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.