The kurtosis coefficient and the linear discriminant function

In this note we analyze the relationship between the direction obtained from the minimization of the kurtosis coefficient of the projections of a mixture of multivariate normal distributions and the linear discriminant function. We show that both directions are closely related and, in particular, that given two vector random variables having symmetric distributions with unknown means and the same covariance matrix, the direction which minimizes the kurtosis coefficient of the projection is the linear discriminant function. This result provides a way to compute the discriminant function between two normal populations in the case in which means and common covariance matrix are unknown.