Spokoiny, Vladimir Parametric estimation. Finite sample theory. (English) Zbl 1296.62051 Ann. Stat. 40, No. 6, 2877-2909 (2012). Summary: The paper aims at reconsidering the famous Le Cam LAN theory. The main features of the approach which make it different from the classical one are as follows: (1) the study is nonasymptotic, that is, the sample size is fixed and does not tend to infinity; (2) the parametric assumption is possibly misspecified and the underlying data distribution can lie beyond the given parametric family. These two features enable to bridge the gap between parametric and nonparametric theory and to build a unified framework for statistical estimation. The main results include large deviation bounds for the (quasi) maximum likelihood and the local quadratic bracketing of the log-likelihood process. The latter yields a number of important corollaries for statistical inference: concentration, confidence and risk bounds, expansion of the maximum likelihood estimate, etc. All these corollaries are stated in a nonclassical way admitting a model misspecification and finite samples. However, the classical asymptotic results including the efficiency bounds can be easily derived as corollaries of the obtained nonasymptotic statements. At the same time, the new bracketing device works well in the situations with large or growing parameter dimension in which the classical parametric theory fails. The general results are illustrated for the i.i.d. setup as well as for generalized linear and median estimation. The results apply for any dimension of the parameter space and provide a quantitative lower bound on the sample size yielding the root-\(n\) accuracy. Cited in 1 ReviewCited in 33 Documents MSC: 62F10 Point estimation 62J12 Generalized linear models (logistic models) 62F25 Parametric tolerance and confidence regions 62H12 Estimation in multivariate analysis Keywords:maximum likelihood; local quadratic bracketing; deficiency; concentration × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Andresen, A. and Spokoiny, V. (2012). Wilks theorem for a quasi profile maximum likelihood. Unpublished manuscript. [2] Bednorz, W. (2006). A theorem on majorizing measures. Ann. Probab. 34 1771-1781. · Zbl 1113.60040 · doi:10.1214/009117906000000241 [3] Birgé, L. (2006). Model selection via testing: An alternative to (penalized) maximum likelihood estimators. Ann. Inst. Henri Poincaré Probab. Stat. 42 273-325. · Zbl 1333.62094 · doi:10.1016/j.anihpb.2005.04.004 [4] Birgé, L. and Massart, P. (1993). Rates of convergence for minimum contrast estimators. Probab. 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