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.