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June 2003 Asymptotics for likelihood ratio tests under loss of identifiability
Xin Liu, Yongzhao Shao
Ann. Statist. 31(3): 807-832 (June 2003). DOI: 10.1214/aos/1056562463


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.


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Xin Liu. Yongzhao Shao. "Asymptotics for likelihood ratio tests under loss of identifiability." Ann. Statist. 31 (3) 807 - 832, June 2003.


Published: June 2003
First available in Project Euclid: 25 June 2003

zbMATH: 1032.62014
MathSciNet: MR1994731
Digital Object Identifier: 10.1214/aos/1056562463

Primary: 62F05
Secondary: 62A10 , 62H30

Keywords: Donsker class , Finite mixture model , Hellinger distance , likelihood ratio test , loss of identifiability

Rights: Copyright © 2003 Institute of Mathematical Statistics

Vol.31 • No. 3 • June 2003
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