×

Minimum disparity estimation: improved efficiency through inlier modification. (English) Zbl 1468.62132

Summary: Inference procedures based on density based minimum distance techniques provide attractive alternatives to likelihood based methods for the statistician. The minimum disparity estimators are asymptotically efficient under the model; several members of this family also have strong robustness properties under model misspecification. Similarly, the disparity difference tests have the same asymptotic null distribution as the likelihood ratio test but are often superior than the latter in terms of robustness properties. However, many disparities put large weights on the inliers, cells with fewer data than expected under the model, which appears to be responsible for a somewhat poor efficiency of the corresponding methods in small samples. Here we consider several techniques which control the inliers without significantly affecting the robustness properties of the estimators and the corresponding tests. Extensive numerical studies involving simulated data illustrate the performance of the methods.

MSC:

62-08 Computational methods for problems pertaining to statistics
62F35 Robustness and adaptive procedures (parametric inference)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agostinelli, M.; Markatou, C., Test of hypotheses based on the weighted likelihood methodology, Statist. Sinica, 11, 2, 499-514, (2001) · Zbl 0980.62012
[2] Alin, A., A note on penalized power-divergence test statistics, Int. J. Math. Sci. (WASET), 1, 3, 209-215, (2007), (electronic)
[3] Basu, A.; Basu, S., Penalized minimum disparity methods for multinomial models, Statist. Sinica, 8, 3, 841-860, (1998) · Zbl 0905.62030
[4] Basu, A.; Harris, I.; Basu, S., Tests of hypotheses in discrete models based on the penalized Hellinger distance, Statist. Probab. Lett., 27, 4, 367-373, (1996) · Zbl 0849.62012
[5] Basu, A.; Harris, I. R.; Basu, S., Minimum distance estimation: the approach using density-based distances, (Robust Inference, Handbook of Statist., vol. 15, (1997), North-Holland Amsterdam), 21-48 · Zbl 0906.62019
[6] Basu, A.; Lindsay, B. G., The iteratively reweighted estimating equation in minimum distance problems, Comput. Statist. Data Anal., 45, 2, 105-124, (2004) · Zbl 1400.62074
[7] Basu, A.; Mandal, A.; Pardo, L., Hypothesis testing for two discrete populations based on the Hellinger distance, Statist. Probab. Lett., 80, 206-214, (2010) · Zbl 1180.62031
[8] Basu, A.; Ray, S.; Park, C.; Basu, S., Improved power in multinomial goodness-of-fit tests, Statistician, 51, 3, 381-393, (2002)
[9] Beran, R., Minimum Hellinger distance estimates for parametric models, Ann. Statist., 5, 3, 445-463, (1977) · Zbl 0381.62028
[10] Bhandari, S. K.; Basu, A.; Sarkar, S., Robust inference in parametric models using the family of generalized negative exponential dispatches, Aust. N. Z. J. Stat., 48, 1, 95-114, (2006) · Zbl 1109.62015
[11] Cressie, N.; Read, T. R.C., Multinomial goodness-of-fit tests, J. R. Stat. Soc. Ser. B, 46, 3, 440-464, (1984) · Zbl 0571.62017
[12] Csiszár, I., Eine informationstheoretische ungleichung und ihre anwendung auf den beweis der ergodizität von markoffschen ketten, Magyar. Tud. Akad. Mat. Kutató Int. Közl., 8, 85-108, (1963) · Zbl 0124.08703
[13] Hampel, F. R.; Ronchetti, E. M.; Rousseeuw, P. J.; Stahel, W. A., (Robust Statistics: The Approach Based on Influence Functions, Wiley Series in Probability and Mathematical Statistics, (1986), John Wiley & Sons Inc. New York)
[14] Harris, I. R.; Basu, A., Hellinger distance as a penalized log likelihood, Comm. Statist. Simulation Comput., 23, 4, 1097-1113, (1994)
[15] Lindsay, B. G., Efficiency versus robustness: the case for minimum Hellinger distance and related methods, Ann. Statist., 22, 2, 1081-1114, (1994) · Zbl 0807.62030
[16] Mandal, A., 2010. Minimum disparity inference: strategies for improvement in efficiency. Ph.D. Thesis. Indian Statistical Institute, Kolkata, India.
[17] Mandal, A.; Bhandari, S. K.; Basu, A., Minimum disparity estimation based on combined disparities: asymptotic results, J. Statist. Plann. Inference, 141, 2, 701-710, (2011) · Zbl 1209.62028
[18] Mandal, A.; Pardo, L.; Basu, A., Minimum disparity inference and the empty cell penalty: asymptotic results, Sankhyā Ser. A, 72, 2, 376-406, (2010) · Zbl 1213.62042
[19] Markatou, M.; Basu, A.; Lindsay, B. G., Weighted likelihood equations with bootstrap root search, J. Amer. Statist. Assoc., 93, 442, 740-750, (1998) · Zbl 0918.62046
[20] Neyman, J.; Pearson, E. S., On the use and interpretation of certain test criteria for purposes of statistical inference: part I, Biometrika, 20, 1, 175-240, (1928) · JFM 54.0565.05
[21] Pardo, L., (Statistical Inference Based on Divergence Measures, Statistics: Textbooks and Monographs, vol. 185, (2006), Chapman & Hall/CRC Boca Raton, FL) · Zbl 1118.62008
[22] Pardo, L.; Pardo, M. C., Minimum power-divergence estimator in three-way contingency tables, J. Stat. Comput. Simul., 73, 11, 819-831, (2003) · Zbl 1053.62073
[23] Park, C.; Basu, A.; Basu, S., Robust minimum distance inference based on combined distances, Comm. Statist. Simulation Comput., 24, 3, 653-673, (1995) · Zbl 0850.62243
[24] Park, C.; Basu, A.; Harris, I. R., Tests of hypotheses in multiple samples based on penalized disparities, J. Korean Statist. Soc., 30, 3, 347-366, (2001)
[25] Patra, R. K.; Mandal, A.; Basu, A., Minimum Hellinger distance estimation with inlier modification, Sankhyā Ser. B, 70, 2, 310-322, (2008) · Zbl 1192.62069
[26] Pearson, K., On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling, Phil. Mag. Ser. 5, 50, 302, 157-175, (1900) · JFM 31.0238.04
[27] Rao, C. R., Theory of the method of estimation by minimum chi-square, Bull. Inst. Internat. Statist., 35, 2, 25-32, (1957) · Zbl 0089.15305
[28] Sarkar, S.; Basu, A., On disparity based robust test for two discrete populations, Sankhyā Ser. B, 57, 3, 353-364, (1995) · Zbl 0856.62025
[29] Simpson, D. G., Hellinger deviance tests: efficiency, breakdown points, and examples, J. Amer. Statist. Assoc., 84, 405, 107-113, (1989)
[30] Vajda, I., Theory of statistical inference and information, (1989), Kluwer Academic Pub.
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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.