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.
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References
M. A. Arcones and E. Giné, “Some Bootstrap Tests of Symmetry for Univariate Continuous Distributions”, Ann. Statist. 19, 1496–1511 (1991).
L. Bordes, S. Mottelet, and P. Vandekerkhove, “Semiparametric Estimation of a Two-Component Mixture Model”, Ann. Statist. 34, 1204–1232 (2006a).
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).
L. Bordes, D. Chauveau, and P. Vandekerkhove, A Stochastic EM Algorithm for a Semiparametric Mixture Model”, Comput. Statist. Data Anal. 51, 5429–5443 (2007).
I. R. Cruz-Medina and T. P. Hettmansperger, “Nonparametric Estimation in Semiparametric Univariate Mixture Models”, J. Statist. Comput. Simul. 74, 513–524 (2004).
B. Efron, “Size, Power and False Discovery Rate”, Ann. Statist. 35, 1351–1377 (2007).
E. Giné and A. Guillou, “Rates of Strong Uniform Consistency for Multivariate Kernel density estimators”, Ann. Inst. H. Poincaré 38, 503–522 (2002).
P. Hall and X. H. Zhou, “Nonparametric Estimation of Component Distributions in a Multivariate Mixture”, Ann. Statist. 31, 201–224 (2003).
P. Hall, A. Neeman, R. Pakyari, and R. Elmore, “Nonparametric Inference in Multivariate Mixtures”, Biometrika 92, 667–678 (2005).
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).
D.R. Hunter, S. Wang, and T. P. Hettmansperger, “Inference for Mixtures of Symmetric Distributions”, Ann. Statist. 35, 224–251 (2007).
B. G. Lindsay and M. L. Lesperance, “A Review of Semiparametric Mixture Models”, J. Statist. Plann. Inf. 47, 29–39 (1995).
G. J. McLachlan and D. Peel, Finite Mixture Models (Wiley, New York, 2000).
R. Maiboroda and O. Sugakova, Generalized Estimating Equations for Symmetric Distributions Observed with Admixture. Preprint (2009).
D. S. Moore, “A Chi-Square Statistic with Random Cell Boundaries”, Ann. Math. Statist. 42, 147–156 (1971).
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).
C. Pal and A. Sengupta, “Optimal Tests for no Contamination in Reliability Models”, Lifetime Data Anal. 6, 281–290 (2000).
F. H. Ruymgaart, “A Note on Chi-Square Statistics with Random Cell Boundaries”, Ann. Statist. 3, 965–968 (1975).
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).
G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics (Wiley, New York, 1986).
E. F. Schuster and R. C. Barker, “Using the Bootstrap in Testing Symmetry Versus Asymmetry”, Comm. Statist. Simulation Comput. 16, 69–84 (1987).
B. W. Silverman, “Weak and Strong Consistency of the Kernel Estimate of a Density and Its Derivatives”, Ann. Statist. 6, 177–184 (1978).
D.M. Titterington, A. F.M. Smith, and U. E. Makov, Statistical Analysis of Finite Mixture Distributions (Wiley, Chichester, 1985).
A.W. van der Vaart, Asymptotic Statistics (Cambridge University Press, New York, 2000).
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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
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DOI: https://doi.org/10.3103/S1066530710010023
Key words
- semiparametric
- two-component mixture model
- functional delta method
- asymptotic normality
- chi-square tests