Two LDF characterizations of the normal as a spherical distribution

AbstractTwo optimal characteristic properties of the normal distribution are shown: (a) Of all the SNM (spherical scale normal mixtures) the normal with the same Mahalanobis distances between Πi:SNM(μi) and Πj:SNM(μj), i ≠ j, maximizes the probabilities of correct classification determined by a certain subclass of the LDF classification rules; (b) The class of LDF (linear discriminant function) rules is the admissible class for the discrimination problem with spherical population alternatives iff the spherical distribution is normal