Nonparametric estimation of mixing densities for discrete distributions. (English) Zbl 1086.62049

Summary: By a mixture density is meant a density of the form \(\pi_\mu(\cdot) =\int \pi_\theta(\cdot)\times \mu(d\theta)\), where \((\pi_\theta)_{\theta \in\Theta}\) is a family of probability densities and \(\mu\) is a probability measure on \(\Theta\). We consider the problem of identifying the unknown part of this model the mixing distribution \(\mu\), from a finite sample of independent observations from \(\pi_\mu\). Assuming that the mixing distribution has a density function, we wish to estimate this density within appropriate function classes.
A general approach is proposed and its scope of application is investigated in the case of discrete distributions. Mixtures of power series distributions are more specifically studied. Standard methods for density estimation, such as kernel estimators, are available in this context, and it has been shown that these methods are rate optimal or almost rate optimal in balls of various smoothness spaces. For instance, these results apply to mixtures of the Poisson distribution parametrized by its mean. Estimators based on orthogonal polynomial sequences have also been proposed and shown to achieve similar rates.
The general approach of this paper extends and simplifies such results. For instance, it allows us to prove asymptotic minimax efficiency over certain smoothness classes of the above-mentioned polynomial estimator in the Poisson case. We also study discrete location mixtures, or discrete deconvolution, and mixtures of discrete uniform distributions.


62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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[1] Barbe, P. (1998). Statistical analysis of mixtures and the empirical probability measure. Acta Appl. Math. 50 253–340. · Zbl 0917.62028
[2] Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186. · Zbl 0673.62033
[3] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications . Jones and Bartlett, Boston. · Zbl 0793.60030
[4] DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation . Springer, Berlin. · Zbl 0797.41016
[5] Fan, J. (1991). Global behavior of deconvolution kernel estimates. Statist. Sinica 1 541–551. · Zbl 0823.62032
[6] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272. · Zbl 0729.62033
[7] Fan, J. (1992). Deconvolution with supersmooth distributions. Canad. J. Statist. 20 155–169. · Zbl 0754.62020
[8] Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600–610. · Zbl 0785.62038
[9] Friedman, A. (1982). Foundations of Modern Analysis . Dover, New York. Reprint of the 1970 original. · Zbl 0557.46001
[10] Gautschi, W. (1990). Computational aspects of orthogonal polynomials. In Orthogonal Polynomials : Theory and Practice (P. Nevai, ed.) 181–216. Kluwer, Dordrecht. · Zbl 0697.42017
[11] Gill, R. D. and Levit, B. Y. (1995). Applications of the Van Trees inequality: A Bayesian Cramér–Rao bound. Bernoulli 1 59–79. · Zbl 0830.62035
[12] Hengartner, N. (1997). Adaptive demixing in Poisson mixture models. Ann. Statist. 25 917–928. · Zbl 0876.62042
[13] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions . Wiley, New York. · Zbl 0869.00044
[14] Lindsay, B. G. (1989). Moment matrices: Applications in mixtures. Ann. Statist. 17 722–740. · Zbl 0672.62063
[15] Lindsay, B. G. (1995). Mixture Models : Theory , Geometry and Applications . IMS, Hayward, CA. · Zbl 0832.62027
[16] Loh, W.-L. and Zhang, C.-H. (1996). Global properties of kernel estimators for mixing densities in discrete exponential family models. Statist. Sinica 6 561–578. · Zbl 0854.62031
[17] Loh, W.-L. and Zhang, C.-H. (1997). Estimating mixing densities in exponential family models for discrete variables. Scand. J. Statist. 24 15–32. · Zbl 0923.62043
[18] McLachlan, G. and Peel, D. (2000). Finite Mixture Models . Wiley, New York. · Zbl 0963.62061
[19] Thorisson, H. (2000). Coupling , Stationarity , and Regeneration . Springer, New York. · Zbl 0949.60007
[20] Titterington, D. M., Smith, A. F. M. and Makov, U. E. (1985). Statistical Analysis of Finite Mixture Distributions . Wiley, New York. · Zbl 0646.62013
[21] van de Geer, S. (1996). Rates of convergence for the maximum likelihood estimator in mixture models. J. Nonparametr. Statist. 6 293–310. · Zbl 0872.62039
[22] Zhang, C.-H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806–831. · Zbl 0778.62037
[23] Zhang, C.-H. (1995). On estimating mixing densities in discrete exponential family models. Ann. Statist. 23 929–945. · Zbl 0841.62027
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