Minimum Contrast Estimators on Sieves: Exponential Bounds and Rates of Convergence

Abstract

This paper, which we dedicate to Lucien Le Cam for his seventieth birthday, has been written in the spirit of his pioneering works on the relationships between the metric structure of the parameter space and the rate of convergence of optimal estimators. It has been written in his honour as a contribution to his theory. It contains further developments of the theory of minimum contrast estimators elaborated in a previous paper. We focus on minimum contrast estimators on sieves. By a `sieve' we mean some approximating space of the set of parameters. The sieves which are commonly used in practice are D-dimensional linear spaces generated by some basis: piecewise polynomials, wavelets, Fourier, etc. It was recently pointed out that nonlinear sieves should also be considered since they provide better spatial adaptation (think of histograms built from any partition of D subintervals of [0,1] as a typical example). We introduce some metric assumptions which are closely related to the notion of finite-dimensional metric space in the sense of Le Cam. These assumptions are satisfied by the examples of practical interest and allow us to compute sharp rates of convergence for minimum contrast estimators.