Estimates of Regression Parameters Based on Rank Tests

Abstract

In the linear regression model $Y_j = \alpha + \beta x_j + Z_j$, it is usual to estimate $\alpha$ and $\beta$ by the method of least squares. This method has, among other things, the nice property of providing "best" linear unbiased estimates, under very general conditions. Various other methods of estimation of the parameters are well known, see for example [12] and [14]. Most of these methods however make use of the actual values of the observations, and the estimates they yield are generally vulnerable to gross errors. For some alternative approaches to the problem, see [7], [11] and [13]. In a recent paper [9], Hodges and Lehmann proposed a general method of obtaining robust point estimates for the location parameter, from statistics used to test the hypothesis that this parameter has a specified value. In Section 1 of this paper, this method is used to define point estimates $\hat\alpha$ and $\hat\beta$ of $\alpha$ and $\beta$, in terms of certain test statistics. It is shown that the least squares estimates are obtainable as special cases from the general method of estimation discussed. In Section 2, the existence of `rank score' estimates is proved, and in Section 3, computing techniques are given and illustrated with an example. Both the small sample and asymptotic properties of the estimates are discussed. It is shown, for example, that the joint distribution of the estimates $\hat\alpha$ and $\hat\beta$ is symmetric with respect to the parameter point $(\alpha, \beta)$--and hence that $\hat\alpha$ and $\hat\beta$ are unbiased--if the underlying distribution of the observations is symmetric. In Section 5, the joint asymptotic normality of $\hat\alpha$ and $\hat\beta$ is proved, and in Section 6, it is shown that the asymptotic efficiency of $(\hat\alpha, \hat\beta)$ is the same as the Pitman efficiency of the rank tests [1], on which they are based, relative to the classical tests. Finally in Section 7, the $(\hat\alpha, \hat\beta)$-estimates are compared with the Brown and Mood median estimates with respect to their efficiencies.