Robust statistical inference based on the C-divergence family
This paper describes a family of divergences, named herein as the C -divergence family, which is a generalized version of the power divergence family and also includes the density power divergence family as a particular member of this class. We explore the connection of this family with other diverg...
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| Vydáno v: | Annals of the Institute of Statistical Mathematics Ročník 71; číslo 5; s. 1289 - 1322 |
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01.10.2019
Springer Nature B.V |
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| ISSN: | 0020-3157, 1572-9052 |
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| Abstract | This paper describes a family of divergences, named herein as the
C
-divergence family, which is a generalized version of the power divergence family and also includes the density power divergence family as a particular member of this class. We explore the connection of this family with other divergence families and establish several characteristics of the corresponding minimum distance estimator including its asymptotic distribution under both discrete and continuous models; we also explore the use of the
C
-divergence family in parametric tests of hypothesis. We study the influence function of these minimum distance estimators, in both the first and second order, and indicate the possible limitations of the first-order influence function in this case. We also briefly study the breakdown results of the corresponding estimators. Some simulation results and real data examples demonstrate the small sample efficiency and robustness properties of the estimators. |
|---|---|
| AbstractList | This paper describes a family of divergences, named herein as the C-divergence family, which is a generalized version of the power divergence family and also includes the density power divergence family as a particular member of this class. We explore the connection of this family with other divergence families and establish several characteristics of the corresponding minimum distance estimator including its asymptotic distribution under both discrete and continuous models; we also explore the use of the C-divergence family in parametric tests of hypothesis. We study the influence function of these minimum distance estimators, in both the first and second order, and indicate the possible limitations of the first-order influence function in this case. We also briefly study the breakdown results of the corresponding estimators. Some simulation results and real data examples demonstrate the small sample efficiency and robustness properties of the estimators. This paper describes a family of divergences, named herein as the C -divergence family, which is a generalized version of the power divergence family and also includes the density power divergence family as a particular member of this class. We explore the connection of this family with other divergence families and establish several characteristics of the corresponding minimum distance estimator including its asymptotic distribution under both discrete and continuous models; we also explore the use of the C -divergence family in parametric tests of hypothesis. We study the influence function of these minimum distance estimators, in both the first and second order, and indicate the possible limitations of the first-order influence function in this case. We also briefly study the breakdown results of the corresponding estimators. Some simulation results and real data examples demonstrate the small sample efficiency and robustness properties of the estimators. |
| Author | Basu, Ayanendranath Ghosh, Abhik Maji, Avijit Pardo, Leandro |
| Author_xml | – sequence: 1 givenname: Avijit surname: Maji fullname: Maji, Avijit email: avijit.maji@hotmail.com organization: Department of Statistics and Information Management, Reserve Bank of India – sequence: 2 givenname: Abhik surname: Ghosh fullname: Ghosh, Abhik organization: Indian Statistical Institute – sequence: 3 givenname: Ayanendranath surname: Basu fullname: Basu, Ayanendranath organization: Indian Statistical Institute – sequence: 4 givenname: Leandro surname: Pardo fullname: Pardo, Leandro organization: Department of Statistics and O.R., Complutense University of Madrid |
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| CitedBy_id | crossref_primary_10_1109_TIT_2024_3366538 crossref_primary_10_1007_s40304_020_00221_8 crossref_primary_10_1007_s10463_025_00939_8 crossref_primary_10_1080_00949655_2020_1783665 crossref_primary_10_1080_03610926_2022_2155788 crossref_primary_10_1007_s11749_024_00956_4 |
| Cites_doi | 10.1109/TIT.2018.2794537 10.1016/j.jspi.2015.01.003 10.1111/j.2517-6161.1984.tb01318.x 10.1111/j.2517-6161.1966.tb00626.x 10.1016/j.jspi.2008.04.022 10.1093/biomet/85.3.549 10.1002/0471725382 10.1080/01621459.1989.10478744 10.1016/j.spl.2016.02.007 10.1239/jap/1269610827 10.1080/01621459.1987.10478501 10.1016/j.jspi.2012.03.019 10.1007/s10463-012-0372-y 10.1201/b10956 10.1080/01621459.1986.10478264 10.3150/16-BEJ826 10.1007/s13571-012-0050-3 10.1214/aos/1176350367 10.1002/em.2860060207 10.1007/BF00773476 10.1214/aos/1176343997 10.1214/aos/1176343842 10.1214/aos/1176325512 10.5109/12576 10.1007/978-1-4757-2769-2 10.1016/j.jmva.2008.02.004 10.1007/s13171-014-0063-2 10.1093/biomet/88.3.865 |
