Penalized maximum likelihood and semiparametric second-order efficiency

A. S. Dalalyan, G. K. Golubev, A. B. Tsybakov

The Annals of Statistics · 2006 · Volume 34 · Issue 1

Abstract

We consider the problem of estimation of a shift parameter of an unknown symmetric function in Gaussian white noise. We introduce a notion of semiparametric second-order efficiency and propose estimators that are semiparametrically efficient and second-order efficient in our model. These estimators are of a penalized maximum likelihood type with an appropriately chosen penalty. We argue that second-order efficiency is crucial in semiparametric problems since only the second-order terms in asymptotic expansion for the risk account for the behavior of the “nonparametric component” of a semiparametric procedure, and they are not dramatically smaller than the first-order terms.

Keywords

62G05, 62G20, estimating a shift of a nonparametric function, exact minimax asymptotics, penalized maximum likelihood, second-order efficiency, Semiparametric estimation

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