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Keywords:
divergence; parametric estimation; robustness
Summary:
This paper deals with four types of point estimators based on minimization of information-theoretic divergences between hypothetical and empirical distributions. These were introduced (i) by Liese and Vajda [9] and independently Broniatowski and Keziou [3], called here power superdivergence estimators, (ii) by Broniatowski and Keziou [4], called here power subdivergence estimators, (iii) by Basu et al. [2], called here power pseudodistance estimators, and (iv) by Vajda [18] called here Rényi pseudodistance estimators. These various criterions have in common to eliminate all need for grouping or smoothing in statistical inference. The paper studies and compares general properties of these estimators such as Fisher consistency and influence curves, and illustrates these properties by detailed analysis of the applications to the estimation of normal location and scale.
References:
[1] D. F. Andrews, P. J. Bickel, F. R. Hampel, P. J. Huber, W. H. Rogers, J. W. Tukey: Robust Estimates of Location. Princeton University Press, Princeton N. J. 1972. MR 0331595 | Zbl 0254.62001
[2] A. Basu, I. R. Harris, N. L. Hjort, M. C. Jones: Robust and efficient estimation by minimizing a density power divergence. Biometrika 85 (1998), 3, 549-559. DOI 10.1093/biomet/85.3.549 | MR 1665873
[3] M. Broniatowski, A. Keziou: Minimization of $\phi$-divergences on sets of signed measures. Studia Sci. Math. Hungar. 43 (2006), 403-442. MR 2273419 | Zbl 1121.28004
[4] M. Broniatowski, A. Keziou: Parametric estimation and tests through divergences and the duality technique. J. Multivariate Anal. 100 (2009), 1, 16-31. DOI 10.1016/j.jmva.2008.03.011 | MR 2460474 | Zbl 1151.62023
[5] M. Broniatowski, A. Toma, I. Vajda: Decomposable pseudodistances and applications in statistical estimation. J. Statist. Plann. Inference. 142 (2012), 9, 2574-2585 DOI 10.1016/j.jspi.2012.03.019 | MR 2922007
[6] M. Broniatowski, I. Vajda: Several applications of divergence criteria in continuous families. arXiv:0911.0937v1, 2009.
[7] F. R. Hampel, E. M. Ronchetti, P. J. Rousseuw, W. A. Stahel: Robust Statistics: The approach Based on Influence Functions. Willey, New York 1986. MR 0829458
[8] F. Liese, I. Vajda: Convex Statistical Distances. Teubner, Leipzig 1987. MR 0926905 | Zbl 0656.62004
[9] F. Liese, I. Vajda: On divergences and informations in statistics and information theory. IEEE Trans. Inform. Theory 52 (2006), 10, 4394-4412. DOI 10.1109/TIT.2006.881731 | MR 2300826
[10] C. Miescke, F. Liese: Statistical Decision Theory. Springer, Berlin 2008. MR 2421720 | Zbl 1154.62008
[11] M. R. C. Read, N. A. C. Cressie: Goodness-of-Fit Statistics for Discrete Multivariate Data. Springer, Berlin 1988. MR 0955054 | Zbl 0663.62065
[12] A. Rényi: On measures of entropy and information. In: Proc. 4th Berkeley Symp. on Probability and Statistics, Vol. 1, University of California Press, Berkeley 1961, pp. 547-561. MR 0132570 | Zbl 0106.33001
[13] A. Toma, M. Broniatowski: Minimum divergence estimators and tests: Robustness results. J. Multivariate Anal. 102 (2011), 1, 20-36. DOI 10.1016/j.jmva.2010.07.010 | MR 2729417
[14] I. Vajda: Minimum divergence principle in statistical estimation. Statist. Decisions (1984), Suppl. Issue No. 1, 239-261. MR 0785211 | Zbl 0558.62004
[15] I. Vajda: Efficiency and robustness control via distorted maximum likelihood estimation. Kybernetika 22 (1986), 47-67. MR 0839344 | Zbl 0603.62039
[16] I. Vajda: Comparison of asymptotic variances for several estimators of location. Probl. Control Inform. Theory 18 (1989), 2, 79-89. MR 0991547 | Zbl 0678.62035
[17] I. Vajda: Estimators asymptotically minimax in wide sense. Biometr. J. 31 (1989), 7, 803-810. DOI 10.1002/bimj.4710310706 | MR 1054736
[18] I. Vajda: Modifications od Divergence Criteria for Applications in Continuous Families. Research Report No. 2230, Institute of Information Theory and Automation, Prague 2008.
[19] A. W. van der Vaart: Asymptotic Statistics. Cambridge University Press, Cambridge 1998. MR 1652247 | Zbl 0910.62001
[20] A. W. van der Vaart, J. A. Wellner: Weak Convergence and Empirical Processes. Springer, Berlin 1996. MR 1385671 | Zbl 0862.60002

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