Discriminant function between composite hypotheses and related problems

Abstract

The paper deals with the problem of constructing discriminant functions when the alternative hypotheses are not simple but composite. Such a problem arises when it is intended to identify an individual as belonging to one of two sets, each set consisting of several populations mixed in unknown proportions. A general approach to this problem, using the concepts of decision theory, sufficient statistics and ancillary statistics is given. In particular, when the means of the alternative popublations within a given set are linearly related and the distributions are p variate normal, the discriminant function comes out in a simpler form. It is linear when the dispersion matrices are the same for all the populations and quadratic when the dispersion matrices within sets differ. Methods of estimating the discriminant function from sample date are fully disussed. The fact that in the situations considered one has observations from a mixture of populations within a set does not create any difficulty.