On Some Asymptotically Nonparametric Competitors of Hotelling's $T^{2 1}$

Abstract

In a previous paper [2] the author investigated some alternative estimates of shift in the $p$-variate one-sample problem. This paper examines the properties of tests for shift similar to Hotelling's $T^2$, based on (I) asymptotically normal estimates, in particular those of the type considered in [2], and (II) the originating univariate test statistics of the latter group. The notation, similar to that used in [2], and the tests are introduced in Section 2. In the third section the asymptotic distribution of such test statistics for sequences of alternatives tending to $\mathbf{0}$ as $n^{-\frac{1}{2}}$ is found to be noncentral $\chi^2$. The tests of type I and, often simpler, tests of type II have the same asymptotic distribution. In Section 4 we find that the Pitman efficiency of two such tests depends, in general, on the direction in which the origin is approached. The efficiency, in terms of "generalized variance," of the estimates of [2] lies between the maximum and minimum Pitman efficiencies of the corresponding tests of type I (maximum and minimum being taken over direction). This "generalized variance" efficiency is found to be equal to the efficiency of the tests as defined in terms of a criterion of goodness introduced by Isaacson [9] ($D$-optimality). In case the coordinates are identically distributed, if correlation effects alone are taken into account, it is shown that whatever two tests are considered, there always exists a direction in which one improves the other. Section 5 continues with a discussion of the tests based on the estimates introduced in [2] in relation to Hotelling's $T^2$. Their desirable properties and pathologies are found to be similar to those of the parent estimates. The remaining sections deal with the case $p = 2$. Under normality the minimum Pitman efficiency with respect to $T^2$ of the tests mentioned above behaves like the efficiency of the parent estimates with respect to the mean.