A Characterization of Multisample Distribution-Free Statistics

Abstract

It has been observed [1], [3] that in the one-sample case the distribution-free statistics in common usage are both SDF (strongly distribution-free) and of the form $\psi\lbrack F(X_1), \cdots, F(X_n)\rbrack$, where $X_1, \cdots, X_n$ is a random sample and $F$ is the hypothesized cpf (cumulative probability function); and it has been proved [1], [3] that the two properties above are equivalent in the one-sample case. In the multi-sample cases, one observes that except for the statistics of the Pitman conditional tests, which are not SDF, a major portion of distribution-free statistics (e.g. Kolmogorov-Smirnov, Cramer-von Mises, Wald-Wolfowitz, Mosteller-Turkey, Epstein-Rosenbaum, empty cell and the rank-sum types) in common usage have both the SDF and rank properties. Z. W. Birnbaum (in a personal communication in March, 1963) asks whether the SDF property implies the rank property in the two-sample case. An affirmative answer to this question and its converse would be of use both in analyzing and constructing multisample distribution-free statistics. In this paper, it is shown that in the multisample case, the rank property implies the SDF property; and that, except for zero-probability sets, the two properties are equivalent if the $k$-sample statistic $T$ satisfies Scheffe's [5] NB (null boundary) condition. The former result follows from the definitions of the rank and SDF properties. In proving the latter result one first shows that a completeness property of the class of strictly increasing continuous cpfs implies that each SDF, $k$-sample statistic $T$ is AI (almost invariant) in the appropriate sense; and, then, that the NB condition and AI property imply invariance and, hence, the rank property almost everywhere.