Multivariate hypothesis testing using generalized and {2}-inverses – with applications

The use of generalized inverses in Wald's-type quadratic forms of test statistics having singular normal limiting distributions does not guarantee to obtain chi-square limiting distributions. In this article, the use of {2} -inverses for that problem is investigated. Alternatively, Imhof-based test statistics can also be defined, which converge in distribution to weighted sum of chi-square variables. The asymptotic distributions of these test statistics under the null and alternative hypotheses are discussed. Under fixed and local alternatives, the asymptotic powers are compared theoretically. Simulation studies are also performed to compare the exact powers of the test statistics in finite samples. A data analysis on the temperature and precipitation variability in the European Alps illustrates the proposed methods.