Choice of the hypothesis matrix for using the Anova-type-statistic

Initially developed in Brunner et al. (1997) , the Anova-type-statistic (ATS) is one of the most used quadratic forms for testing multivariate hypotheses for a variety of different parameter vectors θ ∈ R d . Tests based on a version of the ATS are usually preferable over those based on other quadratic forms, like the Wald-type-statistic. However, the same null hypothesis H θ = y can be expressed by various hypothesis matrices H ∈ R m × d and corresponding vectors y ∈ R m , yielding different values of the test statistic. Since this can entail differing test decisions, we investigate under which conditions certain tests using different hypothesis matrices coincide. In this manuscript, we show that for several versions of the Anova-type-statistic, for each hypothesis H θ = y a companion matrix with a minimal number of rows can be constructed, testing the same hypothesis but also always yielding the same test decisions. This can substantially reduce computation time, as demonstrated in several conducted simulations.