On the Joint Distribution of Friedman's $\chi_r^2$ Statistics

This study is concerned with the joint distribution of Gerig's (1969) statistics when applied to tests for shift in various marginal distributions pertaining to complete two-way multivariate data. The exact small-sample distribution can be found using conditional permutation arguments, and the limiting permutation distribution is shown to belong to a known class of multivariate chi-square distributions. A special case yields the limiting joint distribution of Friedman's (1937) $_{\chi r^2}$ statistics for the one-dimensional marginal distributions. Berry-Esseen bounds are given for the rate of convergence of the joint distribution to its limiting form when the underlying distributions are identical over replications.