Asymptotics for likelihood ratio tests under loss of identifiability
成果类型：
Article
署名作者：
Liu, X; Shao, YZ
署名单位：
Rockefeller University; Columbia University
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
发表日期：
2003
页码：
807-832
关键词：
摘要：
This paper describes the large sample properties of the likelihood ratio test statistic (LRTS) when the parameters characterizing the true null distribution are not unique. It is well known that the classical asymptotic theory for the likelihood ratio test does not apply to such problems and the LRTS may not have the typical chi-squared type limiting distribution. This paper establishes a general quadratic approximation of the log-likelihood ratio function in a Hellinger neighborhood of the true density which is valid with or without loss of identifiability of the true distribution. Under suitable conditions, the asymptotic null distribution of the LRTS under loss of identifiability can be obtained by maximizing the quadratic form. These results extend the work of Chernoff and Le Cam. In particular, applications to testing the number of mixture components in finite mixture models are discussed.