  This paper develops the process of using Richardson Extrapolation to improve
the Kernel Density Estimation method, resulting in a more accurate (lower Mean
Squared Error) estimate of a probability density function for a distribution of
data in $R_d$ given a set of data from the distribution. The method of
Richardson Extrapolation is explained, showing how to fix conditioning issues
that arise with higher-order extrapolations. Then, it is shown why higher-order
estimators do not always provide the best estimate, and it is discussed how to
choose the optimal order of the estimate. It is shown that given n
one-dimensional data points, it is possible to estimate the probability density
function with a mean squared error value on the order of only
$n^{-1}\sqrt{\ln(n)}$. Finally, this paper introduces a possible direction of
future research that could further minimize the mean squared error.
