Asymptotically Efficient Adaptive Rank Estimates in Location Models

Abstract

This paper describes a new construction of uniformly asymptotically efficient rank estimates in the one and two-sample location models. The method adopted differs from van Eeden's (1970) earlier construction in three respects. First, the whole sample, rather than a vanishingly small fraction of the sample, is used in estimating the efficient score function. Secondly, a Fourier series estimator is used for the score function rather than a window estimator. Thirdly, the linearized rank estimates corresponding to the estimated score function provide the uniformly asymptotically efficient location estimates. These estimates are asymptotically efficient over a larger class of distributions than the van Eeden estimates and should approach their asymptotic behavior more rapidly.