Asymptotically minimax estimation of a constrained Poisson vector via polydisc transforms (Q2368148)

Let \((X_ 1,\dots,X_ p)\) be a vector of independent Poisson variates, having means \(\sigma=(\sigma_ 1,\dots,\sigma_ p)\). It is known that \(\sigma\) lies in a subset \(mT\) of \(R^ p\), where \(T\) is a bounded domain and \(m>0\). Employing the information normalized loss function \(L(d,\sigma)=\sum^ p_{i=1} \sigma^{-1}_ i(d_ i-\sigma_ i)^ 2\), the authors consider the asymptotic behavior of the minimax risk \(\rho(mT)\) and the construction of asymptotically minimax estimators as \(m\to\infty\). With the use of the polydisk transform, a many-to-one mapping from \(R^{2p}\) to \(R^ p_ +\), the authors 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 \(\Omega\).