Bootstrap, jackknife and Edgeworth approximations for finite population L-statistics

Andrius Čiginas

doi:10.15388/lmr.2010.71

Abstract

In this paper we give exact bootstrap estimators for the parameters defining one-term Edgeworth expansion of distribution function of finite population L-statistic and compare these estimators with corresponding jackknife estimators. We also compare `````` true’ distribution of L-statistic with its normal approximation, Edgeworth expansion, empirical Edgeworth expansion and bootstrap approximation.

Highlights

Consider a population X = {x1, . . . , xN } of size N
Let X = {X1, . . . , Xn} be the simple random sample of size n < N drawn without replacement from X and let X1:n · · · Xn:n denote the order statistics of Consider a linear combination
We aim to compare ‘true’ distribution function Fn (obtained by Monte Carlo (M–C) method) with its normal approximation, Edgeworth expansion, two empirical Edgeworth expansions and bootstrap approximation of Fn

Summary

Introduction

For samples drawn without replacement from finite population the most general results on one-term Edgeworth expansion and empirical Edgeworth expansions are obtained in [5, 3]. In the case of L-statistics explicit expressions of the kernels g1(·) and g2(· , ·) are available, see [7] Using these expressions one can construct bootstrap estimators. We aim to compare ‘true’ distribution function Fn (obtained by Monte Carlo (M–C) method) with its normal approximation, Edgeworth expansion, two empirical Edgeworth expansions (with parameters estimated in two ways mentioned) and bootstrap approximation of Fn. We note that the accuracy and features of the latter approximation is not completely understood for L-statistics (on bootstrap for U -statistics see [4])

Bootstrap estimators

Jackknife estimators

Approximations to distribution function Fn

Simulation study and conclusions