Moderate deviations for estimators under exponentially stochastic differentiability conditions

Abstract

We introduce two exponentially stochastic differentiability conditions to study moderate deviations for M-estimators. Under a generalized exponentially stochastic differentiability condition, a moderate deviation principle is established. Some sufficient conditions of the exponentially stochastic differentiability and examples are also given.

Keywords

M-estimator / exponentially stochastic differentiability / moderate deviations

References

[1] Arcones M A. Moderate deviations for M-estimators. Test, 2002, 11: 465–500

[2] Bitseki Penda S V, Djellout H. Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models. Ann Inst Henri Poincaré Probab Stat, 2014, 50: 806–844

[3] Dembo A, Zeitouni O. Large Deviations Techniques and Applications.New York: Springer, 1998

[4] Fan J, Gao F Q. Deviation inequalities and moderate deviations for estimators of parameters in TAR models. Front Math China, 2011, 6: 1067–1083

[5] Gao F Q. Moderate deviations for the maximum likelihood estimator. Statist Probab Lett, 2001, 55: 345–352

[6] Gao F Q, Jiang H. Deviation inequalities for quadratic Wiener functionals and moderate deviations for parameter estimator. Sci China Math, 2017, 60: 1181–1196

[7] Gao F Q, Zhao X Q. Delta method in large deviations and moderate deviations for estimators. Ann Statist, 2011, 39: 1211–1240

[8] Giné E, Guillou A. On consistency of kernel density estimators for randomly censored data: Rate holding uniformly over adaptive intervals. Ann Inst Henri Poincaré Probab Stat, 2001, 37: 503–522

[9] Huber P J. Robust estimation of a location parameter. Ann Math Statist, 1964, 35: 73–101

[10] Huber P J. The behavior of maximum likelihood estimates under nonstandard conditions. In: Fifth Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California, 1967

[11] Inglot T, Kallenberg W C M. Moderate deviations of minimum contrast estimators under contamination. Ann Statist, 2003, 31: 852–879

[12] Jensen J L, Wood A T A. Large deviation and other results for minimum contrast estimators. Ann Inst Statist Math, 1998, 50: 673–695

[13] Jureckovoá J, Kallenberg W C M, Veraverbeke N. Moderate and Cramér-type large deviation theorem for M-estimators. Statist Probab Lett, 1988, 6: 191–199

[14] Ledoux M. Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann Inst Henri Poincaré Probab Statist, 1992, 28: 267–280

[15] Miao Y, Wang Y. Moderate deviation principle for maximum likelihood estimator. Statistics, 2014, 48: 766–777

[16] Otsu T. Moderate deviations of generalized method of moments and empirical likelihood estimators. J Multivariate Anal, 2011, 102: 1203–1216

[17] Pollard D. New ways to prove central limit theorems. Econometric Theory, 1985, 1: 295–314

[18] Talagrand M. New concentration inequalities in product spaces. Invent Math, 1996, 126: 505–563

[19] Van der Vaart A W, Wellner J A. Weak Convergence and Empirical Processes with Applications to Statistics.New York: Springer, 1996

[20] Vapnik V N, Cervonenkis A Y. On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab Appl, 1971, 26: 532–553

[21] Wu L M. Large deviations, moderate deviations and LIL for empirical Processes. Ann Probab, 1994, 22: 17–27

[22] Wu L M. Moderate deviations of dependent random variables related to the CLT. Ann Probab, 1995, 23: 420–445

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature