ASYMPTOTICALLY MINIMAX ESTIMATION OF A CONSTRAINED POISSON VECTOR VIA POLYDISK TRANSFORMS

Suppose that the mean sigma of a vector of independent Poisson variates (X1, . . ., X(p)) lies in a subset mT of R(p), where T is a bounded domain and m > 0. We study the asymptotic behavior of the minimax risk rho(mT) and the construction of asymptotic minimax estimators as m arrow pointing up and to the right infinity, using the information normalized loss [GRAPHICS] With the use of the polydisc transform, a many-to-one mapping from R2p to R+p, we show that rho(mT) = p-m-1 lambda(OMEGA)+o(m-1), where lambda(OMEGA) is the principal eigenvalue for the Laplace operator on the pre-image OMEGA of T under this transform. The proofs exploit the connection between p-dimensional Poisson estimation in T and 2p-dimensional Gaussian estimation in OMEGA.