INCIDENTAL VERSUS RANDOM NUISANCE PARAMETERS

• 成果类型: Article
• 署名作者: PFANZAGL, J
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176349392
• 发表日期: 1993
• 页码: 1663-1691
• 关键词: maximum-likelihood-estimation; models

摘要

Let (P(theta,eta):(theta,eta) is-an-element-of THETA x H), with THETA is-an-element-of R and H arbitrary, be a family of mutually absolutely continuous probability measures on a measurable space (X,A). The problem is to estimate theta, based on a sample (x1,...,x(n)) from X1(n)P(theta,etanu). If (eta1,...,eta(n)) are independently distributed according to some unknown prior distribution GAMMA, then the distribution of n1/2(theta(n) - theta) under P(theta,GAMMA)n(P(theta,GAMMA) being the GAMMA-mixture of P(theta,eta) eta is-an-element-of H) cannot be more concentrated asymptotically than a certain normal distribution with mean 0, say N(0,sigma02(theta,GAMMA)). Folklore says that such a bound is also valid if (eta1,...,eta(n)) are just unknown values of the nuisance parameter: In this case, the distribution cannot be more concentrated asymptotically than N(0,sigma02(theta,E(n1,...eta(n))(n)), where E(eta1,...eta(n))(n)) is the empirical distribution of (eta,...,eta(n)). The purpose of the present paper is to discuss to which extent this conjecture is true. The results are summarized at the end of Sections 1 and 3.