# Limitations Of Richardson Extrapolation For Kernel Density Estimation

## TL;DR

This paper investigates the application of Richardson Extrapolation to Kernel Density Estimation, highlighting its limitations and discussing optimal estimator order to improve density estimates with lower mean squared error.

## Contribution

It develops a method to fix conditioning issues in Richardson Extrapolation for KDE and analyzes the optimal order for minimal error, proposing future research directions.

## Key findings

- Higher-order estimators do not always yield better results.
- Estimated density function error can be on the order of $n^{-1}\sqrt{\ln(n)}$.
- Discusses fixing conditioning issues in Richardson Extrapolation.

## Abstract

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.