# zbMATH — the first resource for mathematics

Rates of convergence for the Gaussian mixture sieve. (English) Zbl 1105.62333
Summary: Gaussian mixtures provide a convenient method of density estimation that lies somewhere between parametric models and kernel density estimators.When the number of components of the mixture is allowed to increase as sample size increases, the model is called a mixture sieve.We establish a bound on the rate of convergence in Hellinger distance for density estimation using the Gaussian mixture sieve assuming that the true density is itself a mixture of Gaussians; the underlying mixing measure of the true density is not necessarily assumed to have finite support. Computing the rate involves some delicate calculations since the size of the sieve–as measured by bracketing entropy–and the saturation rate, cannot be found using standard methods.When the mixing measure has compact support, using $$k_n \sim n^{2/3}/(\log n)^{1/3}$$ components in the mixture yields a rate of order $$(\log n)^{(1+\eta)/6}/n^{1/6}$$ for every $$\eta > 0$$. The rates depend heavilyon the tail behavior of the true density.The sensitivity to the tail behavior is dimin- ished byusing a robust sieve which includes a long-tailed component in the mixture.In the compact case,we obtain an improved rate of $$(\log n/n)^{1/4}$$. In the noncompact case, a spectrum of interesting rates arise depending on the thickness of the tails of the mixing measure.

##### MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference
Full Text:
##### References:
  Banfield, J. and Raftery, A. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics 49 803-821. JSTOR: · Zbl 0794.62034  Barron, A. and Yang, Y. (1995). An asymptotic property of model selection criteria. Technical report, Dept. Statistics, Yale Univ. · Zbl 0949.62041  Chen, J. (1995). Optimal rate of convergence for finite mixtures models. Ann. Statist. 23 221-233. · Zbl 0821.62023  Escobar, M. and West, M. (1995). Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc. 90 577-588. JSTOR: · Zbl 0826.62021  Gemen, S. and Hwang, C. (1982). Nonparametric maximum likelihood estimation bythe method of sieves. Ann. Statist. 10 401-414. · Zbl 0494.62041  Ghosal, S. and van der Vaart, A. (2000). Rates of convergence for Bayes and maximum likelihood estimation for mixtures of normal densities. Unpublished manuscript. · Zbl 1043.62025  Grenander, U. (1981). Abstract Inference. Wiley, New York. · Zbl 0505.62069  Hall, P. (1987). On Kullback-Leibler loss and densityestimation. Ann. Statist. 15 1491-1519. · Zbl 0678.62045  Li, J. (1999). Estimation of mixtures models. Ph.D. dissertation, Dept. Statistics. Yale Univ.  Li, J. and Barron, A. (1999). Mixture densityestimation.  Lindsay, B. (1995). Mixture Models: Theory, Geometry and Applications. IMS, Hayward, CA. · Zbl 1163.62326  McLachlan, G. and Basford, K. (1988). Mixture Models: Inference and Applications to Clustering. Dekker, New York. · Zbl 0697.62050  Priebe, C. (1994). Adaptive mixtures. J. Amer. Statist. Assoc. 89 796-806. JSTOR: · Zbl 0825.62445  Robert, C. (1996). Mixtures of distributions: inference and estimation. In Markov Chain Monte Carlo in Practice (W. Gilks, S. Richardson, D. Spiegelhalter, eds.) 441-464. Chapman and Hall, London. · Zbl 0849.62013  Roeder, K. and Wasserman, L. (1997). Practical Bayesian density estimation using mixtures of normals. J. Amer. Statist. Assoc. 92 894-902. JSTOR: · Zbl 0889.62021  Roeder, K. (1992). Semiparametric estimation of normal mixture densities. Ann. Statist. 20 929-943. · Zbl 0746.62044  Tong, B., and Viele, K. (1998). Mixtures of normal linear regressions. Technical report, Univ. Kentucky. · Zbl 0958.62030  van de Geer, S. (1996). Rates of convergence for the maximum likelihood estimator in mixture models. Nonparametric Statist. 6 293-310. · Zbl 0872.62039  van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002  Wong, W. and Shen, X. (1995). Probabilityinequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 339-362. · Zbl 0829.62002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.