Development of modern models and methods of the theory of statistical hypothesis testing

Abstract

Typical problems of the theory of statistical hypothesis testing are considered. All these problems belong to the same object area and are formulated in a single system of axioms and assumptions using a common linguistic thesaurus. However, different approaches are used to solve each of these problems and a unique solution method is developed. In this regard, the work proposes a unified methodological approach for formulating and solving these problems. The mathematical basis of the approach is the theory of continuous linear programming (CLP), which generalizes the known mathematical apparatus of linear programming for the continuous case. The mathematical apparatus of CLP allows passing from a two-point description of the solution of the problem in the form {0; 1} to a continuous one on the segment [0; 1]. Theorems justifying the solution of problems in terms of CLP are proved. The problems of testing a simple hypothesis against several equivalent or unequal alternatives are considered. To solve all these problems, a continuous function is introduced that specifies a randomized decision rule leading to continuous linear programming models. As a result, it becomes possible to expand the range of analytically solved problems of the theory of statistical hypothesis testing. In particular, the problem of making a decision based on the maximum power criterion with a fixed type I error probability, with a constraint on the average risk, the problem of testing a simple hypothesis against several alternatives for given type II error probabilities. The method for solving problems of statistical hypothesis testing for the case when more than one observed controlled parameter is used to identify the state is proposed.