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Asymptotically minimax estimation of a constrained Poisson vector via polydisc transforms. (English) Zbl 0779.62022
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 $$\Omega$$.
##### MSC:
 62F12 Asymptotic properties of parametric estimators 62F10 Point estimation 62C20 Minimax procedures in statistical decision theory 62H12 Estimation in multivariate analysis
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