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Zbl 0779.62022
Johnstone, Iain M.; MacGibbon, K.Brenda
Asymptotically minimax estimation of a constrained Poisson vector via polydisc transforms.
(English)
[J] Ann. Inst. Henri PoincarĂ©, Probab. Stat. 29, No.2, 289-319 (1993). ISSN 0246-0203

Let $(X\sb 1,\dots,X\sb p)$ be a vector of independent Poisson variates, having means $\sigma=(\sigma\sb 1,\dots,\sigma\sb p)$. It is known that $\sigma$ lies in a subset $mT$ of $R\sp p$, where $T$ is a bounded domain and $m>0$. Employing the information normalized loss function $L(d,\sigma)=\sum\sp p\sb{i=1} \sigma\sp{-1}\sb i(d\sb i-\sigma\sb i)\sp 2$, the authors consider the asymptotic behavior of the minimax risk $\rho(mT)$ and the construction of asymptotically minimax estimators as $m\to\infty$.\par With the use of the polydisk transform, a many-to-one mapping from $R\sp{2p}$ to $R\sp p\sb +$, the authors show that $\rho(mT)=p-m\sp{- 1}\lambda(\Omega)+o(m\sp{-1})$ 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 $\Omega$.
[J.Melamed (Los Angeles)]
MSC 2000:
*62F12 Asymptotic properties of parametric estimators
62F10 Point estimation
62C20 Statistical minimax procedures
62H12 Multivariate estimation

Keywords: vector of independent Poisson variates; information normalized loss function; minimax risk; asymptotically minimax estimators; polydisk transform; principal eigenvalue; Laplace operator; $p$-dimensional Poisson estimation; $2p$-dimensional Gaussian estimation

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