A General Approach to Some Screening and Classification Problems

Abstract

This paper is concerned with the problem of classifying items which belong to one of several classes. Each item is characterized by a variableYwhich may lie in one of several sets; the actual class to which the item belongs is determined by the set which containsY.We suppose thatYis costly, inconvenient, or impossible to observe, but that an observation X = (X1,…,Xk)is available which is correlated withY.The problem considered is the optimal use of screening variables in the classification problem. We assume that the conditional distribution ofYgiven x is known, and that optimality is to be judged in terms of a given loss function. A simple characterization of minimum risk procedures is given which suggests useful computational methods. These are illustrated by a number of examples for which specific procedures are obtained. The possibility of observingX1,…,Xksequentially is also considered.