Approximation dans les espaces métriques et théorie de l'estimation

Abstract

We investigate the relations between the speed of estimation and the metric structure of the parameter space Θ, especially in the case when its metric dimension is infinite. Given some distance d on Θ (generally Hellinger distance in the case of n i.i.d. variables), we consider the minimax risk for n observations: {Mathematical expression}, Tn being any estimate of θ. We shall look for functions r such that for positive constants C1(q) and C2(q) C1rq(n)≦Rn(q)≦C2rq(n). r(n) is the speed of estimation and we shall show under fairly general conditions (including i.i.d. variables and regular cases of Markov chains and stationnary gaussian processes) that r(n) is determined, up to multiplicative constants, by the metric structure of Θ. We shall also give a construction for some sort of "universal" estimates the risk of which is bounded by C2rq(n) in all cases where the preceding theory applies. © 1983 Springer-Verlag.