Estimates of Location Based on Rank Tests

Abstract

A serious objection to many of the classical statistical methods based on linear models or normality assumptions is their vulnerability to gross errors. For certain testing problems this difficulty is successfully overcome by rank tests such as the two Wilcoxon tests or the Kruskal-Wallis H-test. Their power is more robust against gross errors than that of the $t$- and $F$-tests, and their efficiency loss is quite small even in the rare case in which the suspicion of the possibility of gross errors is unfounded. For the corresponding problems of point estimation a beginning has been made to attack the difficulty by modifying the classical estimates either through removal or Winsorization of outlying observations; see for example Tukey (1960) and Anscombe (1960). It is the purpose of the present paper to explore a different approach to these problems of point estimation. In Sections 2-5 point estimates of location or shift parameter are defined in terms of rank test statistics such as the Wilcoxon or normal scores statistic, which are successful in providing robust power for the corresponding testing problems. In Sections 6 and 7, certain regularity and invariance properties of these estimates are proved. The distributions of the estimates are shown in Section 8 to be symmetric with respect to the parameter being estimated--and hence in particular to be unbiased--if the underlying distribution of the observations on which the estimate is based is symmetric. Without this assumption, the estimates are shown in Section 9 to be either exactly or approximately median unbiased for small samples and in Section 10 to be approximately normally distributed about the true parameter value for large samples. The variance of this asymptotic distribution depends of course on the underlying distribution of the observations, so that the estimates are not "distribution-free." In Section 9 there is also established a close relationship between the estimates and the corresponding upper and lower confidence bound for the parameter at confidence level $\frac{1}{2}$, with which the estimate coincide in many cases. Finally, in Section 11, it is proved that the asymptotic relative efficiency of the estimates to the classical linear estimates is the same as the Pitman efficiency of the rank tests on which they are based to the corresponding $t$-tests.