Vol 227, No 2 (2017)

In this paper we consider an informative model of competing risks. We offer a new power estimate for the distribution function and prove its asymptotic equivalence to the previously known estimates and its efficiency compared to the well-known multiplicative Kaplan–Meier estimate.

The paper contains an introduction to the asymptotic theory of hypothesis testing and a review of recent results of the author and his students. We consider only the asymptotic approach, for which with increasing sample size n the test size is separated from zero, and the sequences of local alternatives, for which the power is separated from one. This paper focuses on asymptotically efficient tests when testing a simple hypothesis in the case of a one-parameter family. We study the difference between the powers of the best and asymptotically efficient tests. This difference is closely related to the notion of asymptotic test deficiency. We consider the formula for the limiting deviation of the power of asymptotically optimal test from the power of the best test in the case of Laplace distribution. Due to the irregularity of the Laplace distribution, this deviation is of order n^−1/2, in contrast to the usual regular families for which this order is n⁻¹. We also study the Bayesian settings and the case of increasing parameter dimension.

The empirical Bayesian approach is considered for Bayesian stable families of prior parameter distributions. It is shown that in these cases the derivation of the estimate for the Bayesian decision function is considerably simplified. We consider a generalization of the Deely–Lindley’s Bayesian approach to the problem of empirical Bayesian estimation, and present examples of application of the obtained results.

In this paper we propose the median modifications of the EM-algorithm and demonstrate their advantages in comparison with conventional methods by the example dealing with the numerical solution of the problem of decomposition of the volatility of financial indexes. We provide examples of volatility decompositions for AMEX, CAC 40, NIKKEI, and NASDAQ indexes.

In this paper we present an R-mixing test for the sequence of random variables, introduced by A. Rényi, and prove a certain property of R-mixed sequences. It is proved that the statistics of homogeneity test has the R-mixing property with the limit chi-square distribution with k − 1 degrees of freedom. This result strengthens a well-known result on the asymptotic chi-square distribution of the statistics of homogeneity test.

We prove a limit theorem for the joint distribution of the maximum of the initial sample and the maximum of randomly decimated arbitrary stationary random sequence with a trend. The results are illustrated by a numerical example.