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| Keywords | Generalized power divergence Divergence Density power divergence Power divergence |
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| References | Broniatowski, M., Toma, A., Vajda, I. (2012). Decomposable pseudodistances and applications in statistical estimation. Journal of Statistical Planning and Inference, 142, 2574–2585. BeranRJMinimum Hellinger distance estimates for parametric modelsAnnals of Statistics1977544546344870010.1214/aos/11763438420381.62028 Basu, A., Harris, I. R., Hjort, N. L., Jones, M. C. (1998). Robust and efficient estimation by minimising a density power divergence. Biometrika, 85, 549–559. Broniatowski, M., Vajda, I. (2012). Several applications of divergence criteria in continuous families. Kybernetika, 48, 600–636. Ghosh, A., Basu, A., Pardo, L. (2015). On the robustness of a divergence based test of simple statistical hypotheses. Journal of Statistical Planning and Inference, 161, 91–108. Ghosh, A., Maji, A., Basu, A. (2013). Robust inference based on divergences in reliability systems. In I. Frenkel, A. Karagrigoriou, A. Lisnianski, A. Kleyner (Eds.), Applied reliability engineering and risk analysis. probabilistic models and statistical inference. Dedicated to the Centennial of the birth of Boris Gnedenko. New York: Wiley. SimpsonDGMinimum Hellinger distance estimation for the analysis of count dataJournal of the American Statistical Association19878280280790998510.1080/01621459.1987.104785010633.62029 Ghosh, A., Harris, I. R., Maji, A., Basu, A., Pardo, L. (2017). A generalized divergence for statistical inference. Bernoulli, 23, 2746–2783. Woodruff, R. C., Mason, J. M., Valencia, R., Zimmering, S. (1984). Chemical mutagenesis testing in drosophila I: Comparison of positive and negative control data for sex-linked recessive lethal mutations and reciprocal translocations in three laboratories. Environmental and Molecular Mutagenesis, 6, 189–202. Simpson, D. G., Carroll, R. J., Ruppert, D. (1987). M-estimation for discrete data: Asymptotic distribution theory and implications. Annals of Statistics, 15, 657–669. SimpsonDGHellinger deviance test: Efficiency, breakdown points, and examplesJournal of the American Statistical Association19898410711399966710.1080/01621459.1989.10478744 Vonta, F., Karagrigoriou, A. (2010). Generalized measures of divergences in survival analysis and reliability. Journal of Applied Probability, 47, 216–234. LehmannELTheory of point estimation1983New YorkWiley10.1007/978-1-4757-2769-20522.62020 Ghosh, A., Basu, A. (2016). Testing composite null hypotheses based on S-divergences. Statistics and Probability Letters, 114, 38–47. LindsayBGEfficiency versus robustness: The case for minimum Hellinger distance and related methodsAnnals of Statistics19942210811114129255710.1214/aos/11763255120807.62030 Fujisawa, H., Eguchi., S. (2008). Robust parameter estimation with a small bias against heavy contamination. Journal of Multivariate Analysis, 99, 2053–2081. Tamura, R. N., Boos, D. D. (1986). Minimum Hellinger distance estimation for multivariate location and covariance. Journal of the American Statistical Association, 81, 223–229. GhoshAAsymptotic properties of minimum S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document}-divergence estimator for discrete modelsSankhya A201577380407340012010.1007/s13171-014-0063-21322.62093 Ali, S. M., Silvey, S. D. (1966). A general class of coefficients of divergence of one distribution from another. Journal of the Royal Statistical Society B, 28, 131–142. Basu, A., Mandal, A., Martin, N., Pardo, L. (2013). Testing statistical hypotheses based on the density power divergence. Annals of the Institute of Statistical Mathematics, 65, 319–348. PardoLStatistical inference based on divergences2006Boca RatonCRC/Chapman-Hall1118.62008 StiglerSMDo robust estimators work with real data?Annals of Statistics197751055109845520510.1214/aos/11763439970374.62050(with discussion) CsiszárIEine informations theoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen KettenPublications of the Mathematical Institute of the Hungarian Academy of Sciences19633851070124.08703 Basu, A., Shioya, H., Park, C. (2011). Statistical inference: The minimum distance approach. Boca Raton: Chapman & Hall/CRC. Jones, M. C., Hjort, N. L., Harris, I. R., Basu, A. (2001). A comparison of related density based minimum divergence estimators. Biometrika, 88, 865–873. Patra, S., Maji, A., Basu, A., Pardo, L. (2013). The power divergence and the density power divergence families: The mathematical connection. Sankhya B, 75, 16–28. Cressie, N., Read, T. R. C. (1984). Multinomial goodness-of-fit tests. 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Minimum disparity estimation: Asymptotic normality and breakdown point results. Bulletin of Informatics and Cybernetics, 36, 19–33. (special issue in Honor of Professor Takashi Yanagawa). Mattheou, K., Leeb, S., Karagrigoriou, A. (2009). A model selection criterion based on the BHHJ measure of divergence. Journal of Statistical Planning and Inference, 139, 228–235. Hampel, F. R., Ronchetti, E., Rousseeuw, P. J., Stahel, W. (1986). Robust statistics: The approach based on influence functions. New York: Wiley. A Ghosh (678_CR13) 2015; 77 678_CR28 678_CR27 678_CR26 I Csiszár (678_CR11) 1967; 2 678_CR24 678_CR21 678_CR20 678_CR3 678_CR4 678_CR5 678_CR1 SM Stigler (678_CR32) 1977; 5 678_CR2 EL Lehmann (678_CR22) 1983 DG Simpson (678_CR29) 1987; 82 BG Lindsay (678_CR23) 1994; 22 678_CR19 678_CR18 678_CR17 678_CR16 678_CR15 678_CR14 678_CR35 678_CR12 678_CR34 678_CR33 678_CR31 678_CR7 678_CR8 678_CR9 L Pardo (678_CR25) 2006 RJ Beran (678_CR6) 1977; 5 I Csiszár (678_CR10) 1963; 3 DG Simpson (678_CR30) 1989; 84 |
