ASYMPTOTIC EXPANSIONS FOR THE DISTRIBUTION FUNCTIONS OF LINEAR COMBINATIONS OF ORDER STATISTICS

Abstract

This chapter discusses asymptotic expansions and explains their need. It reviews the classical theory of Edgeworth expansions for sums of independent and identically distributed random variables, and indicates the two main techniques for extending this theory to more general statistics. The chapter presents an account of as yet unpublished results of Bjerve and Helmers who establish Berry–Esseen type bounds for linear combinations of order statistics. For many years, mathematical statisticians have spent a great deal of effort and ingenuity toward applying the central limit theorem in statistics. The estimators and test statistics that interest statisticians are as a rule not sums of independent random variables, and much work went into showing that they can often be approximated sufficiently well by such sums to ensure asymptotic normality. This work can be traced throughout the development of mathematical statistics from the proof of the asymptotic normality of the maximum likelihood estimator to much of the recent work in nonparametric statistics.