Prediction with a generalized cost of error function

Abstract

Classical prediction theory limits itself to quadratic cost functions, and hence least-square predictors. However, the cost functions that arise in practice in economics and management situations are not likely to be quadratic in form, and frequently will be non-symmetric. It is the object of this paper to throw light on prediction in such situations and to suggest some practical implications. It is suggested that a useful, although suboptimal, manner of taking into account generalized cost functions is to add a constant bias term to the predictor. Two theorems are proved showing that under fairly general conditions the bias term can be taken to be zero when one uses a symmetric cost function. If the cost function is a non-symmetric linear function, an expression for the bias can be simply obtained. INTRODUCTION Suppose that one predicts some stochastic process and that it is subsequently found that an error of size x has been made. With such an error one can usually determine the cost of having made the error and the amount of this cost will usually increase as the magnitude of the error increases. Let g(x) represent the cost of error function. In both the classical theory of statistical prediction and in practice, this function is usually taken to be of the form g(x) = cx 2 , so that least-squares predictors are considered. However, in the fields of economics and management an assumption that the cost of error function is proportional to x 2 is not particularly realistic.