The Empirical Edgeworth Expansion for a Studentized Trimmed Mean

Abstract

We establish the validity of the empirical Edgeworth expansion (EE) for a studentized trimmed mean, under the sole condition that the underlying distribution function of the observations satisfies a local smoothness condition near the two quantiles where the trimming occurs. A simple explicit formula for the N^{-1/2} term (correcting for skewness and bias; N being the sample size) of the EE is given. In particular our result supplements previous work by P. Hall and A.R. Padmanabhan, On the bootstrap and the trimmed mean}, J. of Multivariate Analysis, v. 41 (1992), pp. 132-153. and H. Putter and W.R. van Zwet, Empirical Edgeworth expansions for symmetric statistics, Ann. Statist., v. 26 (1998), pp. 1540-1569. The proof is based on a U-statistic type approximation and also uses a version of Bahadur's representation for sample quantiles.