"Rates of convergence for estimators of convolutions of densities."
Abstract
The goal of this talk is to give an overview of various
types of convergence results for estimating the convolution
of a density with itself.
The estimator of this convolution is a local U-statistic
based on a random sample from the base density.
The surprising fact is that under rather mild assumptions on the base density
this estimator has a convergence rate of the order root-n,
pointwise and in various norms, and (functional) central limit theorems
can be proved in the corresponding normed spaces.
Key to these results are integrability conditions on the base density.
A violation of these conditions results in slower rates of convergence.
The behavior of the local U-statistic is now similar to that of
kernel estimators with the customary trade-off between bias and variance
terms.
These slower rates of convergence, however, are still faster than
the optimal rates of convergence for kernel estimators based
on a sample from the convolution.
