PENALIZED MAXIMUM LIKELIHOOD AND SEMIPARAMETRIC SECOND-ORDER EFFICIENCY

By Arnak S. Dalalyan, G. Golubev, Alexandre B. Tsybakov. 2006

52 citations recorded by the source corpus.

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-orderefficient in our model. These estimators are of a penalized maximum likelihood type with an appropriately chosen penalty. We argue that secondorder efficiency is crucial in semiparametric problems sinceonly 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. 1. Introduction. Semiparametric statistical models are the ones containing a finite-dimensional parameter of interest θ and an infinite-dimensional nuisance parameter f which is a member of some large functional class. The goal is then to estimate θ efficiently without knowingf. A comprehensive account of the theory of semiparametric estimation is given in the book of Bickel, Klaassen, Ritov and Wellner [3]. In particular, it is shown that for many semiparametric models there exist estimators attaining the same asymptotic performance as efficient parametric estimators c for the problem where f is completely specified. In other words, for such semiparametric models there is no loss of efficiency as compared tothe corresponding parametric models with known f. These semiparametric models are usually called adaptive, but we prefer here to call them S-adaptive, or semiparametrically adaptive, in order to avoid confusion with nonparametric adaptivity to unknown smoothness of f. Estimators attaining parametric

Subfields: Computer Networks and Communications, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1ff2251dc31ef23566e7cb3963ff1527356dc370

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Citations in the corpus

An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation (2009, background): “The rate of convergence of the remainder term is of order ‖pn − p‖(2)∞, which typically depends on unknown properties of p, and here adaptive pn is desirable, see also the recent article [5] on higher order efficiency of semiparametric estimators.”
Asymptotically Optimal Regularization in Smooth Parametric Models (2009, methodology): “Higher-order asymptotics has been used in many statistical settings, for example, in the context of estimating means [6, 24] or estimating shift parameters in semiparametric models [15].”
Noname manuscript No. (will be inserted by the editor) A semiparametric Bernstein- von Mises theorem for Gaussian process priors (unknown): “Model (9) can be seen as an idealized version in continuous time of a signal processing problem where one would observe discretely sampled and noisy observations of a symmetric signal, see for instance [15] and references therein for the parametric case and [7] for frequentist estimators in the semi”
Semi-parametric estimation of shifts (2007, background): “This Bayesian point of view is tackled in [3].”
Adaptation on the space of finite signed measures (2008, result): “The results of the present article also have applications to semiparametric higher order efficiency problems, similar to those studied in Golubev and Levit (1996) and Dalalyan, Golubev and Tsybakov (2006).”