Skip to main content

Semiparametric two-component mixture model with a known component: An asymptotically normal estimator

Mathematical Methods of Statistics volume 19, pages 22–41 (2010)Cite this article

Abstract

In this paper we consider a two-component mixture model, one component of which has a known distribution while the other is only known to be symmetric. The mixture proportion is also an unknown parameter of the model. This mixture model class has proved to be useful to analyze gene expression data coming from microarray analysis. In this paper a general estimation method is proposed leading to a joint central limit result for all the estimators. Applications to basic testing problems related to this class of models are proposed, and the corresponding inference procedures are illustrated through some simulation studies.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Finland)

References

  1. M. A. Arcones and E. Giné, “Some Bootstrap Tests of Symmetry for Univariate Continuous Distributions”, Ann. Statist. 19, 1496–1511 (1991).

    MATH  Article  MathSciNet  Google Scholar 

  2. L. Bordes, S. Mottelet, and P. Vandekerkhove, “Semiparametric Estimation of a Two-Component Mixture Model”, Ann. Statist. 34, 1204–1232 (2006a).

    MATH  Article  MathSciNet  Google Scholar 

  3. L. Bordes, C. Delmas, and P. Vandekerkhove, “Semiparametric Estimation of a Two-Component Mixture Model when a Component is Known”, Scand. J. Statist. 33, 733–752 (2006b).

    MATH  Article  MathSciNet  Google Scholar 

  4. L. Bordes, D. Chauveau, and P. Vandekerkhove, A Stochastic EM Algorithm for a Semiparametric Mixture Model”, Comput. Statist. Data Anal. 51, 5429–5443 (2007).

    MATH  Article  MathSciNet  Google Scholar 

  5. I. R. Cruz-Medina and T. P. Hettmansperger, “Nonparametric Estimation in Semiparametric Univariate Mixture Models”, J. Statist. Comput. Simul. 74, 513–524 (2004).

    MATH  Article  MathSciNet  Google Scholar 

  6. B. Efron, “Size, Power and False Discovery Rate”, Ann. Statist. 35, 1351–1377 (2007).

    MATH  Article  MathSciNet  Google Scholar 

  7. E. Giné and A. Guillou, “Rates of Strong Uniform Consistency for Multivariate Kernel density estimators”, Ann. Inst. H. Poincaré 38, 503–522 (2002).

    Google Scholar 

  8. P. Hall and X. H. Zhou, “Nonparametric Estimation of Component Distributions in a Multivariate Mixture”, Ann. Statist. 31, 201–224 (2003).

    MATH  Article  MathSciNet  Google Scholar 

  9. P. Hall, A. Neeman, R. Pakyari, and R. Elmore, “Nonparametric Inference in Multivariate Mixtures”, Biometrika 92, 667–678 (2005).

    MATH  Article  MathSciNet  Google Scholar 

  10. T. P. Hettmansperger and H. Thomas, Almost Nonparametric Inference for Repeated Measures in Mixture Models”, J. Roy. Statist. Soc. Ser. B 62, 811–825 (2000).

    MATH  Article  MathSciNet  Google Scholar 

  11. D.R. Hunter, S. Wang, and T. P. Hettmansperger, “Inference for Mixtures of Symmetric Distributions”, Ann. Statist. 35, 224–251 (2007).

    MATH  Article  MathSciNet  Google Scholar 

  12. B. G. Lindsay and M. L. Lesperance, “A Review of Semiparametric Mixture Models”, J. Statist. Plann. Inf. 47, 29–39 (1995).

    MATH  Article  MathSciNet  Google Scholar 

  13. G. J. McLachlan and D. Peel, Finite Mixture Models (Wiley, New York, 2000).

    MATH  Book  Google Scholar 

  14. R. Maiboroda and O. Sugakova, Generalized Estimating Equations for Symmetric Distributions Observed with Admixture. Preprint (2009).

  15. D. S. Moore, “A Chi-Square Statistic with Random Cell Boundaries”, Ann. Math. Statist. 42, 147–156 (1971).

    MATH  Article  MathSciNet  Google Scholar 

  16. J.C. Naylor and A. F. M. Smith, “A Contamination Model in Clinical Chemistry: An Illustration of a Method for the Efficient Computation of Posterior Distributions”, The Statistician 32, 82–87 (1983).

    Article  Google Scholar 

  17. C. Pal and A. Sengupta, “Optimal Tests for no Contamination in Reliability Models”, Lifetime Data Anal. 6, 281–290 (2000).

    MATH  Article  MathSciNet  Google Scholar 

  18. F. H. Ruymgaart, “A Note on Chi-Square Statistics with Random Cell Boundaries”, Ann. Statist. 3, 965–968 (1975).

    MATH  Article  MathSciNet  Google Scholar 

  19. S. A. Robin, A. Bar-Hen, J-J. Daudin, and L. Pierre, “Semi-Parametric Approach for Mixture Models: Application to Local False Discovery Rate Estimation”, Comput. Statist. Data Anal. 51, 5483–5493 (2007).

    MATH  Article  MathSciNet  Google Scholar 

  20. G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics (Wiley, New York, 1986).

    MATH  Google Scholar 

  21. E. F. Schuster and R. C. Barker, “Using the Bootstrap in Testing Symmetry Versus Asymmetry”, Comm. Statist. Simulation Comput. 16, 69–84 (1987).

    MATH  Article  MathSciNet  Google Scholar 

  22. B. W. Silverman, “Weak and Strong Consistency of the Kernel Estimate of a Density and Its Derivatives”, Ann. Statist. 6, 177–184 (1978).

    MATH  Article  MathSciNet  Google Scholar 

  23. D.M. Titterington, A. F.M. Smith, and U. E. Makov, Statistical Analysis of Finite Mixture Distributions (Wiley, Chichester, 1985).

    MATH  Google Scholar 

  24. A.W. van der Vaart, Asymptotic Statistics (Cambridge University Press, New York, 2000).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Math. Appl., UMR CNRS 5142, Univ. de Pau et des Pays de l’Adour, l’Adour, France

    L. Bordes

  2. Laboratoire d’Analyse et de Math. Appl., UMR CNRS 8050, Univ. de Marne-la-Vallée, Marne-la-Vallée, France

    P. Vandekerkhove

Authors
  1. L. Bordes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Vandekerkhove
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Bordes.

About this article

Cite this article

Bordes, L., Vandekerkhove, P. Semiparametric two-component mixture model with a known component: An asymptotically normal estimator. Math. Meth. Stat. 19, 22–41 (2010). https://doi.org/10.3103/S1066530710010023

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530710010023

Key words

  • semiparametric
  • two-component mixture model
  • functional delta method
  • asymptotic normality
  • chi-square tests

2000 Mathematics Subject Classification

  • primary 62G05, 62G20
  • secondary 62E10