Document Zbl 1257.62013

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Pap.</a> <a href="/?q=in%3A314354" title="Articles in this Issue">54, No. 1, 177-192 (2013)</a>. </div> <div class="abstract">Summary: The <span class="zbmathjax-textit">P. Kumaraswamy</span> distribution [A generalized probability density function for double-bounded random processes. J. Hydrol. 46, 79–88 (1980)] is very similar to the beta distribution, but has the important advantage of an invertible closed form cumulative distribution function. The parameterization of the distribution in terms of shape parameters and the lack of simple expressions for its mean and variance hinder, however, its utilization with modeling purposes.<br class="zbmathjax-paragraph"> This paper presents two median-dispersion re-parameterizations of the Kumaraswamy distribution aimed at facilitating its use in regression models in which both the location and the dispersion parameters are functions of their own distinct sets of covariates, and in latent-variable and other models are estimated through simulation-based methods. In both re-parameterizations the dispersion parameter establishes a quantile-spread order among Kumaraswamy distributions with the same median and support. The study also describes the behavior of the re-parameterized distributions, determines some of their limiting distributions, and discusses the potential comparative advantages of using them in the context of regression modeling and simulation-based estimation.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6137155">Cited in <strong>37</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62E10" title="MSC2020">62E10</a> </td> <td class="space"> Characterization and structure theory of statistical distributions </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62E15" title="MSC2020">62E15</a> </td> <td class="space"> Exact distribution theory in statistics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J12" title="MSC2020">62J12</a> </td> <td class="space"> Generalized linear models (logistic models) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62E20" title="MSC2020">62E20</a> </td> <td class="space"> Asymptotic distribution theory in statistics </td> </tr><tr> <td class="mono"> 65C60 </td> <td class="space"> Computational problems in statistics (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Abeta+distribution">beta distribution</a>; <a href="/?q=ut%3Amedian-dispersion+parameterization">median-dispersion parameterization</a>; <a href="/?q=ut%3Aquantile-spread+order">quantile-spread order</a>; <a href="/?q=ut%3Alimiting+distributions">limiting distributions</a>; <a href="/?q=ut%3Aregression+modeling">regression modeling</a>; <a href="/?q=ut%3Ageneralized+linear+models">generalized linear models</a>; <a href="/?q=ut%3Alatent-variable+models">latent-variable models</a>; <a href="/?q=ut%3Asimulation-based+estimation+methods">simulation-based estimation methods</a></div> <div class="software"> <h3>Software:</h3><a href="/software/15449">betareg</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 1257.62013" data-ciurl="/ci/06137155" data-biburl="/bibtex/06137155.bib" data-amsurl="/amsrefs/06137155.bib" data-xmlurl="/xml/06137155.