Asymptotic optimality of Hodges-Lehmann inverse rank likelihood estimators

Abstract

Hodges and Lehmann proposed using rank test statistics evaluated at inversely transformed data to construct estimating equations. This paper reframes the resulting estimates in terms of a rank likelihood. In the context of a model of the form Y = h( e; z, β), where Y is a response, z a vector of predictors and e is a random error, this approach corresponds to computing the inverse e = g( Y, z; β) by solving Y = h( e; z, β) for e and using the distribution of the ranks of independent e’s as a likelihood. The properties of the resulting estimators have been developed in many important contexts. This paper will review and extend asymptotic optimality properties of Hodges Lehmann estimators in semiparametric models. In particular, the paper will establish semiparametric optimality of the estimate obtained from a Gaussian linear model. Moreover, it will be shown that the Hodges-Lehmann estimate obtained from the exponential likelihood is asymptotically minimax for the semiparametric accelerated failure time model with increasing hazard rates, and it will be shown that a uniform (Wilcoxon) score estimate applied to log Yi, 1 ≤ i ≤ n, is asymptotically minimax for an accelerated failure time model with increasing logit rate. References to recently developed software in R is provided.