# Average Density Estimators: Efficiency and Bootstrap Consistency

## TL;DR

This paper examines the trade-off between efficiency and bootstrap consistency in average density estimation, showing that simple estimators can be bootstrap consistent despite bias issues, unlike some debiased estimators.

## Contribution

It reveals the tension between achieving semiparametric efficiency and bootstrap consistency, demonstrating that simple estimators can be bootstrap consistent even with bias.

## Key findings

- Simple plug-in estimators are biased but bootstrap consistent.
- Debiased estimators achieve efficiency but lack bootstrap consistency.
- Bootstrap can correct bias in simple estimators under minimal smoothness.

## Abstract

This paper highlights a tension between semiparametric efficiency and bootstrap consistency in the context of a canonical semiparametric estimation problem, namely the problem of estimating the average density. It is shown that although simple plug-in estimators suffer from bias problems preventing them from achieving semiparametric efficiency under minimal smoothness conditions, the nonparametric bootstrap automatically corrects for this bias and that, as a result, these seemingly inferior estimators achieve bootstrap consistency under minimal smoothness conditions. In contrast, several "debiased" estimators that achieve semiparametric efficiency under minimal smoothness conditions do not achieve bootstrap consistency under those same conditions.