Nonparametric estimates of low bias

TL;DR

This paper introduces a simple, nonparametric method for estimating smooth functionals of multiple distributions with low bias, using iterative bias correction based on von Mises derivatives, avoiding complex bootstrap or jackknife procedures.

Contribution

It provides explicit formulas for bias reduction up to fourth order using von Mises derivatives, enabling unbiased estimates with minimal computational effort.

Findings

Explicit bias estimates for p ≤ 4 using von Mises derivatives

Unbiased estimates are simpler than polykay-based methods

Computational complexity is comparable to basic empirical estimates

Abstract

We consider the problem of estimating an arbitrary smooth functional of $k \geq 1 $ distribution functions (d.f.s.) in terms of random samples from them. The natural estimate replaces the d.f.s by their empirical d.f.s. Its bias is generally $\sim n^{-1}$, where $n$ is the minimum sample size, with a {\it $p$th order} iterative estimate of bias $ \sim n^{-p}$ for any $p$. For $p \leq 4$, we give an explicit estimate in terms of the first $2p - 2$ von Mises derivatives of the functional evaluated at the empirical d.f.s. These may be used to obtain {\it unbiased} estimates, where these exist and are of known form in terms of the sample sizes; our form for such unbiased estimates is much simpler than that obtained using polykays and tables of the symmetric functions. Examples include functions of a mean vector (such as the ratio of two means and the inverse of a mean), standard deviation, correlation, return times and exceedances. These $p$th order estimates require only $\sim n $ calculations. This is in sharp contrast with computationally intensive bias reduction methods such as the $p$th order bootstrap and jackknife, which require $\sim n^p $ calculations.