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06137155.pdf" title="Zbl 1257.62013 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/10.1007/s00362-011-0417-y" aria-label="DOI for “The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation”" title="10.1007/s00362-011-0417-y">DOI</a> <a class="btn btn-default btn-xs" type="button" href="http://hdl.handle.net/10.1007/s00362-011-0417-y" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abramowitz M, Stegun IA (eds) (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables (9th edn). New York, Dover · <a href="/0543.33001" class="nowrap">Zbl 0543.33001</a></td> </tr><tr> <td>[2]</td> <td class="space">Balanda K, MacGillivray H (1990) Can J Stat 18(1): 17–30 · <a href="/0692.62010" class="nowrap">Zbl 0692.62010</a> · <a href="https://doi.org/10.2307/3315414" class="nowrap">doi:10.2307/3315414</a></td> </tr><tr> <td>[3]</td> <td class="space">Bickel PJ, Lehmann EL (1979) Descriptive statistics for nonparametric models IV. Spread. In: Jurečková J (ed). Contributions to statistics. D. Reidel, Dordrecht and Boston · <a href="/0415.62015" class="nowrap">Zbl 0415.62015</a></td> </tr><tr> <td>[4]</td> <td class="space">Brehm J, Gates S (1993) Donut shops and speed traps: evaluating models of supervision on police behavior. Am J Political Sci 37(2): 555–581 · <a href="https://doi.org/10.2307/2111384" class="nowrap">doi:10.2307/2111384</a></td> </tr><tr> <td>[5]</td> <td class="space">Cameron A, Trivedi P (2005) Microeconometrics. Cambridge University Press, Cambridge · <a href="/1182.62213" class="nowrap">Zbl 1182.62213</a></td> </tr><tr> <td>[6]</td> <td class="space">Courard-Hauri D (2007) Using Monte Carlo analysis to investigate the relationship between overconsumption and uncertain access to one’s personal utility function. Ecol Econ 64(1): 152–162 · <a href="https://doi.org/10.1016/j.ecolecon.2007.02.018" class="nowrap">doi:10.1016/j.ecolecon.2007.02.018</a></td> </tr><tr> <td>[7]</td> <td class="space">Cribari-Neto F, Souza T (Forthcoming) Testing inference in variable dispersion beta regressions. J Stat Comput Simul · <a href="/1431.62329" class="nowrap">Zbl 1431.62329</a></td> </tr><tr> <td>[8]</td> <td class="space">Cribari-Neto F, Zeileis A (2010) Beta regression in R. J Stat Software 34(2): 1–24</td> </tr><tr> <td>[9]</td> <td class="space">Devroye L (1986) Non-uniform random variate generation. Springer-Verlag, New York · <a href="/0593.65005" class="nowrap">Zbl 0593.65005</a></td> </tr><tr> <td>[10]</td> <td class="space">Espinheira PL, Ferrari SLP, Cribari-Neto F (2008a) Influence diagnostics in beta regression. Comput Stat Data Anal 52(9): 4417–4431 · <a href="/1452.62496" class="nowrap">Zbl 1452.62496</a> · <a href="https://doi.org/10.1016/j.csda.2008.02.028" class="nowrap">doi:10.1016/j.csda.2008.02.028</a></td> </tr><tr> <td>[11]</td> <td class="space">Espinheira PL, Ferrari SLP, Cribari-Neto F (2008b) On beta regression residuals. J Appl Stat 35(4): 407–419 · <a href="/1147.62315" class="nowrap">Zbl 1147.62315</a> · <a href="https://doi.org/10.1080/02664760701834931" class="nowrap">doi:10.1080/02664760701834931</a></td> </tr><tr> <td>[12]</td> <td class="space">Fernandez-Ponce JM, Kochar SC, Muñoz-Perez J (1998) Partial orderings of distributions based on right-spread functions. J Appl Prob 35: 221–228 · <a href="/0898.62124" class="nowrap">Zbl 0898.62124</a> · <a href="https://doi.org/10.1239/jap/1032192565" class="nowrap">doi:10.1239/jap/1032192565</a></td> </tr><tr> <td>[13]</td> <td class="space">Ferrari SLP, Cribari-Neto F (2004) Beta regression for modeling rates and proportions. J Appl Stat 31(7): 799–815 · <a href="/1121.62367" class="nowrap">Zbl 1121.62367</a> · <a href="https://doi.org/10.1080/0266476042000214501" class="nowrap">doi:10.1080/0266476042000214501</a></td> </tr><tr> <td>[14]</td> <td class="space">Ferrari