Minimax estimation of a constrained Poisson vector. (English) Zbl 0761.62006

Johnstone, Iain M.; MacGibbon, K. Brenda

Ann. Stat. 20, No. 2, 807-831 (1992).

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

Keywords:

Bayes risk lower bounds; second-order minimax; loss estimation; Fisher information; minimax linear risk; hardest rectangular subproblem; isoperimetric inequalities; mean vector; vector of Poisson variates; bounded domain; a priori information; precision of estimation; minimax risk; approximations; rectangulary convex; linear estimators; principal eigenvalue; Laplace operator; polydisc transform; simplex; hyperrectangle; Gaussian vector