MINIMAX ESTIMATION OF A CONSTRAINED POISSON VECTOR

• 成果类型: Article
• 署名作者: JOHNSTONE, IM; MACGIBBON, KB
• 署名单位: University of Quebec; University of Quebec Montreal
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176348658
• 发表日期: 1992
• 页码: 807-831
• 关键词: parameter space; RISK

摘要

Suppose that the mean-tau of a vector of Poisson variates is known to lie in a bounded domain T in [0, infinity)p. How much does this a priori information increase precision of estimation of tau? Using error measure SIGMA(i)(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) greater-than-or-equal-to p2/(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.