In nonparametric statistics an optimality criterion for estimation procedures is provided by the minimax rate of convergence. However this classical point of view is subject to controversy as it requires to look for the worst behaviour reached by an estimation procedure in a given space. The purpose of this paper is to show that this is not justified as the minimax risk often coincides with a g...

In this paper, the problem of carrier frequency offset estimation for frequency-selective channels in a data-aided context is considered. The channel is assumed to be contaminated with an unknown noise and the Cramér-Rao bound (CRB) is used as an estimation accuracy criterion. The minimax approach consists in minimizing the worst-case (maximum) CRB of the frequency offset. Under certain assumpt...

We investigate the possibility of finding multi-purpose nonparametric function estimators that are minimax rate optimal for estimating the entire function and are also minimax rate optimal for point estimation. We show that it is impossible to attain the global optimal rate of convergence as well as the local optimal rate at every point. An inequality concerning estimation in a two parameter st...

A central result in statistical theory is Pinsker’s theorem, which characterizes the minimax rate in the normal means model of nonparametric estimation. In this paper, we present an extension to Pinsker’s theorem where estimation is carried out under storage or communication constraints. In particular, we place limits on the number of bits used to encode an estimator, and analyze the excess ris...

When a parameter of interest is de…ned to be a nondi¤erentiable transform of a regular parameter, the parameter does not have an in‡uence function, rendering the existing theory of semiparametric e¢ cient estimation inapplicable. However, when the nondi¤erentiable transform is a known composite map of a continuous piecewise linear map with a single kink point and a translation-scale equivariant...

This paper describes a minimax state estimation approach for linear differential-algebraic equations (DAEs) with uncertain parameters. The approach addresses continuous-time DAEs with non-stationary rectangular matrices and uncertain bounded deterministic input. An observation’s noise is supposed to be random with zero mean and unknown bounded correlation function. Main result is a Generalized ...

We establish minimax lower bounds and maximum likelihood convergence rates of parameter estimation for mean-covariance multivariate Gaussian mixtures, shape-rate Gamma mixtures, and some variants of finite mixture models, including the setting where the number of mixing components is bounded but unknown. These models belong to what we call ”weakly identifiable” classes, which exhibit specific i...

We apply recent results on the minimax risk in density estimation to the related problem of pattern classiication. The notion of loss we seek to minimize is an information theoretic measure of how well we can predict the classiication of future examples, given the classiication of previously seen examples. We give an asymptotic characterization of the minimax risk in terms of the metric entropy...