Nonparametric Estimates of Low Bias

Abstract

We consider the problem of estimating an arbitrary smooth functional of k ≥ 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 ∼ n⁻¹ , where n is the minimum sample size, with a p^th order iterative estimate of bias ∼ n^−p for any p. For p ≤ 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 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 ∼ n calculations. This is in sharp contrast with computationally intensive bias reduction methods such as the p th order bootstrap and jackknife, which require ∼ n^p calculations.