Asymptotic Expansions for Moments of Functions of Random Sequences

Jan Hurt

Charles University of Prague, Czech Republic

Abstract

We consider a smooth real function g and a random sequence (Subscript[{(Subscript[T, n])},  n=1,...,∞]) which converges to a parameter θ in some sense. In Statistics, this sequence is often a sequence of statistics that estimate some unknown parameter based on random sample of size n. We give the asymptotic expansions (theoretically of arbitrary order) of the expected value, variance, and mean squared error of the sequence (Subscript[{g((Subscript[T, n]))} , n=1,...,∞]). The method is also known as the method of statistical differentials or the delta method. A supporting theory may be found in the author's paper in Statistics & Decisions 4, 251-271 (1986), e.g., Mathematica helps a lot to perform rather complex technical calculations.

We will illustrate the methods in applications to estimation problems in reliability theory in complete and censored samples. We will also show how to reduce bias using the asymptotic expansions and how to compare competing efficient estimators by Hodges-Lehmann's deficiency.