Moderate deviations for estimators under exponentially stochastic differentiability conditions

doi:10.1007/s11464-017-0668-6

Front. Math. China

2018, Vol. 13

Issue (1) : 25-40 https://doi.org/10.1007/s11464-017-0668-6

RESEARCH ARTICLE

Moderate deviations for estimators under exponentially stochastic differentiability conditions

Fuqing GAO, Qiaojing LIU(

)

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

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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

Corresponding Author(s): Qiaojing LIU

Issue Date: 12 January 2018

Cite this article:

Fuqing GAO,Qiaojing LIU. Moderate deviations for estimators under exponentially stochastic differentiability conditions[J]. Front. Math. China, 2018, 13(1): 25-40.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0668-6
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I1/25

1	Arcones M A. Moderate deviations for M-estimators. Test, 2002, 11: 465–500 https://doi.org/10.1007/BF02595717
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 https://doi.org/10.1214/13-AIHP545
3	Dembo A, Zeitouni O. Large Deviations Techniques and Applications.New York: Springer, 1998 https://doi.org/10.1007/978-1-4612-5320-4
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 https://doi.org/10.1007/s11464-011-0118-9
5	Gao F Q. Moderate deviations for the maximum likelihood estimator. Statist Probab Lett, 2001, 55: 345–352 https://doi.org/10.1016/S0167-7152(01)00023-2
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 https://doi.org/10.1007/s11425-016-0017-6
7	Gao F Q, Zhao X Q. Delta method in large deviations and moderate deviations for estimators. Ann Statist, 2011, 39: 1211–1240 https://doi.org/10.1214/10-AOS865
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 https://doi.org/10.1016/S0246-0203(01)01081-0
9	Huber P J. Robust estimation of a location parameter. Ann Math Statist, 1964, 35: 73–101 https://doi.org/10.1214/aoms/1177703732
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 https://doi.org/10.1214/aos/1056562465
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 https://doi.org/10.1023/A:1003708829482
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 https://doi.org/10.1016/0167-7152(88)90119-8
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 https://doi.org/10.1080/02331888.2013.822501
16	Otsu T. Moderate deviations of generalized method of moments and empirical likelihood estimators. J Multivariate Anal, 2011, 102: 1203–1216 https://doi.org/10.1016/j.jmva.2011.04.002
17	Pollard D. New ways to prove central limit theorems. Econometric Theory, 1985, 1: 295–314 https://doi.org/10.1017/S0266466600011233
18	Talagrand M. New concentration inequalities in product spaces. Invent Math, 1996, 126: 505–563 https://doi.org/10.1007/s002220050108
19	Van der Vaart A W, Wellner J A. Weak Convergence and Empirical Processes with Applications to Statistics.New York: Springer, 1996 https://doi.org/10.1007/978-1-4757-2545-2
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 https://doi.org/10.1137/1126059
21	Wu L M. Large deviations, moderate deviations and LIL for empirical Processes. Ann Probab, 1994, 22: 17–27 https://doi.org/10.1214/aop/1176988846
22	Wu L M. Moderate deviations of dependent random variables related to the CLT. Ann Probab, 1995, 23: 420–445 https://doi.org/10.1214/aop/1176988393

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