Robustness properties of the Pitman-Morgan test

The classical Pitman-Morgan test is known to be optimal for testing equality of the variances of components of a bivariate normal vector. We first show that it is also optimal for a generalized model involving the matrix spherical distribution. Then we discuss and demonstrate, both analytically and empirically, that it is nonrobust, i.e., its type I error control is inexact both asymptotically and in moderate size bivariate random samples.