Comparison of Constrained Bayesian and Classical Methods of Testing Statistical Hypotheses in Sequential Experiments

What is it about?

The article focuses on the discussion of basic approaches to hypotheses testing in sequential experiments, which are Wald and Berger sequential tests and the test based on Constrained Bayesian Method (CBM). The positive and negative aspects of these approaches are considered and demonstrated on the basis of computed examples.

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Why is it important?

The offered CBM method is a more general method of hypotheses testing than the existing classical Fisher’s, Jeffreys’, Neyman’s, Berger’s and Wald’s methods. It has all positive properties of the mentioned methods. Namely, it is a data-dependent measure like Fisher’s test, for making the decision it uses posteriori probabilities like Jeffreys’ test and computes Type I and Type II error probabilities like Neyman–Pearson’s approach. Like the Berger’s methods, it has no-decision-making regions. Moreover, the regions of making decisions have new, more general properties than the same regions in the considered methods. These properties allow us to make more well-founded and reliable decisions. Particularly, do not accept a unique hypothesis or do not accept any hypothesis when the information on the basis of which the decision must be made is not enough for distinguishing the informationally close hypotheses or for choosing a hypothesis among informationally distant ones. Very interesting peculiarity of CBM is the possibility of its use in parallel and sequential experiments without any changes and when it is necessary smoothly transit from parallel to sequential methodology. In despite of Berger’s and Wald’s methods, the sequential test of Bayesian type is universal and without modification can be used for any number of hypotheses and any dimensionality of observation vector. It is simple and very convenient for use and methodologically practically does not depend on the number of tested hypotheses and dimensionality of the observation space. The computed results, presented in the paper, clearly demonstrate high quality of the sequential test of Bayesian type.