A Unified Technique for Prediction and Optimization of Future Outcomes under Parametric Uncertainty via Pivotal Quantities and Ancillary Statistics

Abstract: Statistical prediction and optimization of future outcomes on the basis of the past and present knowledge represent a fundamental problem of statistics, arising in many contexts and producing varied solutions. In this paper, the novel unified technique of computational intelligence for prediction and optimization of future outcomes in terms of pivots and ancillary statistics under parametric uncertainty is proposed. It is assumed that only the functional form of the underlying distributions is specified, but some or all of its parameters are unspecified. In such cases ancillary statistics and pivotal quantities, whose distribution does not depend on the unknown parameters, are used. Eliminating unknown (nuisance) parameters from a model is universally recognized as a major problem of statistics. A surprisingly large number of elimination methods have been proposed by various writers on the topic. The classical method of elimination of unknown (nuisance) parameters from the model, which is used repeatedly in the large sample theory of statistics, is to replace the unknown (nuisance) parameter by an estimated value. However, this method is not efficient when dealing with small data samples. The novel statistical technique of computational intelligence isolates and eliminates unknown parameters from the underlying model as efficiently as possible. Unlike the Bayesian approach, which is dependent of the choice of priors, the proposed method is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate unknown parameters from the problem and to find the efficient statistical decision rules, which often have smaller risk than any of the well-known decision rules. To illustrate the proposed technique, practical examples are given.