The Asymptotic Theory of Galton's Test and a Related Simple Estimate of Location

Abstract

In [8] Hodges and Lehmann showed that robust estimates of location in the one and two sample problems could be obtained by inverting the known robust nonparametric tests and using as estimates the average of the resulting upper and lower 50 per cent confidence bounds. In particular, they proposed the use of the estimate derived from the Wilcoxon test, the median of averages of pairs of observations. Unfortunately, despite some short cuts viz. [8] and [9] computation of this estimate seems to require on the order of $n^2 \log_2 n$ steps, a prohibitive number. In [7] Hodges proposed a simple alternative estimate $D_n$ given by $D_n = \text{median}_i \frac{1}{2}(Z_{in} + Z_{i(n - i + 1)})$ where $Z_{1n} < \cdots < Z_{nn}$ are the order statistics of the sample under consideration. This procedure as was noted in [7] is related in the sense of Hodges and Lehmann to the one sample analogue of one of the oldest of non parametric tests, the Galton rank order test, viz. [6], [4a]. In this paper we derive the asymptotic theory of $D_n$ by employing an invariance principle due to Bickel [2] and thus relating the limiting distribution to that of certain functionals of Brownian motion. The necessary refinements of the stochastic process convergence results of [2], which may be of use in related problems of asymptotic theory, are gathered in Section 7. Unfortunately, we can only give explicit form to the limiting distribution of $D_n$ in the two cases of rectangular and Laplace parents. Although this limit is not normal we conclude that $D_n$'s scatter is quite close to that of the estimate proposed by Hodges and Lehmann. Using the same techniques we prove in Section 6 the consistency of the Galton test and characterize its power behaviour for alternatives close to the null hypothesis. Finally, Section 4 gives the small sample distribution of the Galton test statistic and of $D_n$ for a rectangular parent. The techniques of this paper carry over quite easily to the two sample situations. Although our evidence is incomplete it would seem that both $D_n$ and the Galton test are robust as well as easily computable non parametric procedures.