Lin'kov Yu. N.

We obtain an asymptotic decomposition of the logarithm of the likelihood ratio for counting processes in the case of similar hypotheses. We establish the properties of the normalized likelihood ratio in the problem of estimation of an unknown parameter.

We consider the problem of discrimination of a finite number of simple hypotheses in the general scheme of statistical experiments. Under conditions of the validity of theorems on large deviations for the logarithm of likelihood ratio, we investigate the asymptotic behavior of probabilities of errors of the Bayes criterion. We obtain the asymptotics of the amount of Shannon information contained in an observation and in the Bayes criterion.

We consider semimartingales with deterministic discontinuous triplets. We obtain properties of the like-lihood ratio for the parametric case in terms of the Hellinger processes.

We study the behavior of the probability of errors of the Neumann-Pearson criterion under various null and alternative hypotheses by the results of observations of autoregressive processes.

Limit theorems on large deviations of the logarithm of the likelihood ratio are proved for the problem of distinguishing two simple hypotheses in the general scheme of statistical experiments under the null hypothesis and under an alternative hypothesis. The theorems obtained are applied to the investigation of a decrease in the probability of errors of the first and second kind for the Neumann-Pearson criterion.

We prove the general limit theorem on probability of large deviations of the logarithm of the likelihood ratio with the null hypothesis and alternative. Weaker versions of the principle of large deviations are obtained in predictable terms for the problem of distinguishing the counting processes. The case of counting processes with deterministic compensators is studied.

A canonical representation is obtained for the logarithm of the likelihood ratio. Limit theorems describing its asymptotic behavior are proved. Using these theorems, we study the rate of decrease of the probability of an error of the second-kind in the Neyman-Pearson test.