| References_xml | – reference: Basu, A., Shioya, H., Park, C. (2011). Statistical inference: The minimum distance approach. Boca Raton: Chapman & Hall/CRC. – reference: Rousseeuw, P. J., Leroy, A. M. (1987). Robust regression and outlier detection. New York: Wiley. – reference: SimpsonDGHellinger deviance test: Efficiency, breakdown points, and examplesJournal of the American Statistical Association19898410711399966710.1080/01621459.1989.10478744 – reference: Basu, A., Mandal, A., Martin, N., Pardo, L. (2013). Testing statistical hypotheses based on the density power divergence. Annals of the Institute of Statistical Mathematics, 65, 319–348. – reference: CsiszárIInformation-type measures of difference of probability distributions and indirect observationsStudia Scientiarum Mathematicarum Hungarica196722993182193450157.25802 – reference: Broniatowski, M., Vajda, I. (2012). Several applications of divergence criteria in continuous families. Kybernetika, 48, 600–636. – reference: Jones, M. C., Hjort, N. L., Harris, I. R., Basu, A. (2001). A comparison of related density based minimum divergence estimators. Biometrika, 88, 865–873. – reference: Ghosh, A., Basu, A., Pardo, L. (2015). On the robustness of a divergence based test of simple statistical hypotheses. Journal of Statistical Planning and Inference, 161, 91–108. – reference: SimpsonDGMinimum Hellinger distance estimation for the analysis of count dataJournal of the American Statistical Association19878280280790998510.1080/01621459.1987.104785010633.62029 – reference: Basu, A., Lindsay, B. G. (1994). Minimum disparity estimation for continuous models: Efficiency, distributions and robustness. Annals of the Institute of Statistical Mathematics, 46, 683–705. – reference: Patra, S., Maji, A., Basu, A., Pardo, L. (2013). The power divergence and the density power divergence families: The mathematical connection. Sankhya B, 75, 16–28. – reference: LindsayBGEfficiency versus robustness: The case for minimum Hellinger distance and related methodsAnnals of Statistics19942210811114129255710.1214/aos/11763255120807.62030 – reference: Mattheou, K., Leeb, S., Karagrigoriou, A. (2009). A model selection criterion based on the BHHJ measure of divergence. Journal of Statistical Planning and Inference, 139, 228–235. – reference: Tamura, R. N., Boos, D. D. (1986). Minimum Hellinger distance estimation for multivariate location and covariance. Journal of the American Statistical Association, 81, 223–229. – reference: Fujisawa, H., Eguchi., S. (2008). Robust parameter estimation with a small bias against heavy contamination. Journal of Multivariate Analysis, 99, 2053–2081. – reference: Hampel, F. R., Ronchetti, E., Rousseeuw, P. J., Stahel, W. (1986). Robust statistics: The approach based on influence functions. New York: Wiley. – reference: Basu, A., Harris, I. R., Hjort, N. L., Jones, M. C. (1998). Robust and efficient estimation by minimising a density power divergence. Biometrika, 85, 549–559. – reference: Broniatowski, M., Toma, A., Vajda, I. (2012). Decomposable pseudodistances and applications in statistical estimation. Journal of Statistical Planning and Inference, 142, 2574–2585. – reference: CsiszárIEine informations theoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen KettenPublications of the Mathematical Institute of the Hungarian Academy of Sciences19633851070124.08703 – reference: StiglerSMDo robust estimators work with real data?Annals of Statistics197751055109845520510.1214/aos/11763439970374.62050(with discussion) – reference: GhoshAAsymptotic properties of minimum S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document}-divergence estimator for discrete modelsSankhya A201577380407340012010.1007/s13171-014-0063-21322.62093 – reference: Simpson, D. G., Carroll, R. J., Ruppert, D. (1987). 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C
-divergence family, which is a generalized version of the power divergence family and also... This paper describes a family of divergences, named herein as the C-divergence family, which is a generalized version of the power divergence family and also... |
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| SubjectTerms | Computer simulation Divergence Economic models Economics Estimators Finance Hypothesis testing Influence functions Insurance Management Mathematics Mathematics and Statistics Statistical inference Statistics Statistics for Business |
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| Title | Robust statistical inference based on the C-divergence family |
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