Abstract
A number of authors have been concerned with constructing large deviation approximations to densities and probabilities associated with minimum contrast estimators (equivalently, M-estimators) using a tilting approach due to Field. These developments are an interesting and important extension of saddlepoint-type methodology. However, in the case of a multivariate parameter, the theoretical picture has remained incomplete in certain respects, as explained below. In this paper we present results which provide rigorous justification of the tilting argument, using conditions which it is feasible to check. These results include a new formulation and proof of Skovgaard's theorem for the intensity of minimum contrast estimators, but under conditions which are typically straightforward to check in practice. Our most detailed application is to multivariate location-scatter models.
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References
Bahadur, R. R. (1961). On the asymptotic efficiency of tests and estimates, Sankhyā, 22, 229–252.
Bhattacharya, R. N. and Rao, R. R. (1976). Normal Approximation and Asymptotic Expansion, Wiley, New York.
Chmielewski, M. A. (1981). Elliptically symmetric distributions: a review and bibliography, Internat. Statist. Rev., 49, 67–74.
Clarke, B. R. (1991). The selection functional, Probab. Math. Statist., 11, 149–156.
Copas, J. B. (1975). On the unimodality of the likelihood function for the Cauchy distribution, Biometrika, 62, 701–704.
Daniels, H. E. (1983). Saddlepoint approximations for estimating equations, Biometrika, 70, 89–96.
Field, C. A. (1982). Small sample asymptotic expansions for multivariate M-estimates, Ann. Statist, 10, 672–689.
Field, C. A. and Ronchetti, E. (1990). Small sample asymptotics, IMS Lecture Notes-Monograph Ser., 13.
Ibragimov, I. A. and Has'minskii, R. Z. (1981). Statistical Estimation: Asymptotic Theory, Springer, New York.
Jensen, J. L. (1995). Saddlepoint Approximations, Oxford University Press, Oxford.
Kester, A. D. M. and Kallenburg, W. C. M. (1986). Large deviations of estimators, Ann. Statist., 14, 648–664.
Khatri, C. G. (1988). Some inferential problems connected with elliptic distributions, J. Multivariate Anal., 27, 319–333.
Mirsky, L. (1955). An Introduction to Linear Algebra, Clarendon Press, Oxford.
Mitchell, A. F. S. (1988). Statistical manifolds of univariate clliptic distributions, Internat. Statist. Rev., 56, 1–16.
Mitchell, A. F. S. (1989). The information matrix, skewness tensor and α-connections for the general multivariate elliptic distribution, Ann. Inst. Statist. Math., 41, 289–304.
Pazman, A. (1986). On the uniqueness of the M.L. estimates in curved exponential families, Kybernetika, 22, 124–132.
Pollard, D. (1984). Convergence of Stochastic Processes, Springer, New York.
Resek, R. W. (1976). Estimation of the parameters of a general student's t distribution, Comm. Statist. Theory Methods., A5(7), 635–645.
Sieders, A. and Dzhaparidze, K. (1987). A large deviation result for parameter estimators and its application to nonlinear regression analysis, Ann. Statist., 15, 1031–1049.
Skovgaard, I. M. (1990). On the density of minimum contrast estimators, Ann. Statist, 18, 779–789.
Varadhan, S. R. S. (1984). Large Deviations and Applications, SIAM, Philadelphia.
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Jensen, J.L., Wood, A.T.A. Large Deviation and Other Results for Minimum Contrast Estimators. Annals of the Institute of Statistical Mathematics 50, 673–695 (1998). https://doi.org/10.1023/A:1003708829482
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DOI: https://doi.org/10.1023/A:1003708829482