On the Expected Stopping Time of the Universal d-Guaranteeing Procedure for Distinguishing Two Hypotheses
D. S. Simushkin, С. В. Симушкин · Lobachevskii Journal of Mathematics · 2023
In the problem of distinguishing between two hypotheses, the stopping time of the universal d-guaranteeing procedure, denoted as $$ u$$ , is defined as the observation number at which the posterior probability of one of the hypotheses becomes greater than a given reliability. This article demonstrates that for a wide range of probabilistic models, $$ u$$ is almost surely finite. The average value of $$ u$$ is analyzed in a special case where $$ u$$ is the first exit time of a random walk beyond two-sided parabolic boundaries. Conditions are identified under which the average value of $$ u$$ is finite or infinite. The problem of distinguishing between hypotheses $$\theta\leq\theta_{0}$$ and $$\theta>\theta_{0}$$ about the mean $$\theta$$ of a normal random variable with a known variance is also examined. In this case, the output parameter $$\theta$$ is a realization of the normal random variable. It is shown that the average value of $$ u$$ is finite only if $$\theta eq\theta_{0}$$ , otherwise it is infinite. Monte Carlo simulation data supports the assumption that the unconditional mean value of $$ u$$ is infinite.