Parametric estimation. Finite sample theory.

*(English)*Zbl 1296.62051Summary: 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.

##### MSC:

62F10 | Point estimation |

62J12 | Generalized linear models (logistic models) |

62F25 | Parametric tolerance and confidence regions |

62H12 | Estimation in multivariate analysis |

**OpenURL**

##### 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 Unpublished manuscript.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 |

[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 |

[4] | Birgé, L. and Massart, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113-150. · Zbl 0805.62037 |

[5] | Birgé, L. and Massart, P. (1998). Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4 329-375. · Zbl 0954.62033 |

[6] | Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Ann. Probab. 31 1583-1614. · Zbl 1051.60020 |

[7] | Ibragimov, I. A. and Khas’minskiĭ, R. Z. (1981). Statistical Estimation : Asymptotic Theory. Applications of Mathematics 16 . Springer-Verlag, New York-Berlin. Translated from the Russian by Samuel Kotz. · Zbl 0467.62026 |

[8] | Le Cam, L. (1960). Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses. Univ. California Publ. Statist. 3 37-98. · Zbl 0104.12701 |

[9] | Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics : Some Basic Concepts , 2nd ed. Springer, New York. · Zbl 0952.62002 |

[10] | McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models , 2nd ed. Chapman & Hall, London. · Zbl 0744.62098 |

[11] | Spokoiny, V. (2012a). Roughness penalty, Wilks phenomenon, and Bernstein-von Mises theorem. Unpublished manuscript. Available at [stat.ME]. |

[12] | Spokoiny, V. (2012b). Supplement to “Parametric estimation. Finite sample theory.” . · Zbl 1296.62051 |

[13] | Spokoiny, V., Wang, W. and Härdle, W. (2012). Local quantile regression. Unpublished manuscript. Available at [math.ST]. · Zbl 1279.62098 |

[14] | Talagrand, M. (1996). Majorizing measures: The generic chaining. Ann. Probab. 24 1049-1103. · Zbl 0867.60017 |

[15] | Talagrand, M. (2001). Majorizing measures without measures. Ann. Probab. 29 411-417. · Zbl 1019.60033 |

[16] | Talagrand, M. (2005). The Generic Chaining : Upper and Lower Bounds of Stochastic Processes . Springer, Berlin. · Zbl 1075.60001 |

[17] | van de Geer, S. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 21 14-44. · Zbl 0779.62033 |

[18] | van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes : With Applications to Statistics . Springer, New York. · Zbl 0862.60002 |

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