Asymptotic theory of weighted F-statistics based on ranks

Using subspaces to describe the nonparametric null hypotheses introduced in Akritas and Arnold [Fully nonparametric hypotheses for factorial designs I: multivariate repeated measures designs, J. Amer. Statist. Assoc. 89 (1994), pp. 336-343.], leads to a natural extension of the models and the class of nonparametric hypotheses considered there. Nonparametric versions of all saturated or unsaturated parametric models for factorial designs, as well as nonparametric versions of all parametric hypotheses considered in such contexts, are included in the new formulation. To test these new hypotheses we introduce a new family of (mid-)rank statistics. The new statistics are modelled after the weighted F-statistics and are appropriate for (possibly) unbalanced designs with independent observations that can be heteroscedastic. Being rank versions of likelihood ratio statistics, the proposed statistics apply in situations where the Wald-type rank statistics of Akritas, Arnold and Brunner [Nonparametric hypotheses and rank statistics for unbalanced factorial designs, J. Amer. Statist. Assoc. 92 (1997), pp. 258-265.] have not been extended and are at least as efficient in the cases where both apply. We show that the new rank statistics converge in distribution to central chi-squared distributions under their respective null hypotheses. Finding the asymptotic distribution of statistics without a closed form expression requires a novel approach, which we introduce. A simulation study compares the achieved level and power of the new statistics with a number of competing procedures.