A note on kernel density estimators with optimal bandwidths

TL;DR

This paper demonstrates that the cumulative distribution function derived from a kernel density estimator with optimal bandwidth almost surely falls outside any confidence interval around the empirical distribution function as the sample size grows.

Contribution

It provides a theoretical result showing the asymptotic behavior of kernel density estimators' CDFs relative to confidence intervals.

Findings

The kernel density estimator's CDF with optimal bandwidth diverges from confidence intervals asymptotically.

The probability that the estimator's CDF lies outside any confidence interval approaches 1 as sample size increases.

This challenges assumptions about the reliability of kernel density estimators for inference in large samples.

Abstract

We show that the cumulative distribution function corresponding to a kernel density estimator with optimal bandwidth lies outside any confidence interval, around the empirical distribution function, with probability tending to 1 as the sample size increases.