The likelihood approach used in this paper leads to quadratic discriminant functions. Classification into one of two known multivariate normal populations with a known and unknown covariance matrix are separately considered, where the two cases depend on the sample size and an unknown squared Mahalanobis distance. Their exact distributions are complicated to obtain. Therefore, moments for the likelihood based discriminant functions are established to express the basic characteristics of respective distribution.

The exact distribution of a classification function is often complicated to allow for easy numerical calculations of misclassification errors. The use of expansions is one way of dealing with this diculty. In this paper, approximate probabilities of misclassification of the maximum likelihood based discriminant function are established via an Edgeworth-type expansion based on the standard normal distribution for discriminating between two multivariate normal populations.

The possibility that a new observation can be allocated to an unknown population is considered. von Rosen and Singull (2022) derived a classi cation rule taking into account this perspective. The classi cation rule consists of two criteria. In this paper, the mean and variance of these criteria needed to discriminate between two growth curves are established.