M\"obius inversion and the iterated bootstrap

TL;DR

This paper provides a new theoretical framework for understanding bootstrap bias correction as an iterative M"obius inversion process, demonstrating its convergence properties and proposing a modified method for unbiased estimation of moment polynomials.

Contribution

It introduces a novel perspective of bootstrap bias correction as an iterative M"obius inversion, characterizes its convergence, and develops a modified iteration for unbiased moment polynomial estimation.

Findings

Bootstrap bias correction can be viewed as an iterative linear solver for M"obius inversion.

The method exhibits linear convergence for moment polynomials.

A modified bootstrap iteration achieves unbiased estimation of unknown order-m moment polynomials.

Abstract

Estimating nonlinear functionals of probability distributions from samples is a fundamental statistical problem. The "plug-in" estimator obtained by applying the target functional to the empirical distribution of samples is biased. Resampling methods such as the bootstrap derive artificial datasets from the original one by resampling. Comparing the outcome of the plug-in estimator in the original and resampled datasets allows estimating and thus correcting the bias. In the asymptotic setting, iterations of this procedure attain an arbitrarily high order of bias correction, but finite sample results are scarce. This work develops a new theoretical understanding of bootstrap bias correction by viewing it as an iterative linear solver for the combinatorial operation of M\"obius inversion. It sharply characterizes the regime of linear convergence of the bootstrap bias reduction for moment polynomials. It uses these results to show its superalgebraic convergence rate for band-limited functionals. Finally, it derives a modified bootstrap iteration enabling the unbiased estimation of unknown order-$m$ moment polynomials in $m$ bootstrap iterations.