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Minimax estimation of a constrained Poisson vector. (English) Zbl 0761.62006
Summary: Suppose that the mean $$\tau$$ of a vector of Poisson variates is known to lie in a bounded domain $$T$$ in $$[0,\infty)^ p$$. How much does this a priori information increase precision of estimation of $$\tau$$? Using error measure $$\Sigma_ i(\hat\tau_ i-\tau_ i)^ 2/\tau_ i$$ 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 “rectangulary 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 $$\Omega$$.

##### MSC:
 62C20 Minimax procedures in statistical decision theory 62F10 Point estimation
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