PROBABILITY-INEQUALITIES FOR LIKELIHOOD RATIOS AND CONVERGENCE-RATES OF SIEVE MLES

• 成果类型: Article
• 署名作者: WONG, WH; SHEN, XT
• 署名单位: University System of Ohio; Ohio State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176324524
• 发表日期: 1995
• 页码: 339-362
• 关键词: estimators; SPACES

摘要

Let Y-1,...,Y-n be independent identically distributed with density p(o) and let F be a space of densities. We show that the supremum of the likelihood ratios Pi(i=1)(n)p(Y-i)/p(o)(Y-i), where the supremum is over p is an element of F with \\p(1/2) - p(o)(1/2)\\(2) greater than or equal to epsilon, is exponentially small with probability exponentially dose to 1. The exponent is proportional to n epsilon(2). The only condition required for this to hold is that epsilon exceeds a value determined by the bracketing Hellinger entropy of F. A similar inequality also holds if we replace F by F-n and p(o) by q(n), where q(n) is an approximation to p(o) in a suitable sense. These results are applied to establish rates of convergence of sieve MLEs, Furthermore, weak conditions are given under which the ''optimal'' rate epsilon(n) defined by H(epsilon(n),F) = n epsilon(n)(2), where H(.,F) is the Hellinger entropy of F, is nearly achievable by sieve estimators.