The Annals of Statistics

Asymptotics for likelihood ratio tests under loss of identifiability

Xin Liu and Yongzhao Shao

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 31, Number 3 (2003), 807-832.

Dates
First available in Project Euclid: 25 June 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1056562463

Digital Object Identifier
doi:10.1214/aos/1056562463

Mathematical Reviews number (MathSciNet)
MR1994731

Zentralblatt MATH identifier
1032.62014

Subjects
Primary: 62F05: Asymptotic properties of tests
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62A10

Keywords
Donsker class finite mixture model Hellinger distance likelihood ratio test loss of identifiability

Citation

Liu, Xin; Shao, Yongzhao. Asymptotics for likelihood ratio tests under loss of identifiability. Ann. Statist. 31 (2003), no. 3, 807--832. doi:10.1214/aos/1056562463. https://projecteuclid.org/euclid.aos/1056562463


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  • NEW YORK, NEW YORK 10021 E-MAIL: liuxin@linkage.rockefeller.edu DEPARTMENT OF STATISTICS COLUMBIA UNIVERSITY 2990 BROADWAY
  • NEW YORK, NEW YORK 10027 E-MAIL: yshao@stat.columbia.edu