SLP, Espinheira PL, Cribari-Neto F (2011) Diagnostic tools in beta regression with varying dispersion. Statistica Neerlandica 65(3): 337–351 · <a href="/1219.62117" class="nowrap">Zbl 1219.62117</a> · <a href="https://doi.org/10.1111/j.1467-9574.2011.00488.x" class="nowrap">doi:10.1111/j.1467-9574.2011.00488.x</a></td> </tr><tr> <td>[15]</td> <td class="space">Fletcher SC, Ponnambalam K (1996) Estimation of reservoir yield and storage distribution using moments analysis. J Hydrol 182(1–4): 259–275 · <a href="https://doi.org/10.1016/0022-1694(95)02946-X" class="nowrap">doi:10.1016/0022-1694(95)02946-X</a></td> </tr><tr> <td>[16]</td> <td class="space">Gallant AR, Tauchen G (1996) Which moments to match. Econom Theory 12: 657–681 · <a href="https://doi.org/10.1017/S0266466600006976" class="nowrap">doi:10.1017/S0266466600006976</a></td> </tr><tr> <td>[17]</td> <td class="space">Ganji A, Ponnambalam K, Khalili D, Karamouz M (2006) Grain yield reliability analysis with crop water demand uncertainty. Stoch Environ Res Risk Assess 20(4): 259–277 · <a href="https://doi.org/10.1007/s00477-005-0020-7" class="nowrap">doi:10.1007/s00477-005-0020-7</a></td> </tr><tr> <td>[18]</td> <td class="space">Garg M (2008) On distribution of order statistics from Kumaraswamy distribution. Kyunpook Math J 48: 411–417 · <a href="/1158.62037" class="nowrap">Zbl 1158.62037</a> · <a href="https://doi.org/10.5666/KMJ.2008.48.3.411" class="nowrap">doi:10.5666/KMJ.2008.48.3.411</a></td> </tr><tr> <td>[19]</td> <td class="space">Gouriéroux C, Monfort A (1996) Simulation-based econometric methods. Oxford University Press, Oxford</td> </tr><tr> <td>[20]</td> <td class="space">Green WH (2007) Censored data and truncated distributions. In: Mills T, Patterson K (eds) The handbook of econometrics, Vol 1. Palgrave, London</td> </tr><tr> <td>[21]</td> <td class="space">Hao L, Naiman D (2007) Quantile regression. Sage, Thousand Oaks</td> </tr><tr> <td>[22]</td> <td class="space">He X (1997) Quantile curves without crossing. Am Stat 51(2): 186–192</td> </tr><tr> <td>[23]</td> <td class="space">Hickey R (1986) Concepts of dispersion in distributions: a comparative note. J Appl Prob 23(4): 914–921 · <a href="/0607.60016" class="nowrap">Zbl 0607.60016</a> · <a href="https://doi.org/10.2307/3214465" class="nowrap">doi:10.2307/3214465</a></td> </tr><tr> <td>[24]</td> <td class="space">Johnson N, Kotz S, Balakrishnan N (1995) Continuous univariate distributions. Wiley, New York · <a href="/0821.62001" class="nowrap">Zbl 0821.62001</a></td> </tr><tr> <td>[25]</td> <td class="space">Jones MC (2009) Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages. Stat Methodol 6(1): 70–81 · <a href="/1215.60010" class="nowrap">Zbl 1215.60010</a> · <a href="https://doi.org/10.1016/j.stamet.2008.04.001" class="nowrap">doi:10.1016/j.stamet.2008.04.001</a></td> </tr><tr> <td>[26]</td> <td class="space">Kieschnick R, McCullough B (2003) Regression analysis of variates observed on (0, 1): percentages, proportions and fractions. Stat Model 3: 193–213 · <a href="https://doi.org/10.1191/1471082X03st053oa" class="nowrap">doi:10.1191/1471082X03st053oa</a></td> </tr><tr> <td>[27]</td> <td class="space">King G, Tomz M, Wittenberg J (2000) Making the most of statistical analyses: improving interpretation and presentation. Am J Political Sci 44(2): 341–355</td> </tr><tr> <td>[28]</td> <td class="space">Koenker R (2005) Quantile regression. Cambridge University Press, Cambridge · <a href="/1111.62037" class="nowrap">Zbl 1111.62037</a></td> </tr><tr> <td>[29]</td> <td