Biometrika Advance Access originally published online on April 27, 2009
Biometrika 2009 96(2):411-426; doi:10.1093/biomet/asp011
Article |
Non-finite Fisher information and homogeneity: an EM approach
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada pengfei{at}stat.ualberta.ca
Department of Statistics, University of British Columbia, Vancouver, V6T 1Z2, Canada jhchen{at}stat.ubc.ca
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, N2L 3G1, Canada pmarriot{at}math.uwaterloo.ca
Received for publication 1 November 2007. Revision received 1 August 2008.
Even simple examples of finite mixture models can fail to fulfil the regularity conditions that are routinely assumed in standard parametric inference problems. Many methods have been investigated for testing for homogeneity in finite mixture models, for example, but all rely on regularity conditions including the finiteness of the Fisher information and the space of the mixing parameter being a compact subset of some Euclidean space. Very simple examples where such assumptions fail include mixtures of two geometric distributions and two exponential distributions, and, more generally, mixture models in scale distribution families. To overcome these difficulties, we propose and study an EM-test statistic, which has a simple limiting distribution for examples in this paper. Simulations show that the EM-test has accurate Type I errors and is more efficient than existing methods when they are applicable. A real example is included.
Key Words: Chi-squared limiting distribution • Compactness • Exponential mixture • Finite mixture model • Homogeneity • Likelihood ratio test • Score test
References
-
Anaya-Izquierdo K. A., Marriott P. Local mixture models of exponential families. Bernoulli (2007a) 13:623–40.[CrossRef][Web of Science]
Anaya-Izquierdo K. A., Marriott P. Local mixtures of the exponential distribution. Ann. Inst. Statist. Math. (2007b) 59:111–34.[CrossRef]
Bickel P., Chernoff H. Asymptotic distribution of the likelihood ratio statistic in a prototypical non regular problem. In Statistics and Probability: A Raghu Raj Bahadur Festschrift. Ghosh J. K., Mitra S. K., Parthasarathy K. R., Prakasa Rao B. L. S., eds. (1993) New Delhi: Wiley Eastern. 83–96.
Charnigo R., Sun J. Testing homogeneity in a mixture distribution via the L2-distance between competing models. J. Am. Statist. Assoc. (2004) 99:488–98.[CrossRef][Web of Science]
Chen H., Chen J. The likelihood ratio test for homogeneity in the finite mixture models. Can. J. Statist. (2001) 29:201–15.[CrossRef]
Chen H., Chen J., Kalbfleisch J. D. A modified likelihood ratio test for homogeneity in finite mixture models. J. R. Statist. Soc. (2001) B 63:19–29.[CrossRef]
Chen H., Chen J., Kalbfleisch J. D. Testing for a finite mixture model with two components. J. R. Statist. Soc. (2004) B 66:95–115.[CrossRef]
Chen J. Penalized likelihood ratio test for finite mixture models with multinomial observations. Can. J. Statist. (1998) 26:583–99.[CrossRef]
Chen J., Cheng P. The limit distribution of the restricted likelihood ratio statistic for finite mixture models. Northeast. Math. J. (1995) 11:365–74.
Dacunha-Castelle D., Gassiat E. Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes. Ann. Statist. (1999) 27:1178–209.[CrossRef]
Davies R. B. Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika (1977) 64:247–54.
Dempster A. P., Laird N. M., Rubin D. B. Maximum likelihood from incomplete data via em algorithm (with Discussion). J. R. Statist. Soc. (1977) B 39:1–38.
Giné E., Götze F., Mason D. M. When is the Student t-statistic asymptotically standard normal. Ann. Prob. (1997) 25:1514–31.[CrossRef]
Hall P. The Bootstrap and Edgeworth Expansion (1992) New York: Springer.
Hartigan J. A. A failure of likelihood asymptotics for normal mixtures. In: Proc. Berkeley Conf. in Honor of J. Neyman and Kiefer—LeCam L., Olshen R. A., eds. (1985) 2. Belmont, CA: Wadsworth. 807–10.
Lemdani M., Pons O. Tests for genetic linkage and homogeneity. Biometrics (1995) 51:1033–41.[CrossRef][Web of Science][Medline]
Liang K. Y., Rathouz P. J. Hypothesis testing under mixture models: application to genetic linkage analysis. Biometrics (1999) 55:65–74.[CrossRef][Web of Science][Medline]
Liu X., Shao Y. Z. Asymptotics for likelihood ratio tests under loss of identifiability. Ann. Statist. (2003) 31:807–32.[CrossRef]
Liu X., Shao Y. Z. Asymptotics for the likelihood ratio test in a two-component normal mixture model. J. Statist. Plan. Infer. (2004) 123:61–81.[CrossRef]
Marriott P. Extending local mixture models. Ann. Inst. Statist. Math. (2007) 59:95–110.[CrossRef]
McLachlan G. J., Peel D. Finite Mixture Models (2000) New York: Wiley.
Proschan F. Theoretical explanation of observed decreasing failure rate. Technometrics (1963) 5:375–83.[CrossRef][Web of Science]
Serfling R. J. Approximation Theorem of Mathematical Statistics (1980) New York: Wiley.
Tibshirani R. J. Regression shrinkage and selection via the lasso. J. R. Statist. Soc. (1996) B 58:267–88.
Titterington D. M., Smith A. F. M., Makov U. E. Statistical Analysis of Finite Mixture Distributions (1985) New York: Wiley.
Wald A. Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. (1949) 20:595–601.[CrossRef]
Wilks S. S. The large sample distribution of the likelihood ratio for testing composition hypotheses. Ann. Math. Statist. (1938) 9:60–2.[CrossRef]
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||