Contributions to the $k$-Sample Problem: A Symmetric Statistic

Abstract

The $k$-sample problem is studied. Given a set of independent observations from $k$ populations, the pooled sample is ordered, and certain predetermined order statistics are chosen to form the endpoints of random intervals. A random vector is formed, whose components represents the number of observations from each sample in each of these intervals. The exact and limiting distributions of this vector are derived, both under the null-hypothesis of a single underlying distribution and under the alternative of unequal distributions for the $k$ populations. This leads to the definition of a test-statistic for the null-hypothesis, and its limiting null-distribution and limiting distribution under a suitable sequence of alternative hypotheses are obtained. Hence a consistent test of the null-hypothesis is determined and shown to be an extension of the Mood-Brown test. Asymptotic efficiencies are calculated for this test relative to a family of common tests for the $k$-sample problem. To this end, a method is suggested for comparing efficiencies of tests whose limiting distributions under an alternative sequence are non-central chi-square, with unequal degrees of freedom. Some consideration is given to the problem of design for high relative efficiency.