class="space">Kumaraswamy P (1980) A generalized probability density function for double-bounded random processes. J Hydrol 46: 79–88 · <a href="https://doi.org/10.1016/0022-1694(80)90036-0" class="nowrap">doi:10.1016/0022-1694(80)90036-0</a></td> </tr><tr> <td>[30]</td> <td class="space">Lewis T, Thompson JW (1981) Dispersive distributions and the connections between dispersivity and strong unimodality. J Appl Prob 18(1): 76–90 · <a href="/0453.60023" class="nowrap">Zbl 0453.60023</a> · <a href="https://doi.org/10.2307/3213168" class="nowrap">doi:10.2307/3213168</a></td> </tr><tr> <td>[31]</td> <td class="space">Little R, Rubin D (2002) Statistical analysis with missing data (2nd edn). Wiley, Hoboken · <a href="/1011.62004" class="nowrap">Zbl 1011.62004</a></td> </tr><tr> <td>[32]</td> <td class="space">Machado J, Santos Silva JMC (2010) Quantiles with corners. Unpublished.</td> </tr><tr> <td>[33]</td> <td class="space">Manski C (1991) Regression. J Econ Lit 29(1): 34–50</td> </tr><tr> <td>[34]</td> <td class="space">McCullagh P, Nelder J (1989) Generalized linear models. Chapman &amp; Hall, London · <a href="/0744.62098" class="nowrap">Zbl 0744.62098</a></td> </tr><tr> <td>[35]</td> <td class="space">McDonald J (1984) Some generalized functions for the size distribution of income. Econometrica 52(3): 647–665 · <a href="/0557.62098" class="nowrap">Zbl 0557.62098</a> · <a href="https://doi.org/10.2307/1913469" class="nowrap">doi:10.2307/1913469</a></td> </tr><tr> <td>[36]</td> <td class="space">Mielke PW Jr. (1975) Convenient beta distribution likelihood techniques for describing and comparing meteorological data. J Appl Meteorol 14: 985–990 · <a href="https://doi.org/10.1175/1520-0450(1975)014<0985:CBDLTF>2.0.CO;2" class="nowrap">doi:10.1175/1520-0450(1975)014<0985:CBDLTF>2.0.CO;2</a></td> </tr><tr> <td>[37]</td> <td class="space">Mitnik P (Forthcoming). New properties of the Kumaraswamy distribution. Commun Stat: Theory Method</td> </tr><tr> <td>[38]</td> <td class="space">Nadarajah S (2008) On the distribution of Kumaraswamy. J Hydrol 348: 568–569 · <a href="https://doi.org/10.1016/j.jhydrol.2007.09.008" class="nowrap">doi:10.1016/j.jhydrol.2007.09.008</a></td> </tr><tr> <td>[39]</td> <td class="space">Nagypál É (2007) Learning by doing vs.. learning about match quality: can we tell them apart? Rev Econ Stud 74: 537–566 · <a href="/1297.91099" class="nowrap">Zbl 1297.91099</a></td> </tr><tr> <td>[40]</td> <td class="space">Ospina R, Cribari-Neto F, Vasconcellos K (2006) Improved point and interval estimation for a beta regression model. Comput Stat Data Anal 51: 960–981 · <a href="/1157.62346" class="nowrap">Zbl 1157.62346</a> · <a href="https://doi.org/10.1016/j.csda.2005.10.002" class="nowrap">doi:10.1016/j.csda.2005.10.002</a></td> </tr><tr> <td>[41]</td> <td class="space">Paolino P (2001) Maximum likelihood estimation of models with beta-distributed dependent variables. Political Anal 9(4): 325–346 · <a href="https://doi.org/10.1093/oxfordjournals.pan.a004873" class="nowrap">doi:10.1093/oxfordjournals.pan.a004873</a></td> </tr><tr> <td>[42]</td> <td class="space">Pham-Gia T, Hung TL (2001) The mean and median absolute deviations. Math Comput Model 34: 921–936 · <a href="/0996.60008" class="nowrap">Zbl 0996.60008</a> · <a href="https://doi.org/10.1016/S0895-7177(01)00109-1" class="nowrap">doi:10.1016/S0895-7177(01)00109-1</a></td> </tr><tr> <td>[43]</td> <td class="space">Roberto M, Schuermann T, Weeks M (2000) Simulation-based inference in econometrics: methods and