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- W2060676875 abstract "Suppose that the mean $tau$ of a vector of Poisson variates is known to lie in a bounded domain $T$ in $lbrack 0,infty)^p$. How much does this a priori information increase precision of estimation of $tau$? Using error measure $sum_i(hattau_i - tau_i)^2/tau_i$ and minimax risk $rho(T)$, we give analytical and numerical results for small intervals when $p = 1$. Usually, however, approximations are needed. If $T$ is rectangularly convex at 0, there exist linear estimators with risk at most 1.26$rho(T)$. For general $T, rho(T) geq p^2/(p + lambda(Omega))$, where $lambda(Omega)$ is the principal eigenvalue of the Laplace operator on the polydisc transform $Omega = Omega(T)$, a domain in twice-$p$-dimensional space. The bound is asymptotically sharp: $rho(mT) = p - lambda(Omega)/m + o(m^{-1})$. Explicit forms are given for $T$ a simplex or a hyperrectangle. We explore the curious parallel of the results for $T$ with those for a Gaussian vector of double the dimension lying in $Omega$." @default.
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- W2060676875 date "1992-06-01" @default.
- W2060676875 modified "2023-10-02" @default.
- W2060676875 title "Minimax Estimation of a Constrained Poisson Vector" @default.
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- W2060676875 doi "https://doi.org/10.1214/aos/1176348658" @default.
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