A New Technique of Invariant Statistical Embedding and Averaging Via Pivotal Quantities for Intelligent Constructing Efficient Statistical Decisions under Parametric Uncertainty

Abstract: In the present paper, a new technique of invariant embedding of sample statistics in a decision criterion (performance index) and averaging this criterion via pivotal quantities is proposed for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, the technique of invariant statistical embedding and averaging via pivotal quantities (ISE&APQ) 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. The aim of the present paper is to show how the technique of ISE&APQ may be employed in the particular case of optimization, estimation, or improvement of statistical decisions under parametric uncertainty. To illustrate the proposed technique of ISE&APQ, application examples are given.