Minimax estimation over hyperrectangles with implications in the Poisson case

Abstract

The purpose of this research is to extend the results of Johnstone and MacGibbon [18, 19] to study the asymptotic behavior of the ratio of linear minimax risk to nonlinear minimax risk for the estimation of a Poisson mean that lies in a rescaled compact l¹-ellipsoid using the information-normalized quadratic loss function. This would be an analogue of Pinsker’s [24] result for l²-ellipsoids with quadratic loss in the Gaussian case. In view of the work of Brown et al. [7] which demonstrated the asymptotic equivalence of the problems of estimation under the same loss function of the intensity of a non-homogeneous Poisson process and estimation in the Gaussian white noise model with drift, some results concerning Poisson mean estimation with respect to quadratic loss are also included.