A general method for constructing pseudo-Gaussian tests—reducing to traditional Gaussian tests under Gaussian densities but remaining valid under non-Gaussian ones—is proposed. This method provides a solution to several open problems in classical multivariate analysis. One of them is the test of the homogeneity of covariance matrices, an assumption that plays a crucial role in multivariate analysis of variance, under elliptical, and possibly heterokurtic densities with finite fourth-order moments.