A SMALL SAMPLE ROBUST COMPETITOR OF HODGES-LEHMANN ESTIMATE

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The purpose of this paper is to present a new robust estimate of location in small sample. The pencils considered are normal and double exponential. With respect to this family of pencils, the proposed estimate is, in the case of sample sizes N= shown to be the most robust in the class of estimates which include sample mean, best linear unbiased estimate for double exponential distribution and DN,„ 1, 2, 3 which are closely related to the Hodges-Lehmann estimate and defined in section 3. Let X1, X2, ••• , X, be a random sample from a population with symmetric distribution F(x — 0), where 0 is a location parameter and let XN,1 XN,2 XN,N be its order statistics. Denote by VI the set of all pairs (i, j) such that 1 <j and by the subset of VI such that 1 i < j . N, id-j=N-I-1. For each (i, j) c VT, T, form the mean Mii = (XN,i+XN,j)/2. Among many robust estimates of location, an interesting one would be the HodgesLehmann estimate T N= med ; (i, j) E %I. Though the asymptotic robustness of the estimate has been investigated by many authors such as Hodges-Lehmann [4], HOyland [6], Bickel [1] and others, the investigation of its small sample robustness is still insufficient. Generally, in large sample it would not be so difficult for us to know the population pencil from which the random sample has obtained. Therefore in this case the problem of robust estimation would not be considered so significant. To investigate the small sample robustness of TN, Hodges [4] proposed a closely related estimate I), ;