applications. Cambridge University Press, Cambridge · <a href="/0959.00008" class="nowrap">Zbl 0959.00008</a></td> </tr><tr> <td>[44]</td> <td class="space">Rocha AV, Simas AB (2011) Influence diagnostics in a general class of beta regression models. Test 20: 95–119 · <a href="/1331.62376" class="nowrap">Zbl 1331.62376</a> · <a href="https://doi.org/10.1007/s11749-010-0189-z" class="nowrap">doi:10.1007/s11749-010-0189-z</a></td> </tr><tr> <td>[45]</td> <td class="space">Sanchez S, Ancheyta J, McCaffrey WC (2007) Comparison of probability distribution functions for fitting distillation curves of petroleum. Energy &amp; Fuels 21(5): 2955–2963 · <a href="https://doi.org/10.1021/ef070003y" class="nowrap">doi:10.1021/ef070003y</a></td> </tr><tr> <td>[46]</td> <td class="space">Seifi A, Ponnambalam K, Vlach J (2000) Maximization of manufacturing yield of systems with arbitrary distributions of component values. Ann Oper Res 99: 373–383 · <a href="/0990.90054" class="nowrap">Zbl 0990.90054</a> · <a href="https://doi.org/10.1023/A:1019288220413" class="nowrap">doi:10.1023/A:1019288220413</a></td> </tr><tr> <td>[47]</td> <td class="space">Shaked M, Shanthikumar G (2007) Stochastic orders. Springer, New York · <a href="/0806.62009" class="nowrap">Zbl 0806.62009</a></td> </tr><tr> <td>[48]</td> <td class="space">Simas AB, Barreto-Souza W, Rocha AV (2010) Improved estimators for a general class of beta regression models. Comput Stat Data Anal 54(2): 348–366 · <a href="/1465.62019" class="nowrap">Zbl 1465.62019</a> · <a href="https://doi.org/10.1016/j.csda.2009.08.017" class="nowrap">doi:10.1016/j.csda.2009.08.017</a></td> </tr><tr> <td>[49]</td> <td class="space">Smithson M, Verkuilen J (2006) A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychol Method 11(1): 54–71 · <a href="https://doi.org/10.1037/1082-989X.11.1.54" class="nowrap">doi:10.1037/1082-989X.11.1.54</a></td> </tr><tr> <td>[50]</td> <td class="space">Smyth G (1989) Generalized linear models with varying dispersion. J R Stat Soc, Ser B 51(1): 47–60</td> </tr><tr> <td>[51]</td> <td class="space">Sundar V, Subbiah K (1989) Application of double bounded probability density-function for analysis of ocean waves. Ocean Eng 16(2): 193–200 · <a href="https://doi.org/10.1016/0029-8018(89)90005-X" class="nowrap">doi:10.1016/0029-8018(89)90005-X</a></td> </tr><tr> <td>[52]</td> <td class="space">Townsend J, Colonius H (2005) Variability of the max and min statistic: a theory of the quantile spread as a function of sample size. Psychometrika 70(4): 759–772 · <a href="/1306.62508" class="nowrap">Zbl 1306.62508</a> · <a href="https://doi.org/10.1007/s11336-001-0916-1" class="nowrap">doi:10.1007/s11336-001-0916-1</a></td> </tr><tr> <td>[53]</td> <td class="space">van Dorp JR, Mazzuchi T (2004) Parameter specification of the beta distribution and its dirichlet extensions utilizing quantiles. In: Gupta A, Nadarajah S (eds). Handbook of beta distribution and its applications. Marcel Dekker, New York</td> </tr><tr> <td>[54]</td> <td class="space">Vasconcellos K, Cribari-Neto F (2005) Improved maximum likelihood estimation in a new class of beta regression models. Braz J Prob Stat 19: 13–31 · <a href="/1272.62050" class="nowrap">Zbl 1272.62050</a></td> </tr><tr> <td>[55]</td> <td class="space">Yamaguchi S (2007) Job search, bargaining, and wage dynamics, ISER Discussion Paper # 658. Institute of Social and Economic Research, Osaka University, Osaka</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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