ON ESTIMATING MIXING DENSITIES IN DISCRETE EXPONENTIAL FAMILY MODELS

• 成果类型: Article
• 署名作者: ZHANG, CH
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176324629
• 发表日期: 1995
• 页码: 929-945
• 关键词: maximum likelihood estimation; deconvolution problems; mixture likelihoods; Optimal Rates; CONVERGENCE; geometry

摘要

This paper concerns estimating a mixing density function g and its derivatives based on lid observations from f(x) = integral f(x \ theta)g(theta) d theta, where f(x \ theta) is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive kernel estimators, upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence. If f(x \ theta(o)) greater than or equal to epsilon(x+1) For All x, for some positive numbers theta(o) and epsilon, then our estimators achieve the optimal rate of convergence (log n)(-alpha+m) for estimating the mth derivative of g under a Lipschitz condition of order alpha > m. The optimal rate of convergence is almost achieved when (x\)(beta)f(x \ theta(0)) greater than or equal to epsilon(x+1). Estimation of the mixing distribution function is also considered.