This paper demonstrates that in multivariate statistical classification the decision procedure optimal for multivariate normal distributions, wherein an unknown is assigned on the basis of a comparison of quadratic forms, is in fact fully optimum for much broader classes of distributions. These classes are the multivariate extensions of the Pearson Types II and VII distributions, which can approximate remarkably well a broad spectrum of distributions, varying in shape from the Cauchy to the rectangular, which could arise in real problems. Classification with quadratic discriminant functions, which has already been extensively explored and computer programmed for normal distributions, can then be optimally applied for these wider classes of distributions.