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Minimal penalties for Gaussian model selection. (English) Zbl 1112.62082
Summary: This paper is mainly devoted to a precise analysis of what kind of penalties should be used in order to perform model selection via the minimization of a penalized least-squares type criterion within some general Gaussian framework including the classical ones. As compared to our previous paper on this topic [J. Eur. Math. Soc. (JEMS) 3, No. 3, 203–268 (2001; Zbl 1037.62001)], more elaborate forms of the penalties are given which are shown to be, in some sense, optimal. We indeed provide more precise upper bounds for the risk of the penalized estimators and lower bounds for the penalty terms, showing that the use of smaller penalties may lead to disastrous results. These lower bounds may also be used to design a practical strategy that allows to estimate the penalty from the data when the amount of noise is unknown. We provide an illustration of the method for the problem of estimating a piecewise constant signal in Gaussian noise when neither the number, nor the location of the change points are known.

62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
46N30 Applications of functional analysis in probability theory and statistics
Full Text: DOI
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