Robust Estimation of Location Using Optimally Chosen Sample Quantiles

Robust estimation of the location parameter α based on selected order statistics is considered. The distribution function is only known to belong to a subset of a set D of distributions consisting of Tukey's lambda family of symmetric distributions with the inverse distribution function of the form α + β/γ) [λγ – (1 – λ)γ], 0 ≤ λ ≤ 1, and the normal, Cauchy and double exponential distributions. The scale parameter β can be unknown. In the set D, distributions with tail length varying from short to extremely long, when the shape parameter γ varies, are included. The asymptotically best linear estimate (ABLE) based on k (k = 2(1)5) optimally chosen symmetrical sample quantiles is considered. It can be used as a robust estimate and is shown to compete favorably with optimally trimmed and Winsorized means, in the sense of giving a higher guaranteed relative asymptotic efficiency (GRAE) for subsets of D. Tables are provided so that the robust k-ABLE giving the highest GRAE for any subset of D can easily be obtained. An adaptive procedure of trying to identify the prototype by means of estimating the tail length is suggested.