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As noted by Brown(2000), minimax estimation, especially for the Gaussian sequence model, has been influential in many applications including nonparametric function estimation. However, Poisson estimation has not received the same attention. For example, as early as 1980, Pinsker gave an ingenious proof of the asymptotic equivalence of the linear and nonlinear minimax risk with respect to quadratic loss for the estimation of a Gaussian mean vector constrained to lie in an l2-ellipsoid under specific conditions on the relationship between the volume of the ellipsoid and the dimension of the vector as the latter increases. The purpose of this research is to prove a result analogous to that of Pinsker in the Poisson case. More specifically, it is shown that if a Poisson mean vector is constrained to lie in an l1-ellipsoid under specific, but different, conditions on the relationship between its volume and dimension, then the linear and nonlinear minimax risk using the information-normalized quadratic loss function are asymptotically equivalent. The polydisc transform, a many–to–one mapping of R2n → R+, plays a fundamental role in proving this result as it did in the asymptotic minimax results for Poisson mean estimation obtained by Johnstone and MacGibbon (1993). In view of the work of Brown, Carter, Low and Zhang (2004) which demonstrated the global asymptotic equivalence between a Poisson process with variable intensity and white noise with drift under sharp smoothness conditions on the unknown function, the differences between Poisson mean estimation with respect to quadratic loss and information-normalized quadratic loss will also be discussed.

Minimax estimation over hyperrectangles with implications in the Poisson case