ADAPTIVE CHOICE OF TRIMMING PROPORTIONS

JANA JUREČKOVÁ^{1}, ROGER KOENKER^{2} AND A. H. WELSH^{3}

1Department of Probability and Statistics, Charles University, Sokolovska 83, 18600 Prague, Czech Republic

2Department of Economics, University of Illinois, Champaign, IL 61820, U.S.A.

3Department of Statistics, The Australian National University, GPO Box 4, Canberra, ACT 2601, Australia

(Received December 20, 1993; revised May 30, 1994)

Abstract. We consider Jaeckel’s (1971, Ann. Math. Statist., 42, 1540–1552) proposal for choosing the trimming proportion of the trimmed mean in the more general context of choosing a trimming proportion for a trimmed L-estimator of location. We obtain higher order expansions which enable us to evaluate the effect of the estimated trimming proportion on the adaptive estimator. We find that L-estimators with smooth weight functions are to be preferred to those with discontinuous weight functions (such as the trimmed mean) because the effect of the estimated trimming proportion on the estimator is of order n^{−1} rather than n^{−3/4}. In particular, we find that valid inferences can be based on a particular “smooth” trimmed mean with its asymptotic standard error and the Student t distribution with degrees of freedom given by the Tukey and McLaughlin (1963, Sankhyā Ser. A, 25, 331–352) proposal.

Key words and phrases: Trimmed mean, adaptive estimation, L-statistics.