Groupe d’études et de recherche en analyse des décisions

# Asymptotically Minimax Estimation of Order-Constrained Parameters and Eigenfunctions of the Laplacian on the Ball

## A Korányi et Brenda MacGibbon

Inspired by previous results on asymptotic minimax estimation for a ball of increasing radius in Rn, we study the analogous problem for domains of importance in order-restricted inference. In particular, we study domains that are formed by the intersection of a ball and a fundamental chamber of a finite reflection group in Rn. We show (1) how to obtain the principal eigenfunction of such a domain and asymptotically, the related least favourable distribution for the associated minimax problem, (2) that the order and positivity constraints in the usual statistical problems generate such chambers and (3) and in an analogous way to the work of Bickel (1981) in the one dimensional case, how to find the asymptotic minimax risk and the second order asymptotic minimax estimate for such a domain.

, 17 pages

Ce cahier a été révisé en mai 2001