Abstract
The family of density power divergences is an useful class which generates robust parameter estimates with high efficiency. None of these divergences require any non-parametric density estimate to carry out the inference procedure. However, these divergences have so far not been used effectively in robust testing of hypotheses. In this paper, we develop tests of hypotheses based on this family of divergences. The asymptotic variances of the estimators are generally different from the inverse of the Fisher information matrix, so that the usual drop-in-divergence type statistics do not lead to standard Chi-square limits. It is shown that the alternative test statistics proposed herein have asymptotic limits which are described by linear combinations of Chi-square statistics. Extensive simulation results are presented to substantiate the theory developed.
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References
Basu A., Harris I. R., Hjort N. L., Jones M. C. (1998) Robust and efficient estimation by minimizing a density power divergence. Biometrika, 85: 549–559
Basu, A., Shioya, H., Park, C. (2011). Statistical inference: the minimum distance approach. Boca Raton: Chapman Hall/CRC.
Csiszár I. (1991) Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. The Annal of Statistics 19: 2032–2066
De Angelis D., Young G. (1992) Smoothing the bootstrap. International Statistical Review 60: 45–56
Dik J. J., de GunstM. C. M. (1985) The distribution of general quadratic forms in normal variables. Statistica Neerlandica 39: 14–26
Eckler A. R. (1969) A survey of coverage problems associated with point and area targets. Technometrics 11: 561–589
Fraser D. A. S. (1956) Sufficient statistics with nuisance parameters. The Annal of Mathematical Statistics 27: 838–842
Gupta S. S. (1963) Bibliography on the multivariate normal integrals and related topics. The Annal of Mathematical Statistics 34: 829–838
Harville D. A. (2008) Matrix Algebra from a statistician’s perspective. Springer, New York
Heritier S., Ronchetti E. (1994) Robust Bounded-influence Tests in General Parametric Models. Journal of the American Statistical Association 89: 897–904
Jensen D. R., Solomon H. (1972) A Gaussian approximation to the distribution of a definite quadratic form. Journal of the American Statistical Association 67(340): 898–902
Johnson N. L., Kotz S. (1968) Tables of distributions of positive definite quadratic forms in central normal variables. Sankhya B—The Indian Journal of Statistics 30: 303–314
Jones M. C., Hjort N. L., Harris I. R., Basu A. (2001) A comparison of related density-based minimum divergence methods. Biometrika, 88: 865–873
Lindsay B. G. (1994) Efficiency versus robustness: the case for minimum Hellinger distance and related methods. The Annals of Statistics 22: 1081–1114
Modarres R., Jernigan R. W. (1992) Testing the equality of correlation matrices. Communications in Statistics—Theory and Methods 21: 2107–2125
Pardo L. (2006) Statistical inference based on divergence measures. Chapman Hall/CRC, Boca Raton
Rao J. N. K., Scott A. J. (1981) The analysis of categorical data from complex sample surveys: Chi-squared tests for goodness of fit and independence in two-way tables. Journal of the American Statistical Association 76: 221–230
Satterthwaite F. E. (1946) An approximate distribution of estimates of variance components. Biometrics 2: 110–114
Simpson D. G. (1989) Hellinger deviance tests: Efficiency, breakdown points and examples. Journal of the American Statistical Association 84: 107–113
Solomon, H. (1960). Distribution of quadratic forms: tables and applications. Applied Mathematics and Statistics Laboratories, Technical Report 45, Stanford University, Stanford.
Welch W. J. (1987) Rerandomizing the median in matched-pair designs. Biometrika 74: 609–614
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 mutagenesis 6: 189–202
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Basu, A., Mandal, A., Martin, N. et al. Testing statistical hypotheses based on the density power divergence. Ann Inst Stat Math 65, 319–348 (2013). https://doi.org/10.1007/s10463-012-0372-y
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DOI: https://doi.org/10.1007/s10463-012-0372-y