 2015 · <a href="/1395.62267" class="nowrap">Zbl 1395.62267</a></td> </tr><tr> <td>[20]</td> <td class="space">Gösmann, J.; Kley, T.; Dette, H., Online appendix to: a new approach for open‐end sequential change point monitoring, 2020</td> </tr><tr> <td>[21]</td> <td class="space">Hampel, F.; Ronchetti, E.; Rousseeuw, P.; Stahel, W., Robust statistics: the approach based on influence functions, Wiley Series in Probability and Mathematical Statistics, 1986, John Wiley & Sons, Inc.: New York · <a href="/0593.62027" class="nowrap">Zbl 0593.62027</a></td> </tr><tr> <td>[22]</td> <td class="space">Hoga, Y., Monitoring multivariate time series, Journal of Multivariate Analysis, 155, 105-121, 2017 · <a href="/1359.62369" class="nowrap">Zbl 1359.62369</a></td> </tr><tr> <td>[23]</td> <td class="space">Horváth, L.; Hušková, M.; Kokoszka, P.; Steinebach, J., Monitoring changes in linear models, Journal of Statistical Planning and Inference, 126, 225-251, 2004 · <a href="/1075.62054" class="nowrap">Zbl 1075.62054</a></td> </tr><tr> <td>[24]</td> <td class="space">Horváth, L.; Kühn, M.; Steinebach, J., On the performance of the fluctuation test for structural change, Sequential Analysis, 27, 126-140, 2008 · <a href="/1274.62513" class="nowrap">Zbl 1274.62513</a></td> </tr><tr> <td>[25]</td> <td class="space">Hušková, M.; Koubková, A., Monitoring jump changes in linear models, Journal of Statistical Research, 39, 51-70, 2005</td> </tr><tr> <td>[26]</td> <td class="space">Hušková, M.; Prášková, Z.; Steinebach, J., On the detection of changes in autoregressive time series I, Asymptotics. Journal of Statistical Planning and Inference, 137, 1243-1259, 2007 · <a href="/1107.62090" class="nowrap">Zbl 1107.62090</a></td> </tr><tr> <td>[27]</td> <td class="space">Kirch, C.; Stoehr, C., Sequential change point tests based on U‐statistics, 2019</td> </tr><tr> <td>[28]</td> <td class="space">Kirch, C.; Weber, S., Modified sequential change point procedures based on estimating functions, Electronic Journal of Statistics, 12, 1579-1613, 2018 · <a href="/1392.62241" class="nowrap">Zbl 1392.62241</a></td> </tr><tr> <td>[29]</td> <td class="space">Lai, TL, Sequential changepoint detection in quality control and dynamical systems, Journal of the Royal Statistical Society. Series B (Methodological), 57, 613-658, 1995 · <a href="/0832.62072" class="nowrap">Zbl 0832.62072</a></td> </tr><tr> <td>[30]</td> <td class="space">Lai, TL, Sequential analysis: some classical problems and new challenges, Statistica Sinica, 11, 303-408, 2001 · <a href="/1037.62081" class="nowrap">Zbl 1037.62081</a></td> </tr><tr> <td>[31]</td> <td class="space">Leisch, F.; Hornik, K.; Kuan, C‐M, Monitoring structural changes with the generalized fluctuation test, Econometric Theory, 16, 835-854, 2000 · <a href="/0967.62067" class="nowrap">Zbl 0967.62067</a></td> </tr><tr> <td>[32]</td> <td class="space">Otto, S.; Breitung, J., Backward CUSUM for testing and monitoring structural change, 2020</td> </tr><tr> <td>[33]</td> <td class="space">Page, ES, Continuous inspection schemes, Biometrika, 41, 100-115, 1954 · <a href="/0056.38002" class="nowrap">Zbl 0056.38002</a></td> </tr><tr> <td>[34]</td> <td class="space">Page, ES, Control charts with warning lines, Biometrika, 42, 243-257, 1955 · <a href="/0067.37204" class="nowrap">Zbl 0067.37204</a></td> </tr><tr> <td>[35]</td> <td class="space">Pape, K.; Wied, D.; Galeano, P., Monitoring multivariate variance changes, Journal of Empirical Finance, 39, 54-68, 2016</td> </tr><tr> <td>[36]</td> <td class="space">Schmitz, A.; Steinebach, J. G.; DoukhanP. (ed.); LangG. (ed.); SurgailisD. (ed.); TeyssièreG. (ed.), A note on the monitoring of changes in linear models with dependent errors, Dependence in Probability and Statistics. Lecture Notes in Statistics, 2010, Springer: Berlin, Heidelberg · <a href="/1196.60011" class="nowrap">Zbl 1196.60011</a></td> </tr><tr> <td>[37]</td> <td class="space">Shao, X.; Zhang, X., Testing for change points in time series, Journal of the American Statistical Association, 105, 1228-1240, 2010 · <a href="/1390.62184" class="nowrap">Zbl 1390.62184</a></td> </tr><tr> <td>[38]</td> <td class="space">Wied, D.; Galeano, P., Monitoring correlation change in a sequence of random variables, Journal of Statistical Planning and Inference, 143, 186-196, 2013 · <a href="/1251.62025" class="nowrap">Zbl 1251.62025</a></td> </tr><tr> <td>[39]</td> <td class="space">Zeileis, A., Econometric computing with HC and HAC covariance matrix estimators, Journal of Statistical Software, 11, 1-17, 2004</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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