RATES OF CONVERGENCE FOR THE GAUSSIAN MIXTURE SIEVE

Gaussian mixtures provide a convenient method of densityestimation that lies somewhere between parametric models and kernel densityestimators. 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 densityestimation using the Gaussian mixture sieve assuming that the true density is itself a mixture of Gaussians; the underlying mixing measure of the true densityis not necessarilyassumed to have finite support. Computing the rate involves some delicate calculations since the size of the sieve—as measured bybracketing entropy—and the saturation rate, cannot be found using standard methods. When the mixing measure has compact support, using kn ∼ n 2/3 /� log n� 1/3 components in the mixture yields a rate of order � log n� � 1+η� /6/n1/6 for every η> 0� The rates depend heavilyon the tail behavior of the true density. The sensitivity to the tail behavior is diminished byusing a robust sieve which includes a long-tailed component in the