A preliminary test in classification and probabilities of misclassification

摘要

Consider f_θ to be a probability density function with parameter θ. A set of k populations can now be defined such that the ith population ∏_i is the set of density functions f_(θ_1~((i))),…,f_(θ_(m_i)~((i))) This paper proposes a test, based on the φ-dissimilarity, of the hypothesis that a new individual from a population ∏_0 with a density function f_(θ_0), belongs to the ith population. The probabilities of misclassification of the minimum φ-dissimilarity classification rule are also obtained. In this paper, it is assumed that the parameters θ_1~((i)),.....,θ_(m_i)~((i)) and may be θ_0 are unknown and must be estimated from a set of training samples. Explicit expressions for the hypothesis test and the probabilities of misclassification are derived for the case where the populations ∏_i consist of homoscedastic normal, as well as for gamma distributions.