ASYMPTOTICALLY EFFICIENT ADAPTIVE L-ESTIMATORS IN LINEAR-MODELS

An asymptotically efficient adaptive L-estimator of the slope in a linear model is proposed and investigated. The estimator is a one-step L-estimator of the type discussed by Welsh (1987 a,b) with an estimate of the optimal "score" function. The optimal "score" function is related to the integral of (and hence should be easier to estimate than) the usual optimal L-estimator weight function. In constructing the estimator, the data is convolved with a vanishingly small Cauchy contaminant and then the conditional expectation given the data is taken. The "score" function can be treated as constant with respect to the conditional expectation. This means that the conditional expectation can be evaluated explicitly so that calculation of the estimator does not involve the numerical evaluation of an integral. A particular kernel based estimator of the optimal "score" function is examined.