Annals of the Institute of Statistical Mathematics

, Volume 54, Issue 3, pp 585–594 | Cite as

Likelihood Ratio Statistic for Exponential Mixtures

  • Gabriela Ciuperca
Article

Abstract

Let f0(x) be the exponential density and fγ(x) the translation model. Let (Xi)i=1,n be i.i.d. random variables, with density g. We test that g is f0 against g is a simple mixture, using the LRT statistic. We prove that the LRT diverges to infinity with probability 1/2 and it is equal to 0 with probability 1/2. Therefore, the classical likelihood limiting theory does not hold.

Mixture models likelihood test exponential distribution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bickel, P. and Chernoff, H. (1993). Asymptotic Distribution of the Likelihood Ratio Statistic in a Prototypical Non Regular Problem, Statistics and Probability: A Raghu Raj Bahabur Festschrift, 83–96, Wiley, New York.Google Scholar
  2. Ciuperca, G. (1999). Sur le test de maximum de vraisemblance pour le mélange de populations, Comptes-Rendus de l'Académie des Sciences, Serie I, 328(4), 351–356.Google Scholar
  3. Dacunha-Castelle, D. and Duflo, M. (1990). Probabilités et Statistiques, Tome 1, Problèmes à Temps Fixe, Masson, Paris.Google Scholar
  4. Dacunha-Castelle, D. and Gassiat, E. (1997). Testing in locally conic model and application to mixture models, ESAIM Probab. Statist., 1, 285–317.Google Scholar
  5. Everitt, B. S. and Hand, D. J. (1981). Finite Mixture Distributions, Chapman and Hall, New York.Google Scholar
  6. Ghosh, J. and Sen, P. (1985). On the asymptotic performance of the log-likelihood ratio statistic for the mixture model and related results, Proc. Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II, 789–806, Wadsworth, Belmont, California.Google Scholar
  7. Hanson, D. L. and Russo, R. P (1983). Some results on increments of the Wiener process with applications to lag sums of i.i.d. random variables, Ann. Probab., 11(3), 609–623.Google Scholar
  8. Hartigan, J. A. (1985). A failure of likelihood ratio asymptotics for normal mixtures, Proc. Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II, 807–810, Wadsworth, Belmont, California.Google Scholar
  9. Lindsay, B. G. (1995). Mixture models: Theory, geometry and applications, NSF-CBMS Regional Conference Series in Probability and Statistics, 5, IMS, Hayward, California; ASA, Alexandria, Virginia.Google Scholar
  10. McLachlan, G. J. and Basford, K. E. (1988). Mixture Models: Inference and Applications to Clustering, Marcel Dekker, New York.Google Scholar
  11. Redner, R. (1981). Note on the Consistency of the maximum likelihood estimate for nonidentifiable distributions, Ann. Statist., 9, 225–228.Google Scholar
  12. Richardson, S. and Green, P. J. (1997). On Bayesian analysis of mixtures with an unknown number of components, J. Roy. Statist. Soc. Ser. B, 59(4), 731–792.Google Scholar
  13. Roberts, S. J., Husmeier, D., Rezek, I. and Penny, W. (1998). Bayesian approaches to Gaussian mixture modeling, IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(11), 887–906.Google Scholar
  14. Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statitics, Wiley, New York.Google Scholar
  15. Titterington, D. M., Smith, A. F. M. and Makov, U. E. (1985). Statistical Analysis of Finite Mixture Models, Wiley, New York.Google Scholar
  16. Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes, Springer, New York.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 2002

Authors and Affiliations

  • Gabriela Ciuperca
    • 1
  1. 1.Laboratoire de Probabilités, Combinatoire et Statistique, Domaine de Gerland, Bât. Recherche BUniv. Lyon 1Lyon cedex 07France

Personalised recommendations