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Biometrika Advance Access published online on April 27, 2009
Biometrika, doi:10.1093/biomet/asp011
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© 2009 Biometrika Trust

Article

Non-finite Fisher information and homogeneity: an EM approach

P. Li

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada pengfei{at}stat.ualberta.ca

J. Chen

Department of Statistics, University of British Columbia, Vancouver, V6T 1Z2, Canada jhchen{at}stat.ubc.ca

P. Marriott

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.
  Abstract

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


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