Том 228, № 5 (2018)

To make calculations in the Bayesian analysis, the formalism of which is based on the layering of a probability measure defined on the product of measurable spaces, it is useful to have a summary of the properties of this layering. In this paper we formulate and prove those of them that are used in calculations more often than others. Particularly, we prove Hoeffding-type inequalities using direct elementary techniques.

We consider the problem of restoring a nonparametric dependence described by the sum of linear trend and seasonal component, i.e., a periodic function with the known period. We obtain the asymptotic distribution of the parameter estimates and the trend component. We find the mathematical expectation of the residual sum of squares. We also develop the methods of estimation of the seasonal component and construction of the interval forecast.

We study the properties of tests constructed by a simple modification of the studentized range of the sample from a symmetric stable population in the problem of testing the hypothesis \( \mathrm{\mathscr{H}} \)_α (the stability index equals α, α ∈ (1, 2)) against the alternative \( \mathrm{\mathscr{H}} \)₂. We obtain approximate formulas for the calculation of critical values and estimation of the test power and develop a method for the estimation of the accuracy of these approximations. The major part of the paper deals with the construction of the approximations to the distribution function of the normalized sum of squares of symmetric stable random variables and estimation of the accuracy of approximations.

We compare two sequential d-guaranteed tests and an optimal d-guaranteed test based on a fixed number of observations with respect to the average number of observations within the most accepted practical applications of Bayesian probabilistic models. We consider the sequential “first skipping” test, the sequential locally efficient test based on the score statistic, and the test based on a fixed number of observations that minimizes the necessary sample size. We study various characteristics of these tests connected with the number of observations within three probabilistic models, namely, the normal (ϑ, σ²) distribution of the observed random variable and the normal a priori distribution of ϑ with fixed σ²; the exponential distribution with the intensity parameter ϑ and the a priori gamma distribution of ϑ; and the Bernoulli sampling with the success probability ϑ and the a priori beta distribution of ϑ. We discuss the connection of the d-posterior approach with the compound decision problem as applied to the analysis of data provided by microchips (when the false discovery rate, or FDR, for short, is treated as the d-risk of the first kind). We present the vast data on characteristics of the mentioned tests obtained by the method of mathematical modeling in several tables. We discuss unsolved problems of the d-guaranteed discrimination of hypotheses with the minimal number of observations and approaches to their solution.

This paper continues the previous work of the author and contains the results concerning the relations for a wide class of special functions related to multidimensional Poisson and polynomial random walks.

Nonstationary countable Markov chains with continuous time and absorption at zero are considered. We study the convergence rate to the limit mode. As examples, we consider simple nonstationary random walks.