Bootstrap FiniteSample DistributionUniversity of Notre Dame引导的有限样本分布圣母大学.doc

The Bootstrap’s Finite Sample Distribution:
An Analytical Approach
Lawrence C. Marsh I thank Tom Cosimano, Ken Kelley, Nelson Mark, Ke-Hai Yuan and my colleagues in the Department of Economics and Econometrics for ments and suggestions.
Department of Economics and Econometrics
University of Notre Dame, Notre Dame, IN 46556-5611
Abstract
This paper provides a method for determining the exact finite sample properties of the bootstrap. Previously, information about bootstrap performance has been primarily limited to asymptotic properties and Monte Carlo experiments. The exact small sample properties of most bootstrap procedures have not been determined. We show how to transform the empirical process into an analytical process and separate the bootstrap-induced randomness from the randomness of the underlying random variable. We derive the variances of some selected bootstrap estimators. Other exact properties such as bias, skewness, kurtosis, and mean squared error could be readily derived using this approach. Bootstrap estimators that are nonlinear functions of the bootstrap sample values (including ratios and matrix inverses) can be represented by Taylor series as polynomials in their bootstrap-induced frequencies. Consequently, their finite sample distributions can be analyzed up to any desired degree of accuracy.
Keywords: multinomial, wild bootstrap, block bootstrap, nonlinear, variance, bias
Initial Version: May 2003
Current Version: Monday, September 20, 2004
1. Introduction
The purpose of this paper is to propose an analytical approach to understanding the exact finite sample distribution induced by particular bootstrap procedures. We define the class of bootstrap estimators whose finite sample distributions are “directly analyzable” with this approach. The term “directly analyzable” will be defined in section 2 of this paper.
A Taylor series expansion can be used to transform nonlinear bootstrap structures into polynomials which are easily analyz