Moment Expansions for Robust Statistics

Abstract : Our objective is to give asymptotic expansions for moments of standardized statistics based on n independent, identically distributed random variables as n approaches infinity. The basic premise is that a simple tail condition on the underlying distribution which implies the moments of a standardized quantile converge to the moments of an appropriate normal distribution is sufficient to assure the validity of asymptotic moment expansions for many statistics which are resistant to ouliers. The primary result we present gives sufficient conditions for the validity of moment approximations based on moments of Taylor's series approximations which are obtained by using functional differentiation. We apply the theory to some L- and M-estimates and present a Monte Carlo study to show that the approximations for the variance of statistics based on small to moderate sample sizes can be quite good. Prior to studying the above general problem we consider the problem of the convergence of the moments of a standardized quantile to those of an appropriate normal distribution. Our proof of moment convergence requires fewer non-tail conditions on the underlying distribution than were used in previously published results. We also extend the result to show necessary and sufficient tail conditions on the underlying distribution for convergence of the moment generating function of a standardized quantile to that of a normal distribution.