Minimax‐variance L‐ and R‐estimators of location

Abstract

AbstractWe consider the problem of minimax‐variance, robust estimation of a location parameter, through the use of L‐ and R‐estimators. We derive an easily checked necessary condition for L‐estimation to be minimax, and a related sufficient condition for R‐estimation to be minimax. Those cases in the literature in which L‐estimation is known not to be minimax, and those in which R‐estimation is minimax, are derived as consequences of these conditions. New classes of examples are given in each case. As well, we answer a question of Scholz (1974), who showed essentially that the asymptotic variance of an R‐estimator never exceeds that of an L‐estimator, if both are efficient at the same strongly unimodal distribution. Scholz raised the question of whether or not the assumption of strong unimodality could be dropped. We answer this question in the negative, theoretically and by examples. In the examples, the minimax property fails both for L‐estimation and for R‐estimation, but the variance of the L‐estimator, as the distribution of the observation varies over the given neighbourhood, remains unbounded. That of the R‐estimator is unbounded.