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**Rates of Convergence for the Gaussian Mixture Sieve**

**Chris Genovese and Larry Wasserman**

### Abstract:

*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 assumed to
necessarily 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 components in the mixture
yields a rate of
order for every .When the mixing measure is not compactly supported,
we find that the rates depend heavily on the tail
behavior of the true density.
The sensitivity to the tail behavior is
diminished by using a robust sieve which
includes a long-tailed component
in the mixture.
Then, a spectrum of interesting rates arise
depending on the thickness of the tails of the mixing measure.**
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*Keywords:* Density Estimation; Mixtures, Rates of Convergence, Sieves

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