MINIMUM CONTRAST ESTIMATORS ON SIEVES - EXPONENTIAL BOUNDS AND RATES OF CONVERGENCE

This paper, which we dedicate to Lucien Le Cam for his seventieth birt hday, has been written in the spirit of his pioneering works on the re lationships between the metric structure of the parameter space and th e rate of convergence of optimal estimators. It has been written in hi s honour as a contribution to his theory. It contains further developm ents of the theory of minimum contrast estimators elaborated in a prev ious paper. We focus on minimum contrast estimators on sieves. By a 's ieve' we mean some approximating space of the set of parameters. The s ieves which are commonly used in practice are D-dimensional linear spa ces 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 his tograms built from any partition of D subintervals of [0, 1] as a typi cal example). We introduce some metric assumptions which are closely r elated to the notion of finite-dimensional metric space in the sense o f Le Cam. These assumptions are satisfied by the examples of practical interest and allow us to compute sharp rates of convergence for minim um contrast estimators.