Classification of an observation into linear manifolds,their closed subsets and closed convex subsets

In this paper, we generalize the notion of classification of an observation (sample), into one of the given n populations to the case where some or all of the populations into which the new observation is to be classified may be new but related in a simple way to the given n populations. The discussion is in the frame-work of the given set of observations obeying the usual multivariate general linear hypothesis model. The set ofpopulations into which the new observation may be classified could be linear manifolds of the parameter space or their closed subsets or closed convex subsets or a combination of them or simply t subsets of the parameter space each of which has a finite number of elements. In the last case alikelihood ratio procedure can be obtained easily. Classification procedures given here are based on Mahalanobis distance. Bonferroni lower bound estimate of the probability of correctly classifying an observation is given for the case when the covariance matrix is known or is estimated from a large sample. A numerical example relating to the classification procedures suggested her is given.