The Annals of Statistics
- Ann. Statist.
- Volume 31, Number 3 (2003), 807-832.
Asymptotics for likelihood ratio tests under loss of identifiability
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.
Ann. Statist., Volume 31, Number 3 (2003), 807-832.
First available in Project Euclid: 25 June 2003
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62F05: Asymptotic properties of tests
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62A10
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