Asymptotic Statistics pp 54-59 | Cite as

# Bootstrp and Edgeworth Expansion

## Abstract

Suppose that *T(P)* is a functional, say real valued, on some subset P of the set of all probability measures on a measurable space (*χ*, B), and one wishes to obtain a confidence interval for *T(P)* based on *n* i.i.d. observations *X*_{ 1 },..., *X*_{ n } with common distribution *P*. For example, if P is a parametric family then *T(P)* is a function of the parameter, and one may use the maximum likelihood estimator *θ̂* of *T(P)* and an estimate *s*_{ n } of its standard error *σ*_{ n } to form a confidence interval using normal approximation. Under appropriate assumptions (stated below) one may do better than normal approximation for the *studentized statistic* \( ({\hat{\theta }_{n}} - T(P))/{s_{n}} \). In this subsection we consider two procedures for improvement over the normal approximation: (1) the *bootstrap* proposed by Efron [36], and (2) the *empirical Edgeworth expansion.*

## Keywords

Maximum Likelihood Estimator Normal Approximation Common Distribution Borel Measurable Function Sample Moment## Preview

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