Statistical Bayesian classifiers of normal populations and their comparative analysis

In the present paper, direct and substitutional Bayesian classification rules for normal populations are described and their properties are compared. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III     

On optimal estimators of shift population density functions and conditions of their consistency

A new technique of optimal estimation of density functions for exponential shift families on a homogeneous space of a Lie group is proposed. In contrast to traditional methods, the approach considered is essentially based on the algebraic properties of shift families. Here we give a universal formula for consistent estimators of density functions covering different classes of estimators such as unbiased estimators with uniformly minimum variance and Bayesian estimators under two popular loss functions. The representations of some maximal invariant density functions are derived and simultaneously a close connection between the estimators and these density functions is established. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.     

On the rate of strong convergence for convolutions

An estimate of the rate of strong convergence of convolutions of probability measures is given under the assumption that the components of each convolution converge weakly and in total variation, respectively. Reported at the XVI Seminar on Stability Problems for Stochastic Models, Eger, Hungary, 29 August–3 September 1994. Received by the Editorial Board 15 January, 1996. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

Completeness conditions for sufficient statistics in shift families and unification of estimation procedures

We show that the consideration of algebraic properties of exponential shift families allows one to simplify and unify the procedures of determination of the distribution of a sufficient statistic. We also present a new method for the estimation of a density function based on the determination of the distribution of a sufficient statistic. Reported at the XVI Seminar on Stability Problems for Stochastic Models, Eger, Hungary, 29 August – 3 September 1994. Received by the Editorial Board 1 December, 1994. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part II.     

Statistical rules of group classification of bivariate exponential populations

We consider maximum likelihood estimators and unbiased estimators for bivariate exponential distributions under various a priori hypotheses about the parameters. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Estimation of parameters for a scheme of random allocation

Unbiased estimators with minimum variance are considered for a scheme of random allocation. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

Asymptotics of the necessary sample size under small error probabilities

The asymptotics of the necessary sample size is considered in testing close hypotheses when the error probabilities vanish. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.     

The asymptotics of multidimensional infinitely divisible distributions

The present article deals with the asymptotics at infinity of multidimensional infinitely divisible distributions with the support in a cone. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part III.     

Lower and upper bounds for characteristic functions

In the paper, several known inequalities for characteristic functions are sharpened and several new ones are derived. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.     

Asymptotic analysis of an individual risk model with random insurance premiums

The factorization model of an individual claim is introduced to investiate an insurer's surplus within a general statistical individual risk model. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Statistical problems of the neyman type and decomposable statistics

We consider estimators of distribution parameters, obtained by the moment method and the maximum likelihood method from a sample of specially dependent random variables; namely, we assume that the sample forms a scheme of decomposable statistics. Supported by the Russian Foundation for Fundamental Research (grant No. 94-01-01273) and by the Grant Center on Mathematical Research of the Novosibirsk State University. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996. Part II.     

Limit distributions of random symmetrical polynomials

Conditions are studied which should be imposed on the coefficients of a homogeneous random polynomial of the fourth degree to provide its convergence to some nondegenerate random variable. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

Explicit analytical expressions for the densities of stable laws

Explicit representations are given for the densities of strictly stable laws with α=2/3 and α=3/2 for all admissible values of the parameter ϑ. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part III.     

Transition phenomena for a Galton-Watson process with immigration in a Markovian environment

The behavior of a Galton-Watson process with state-dependent immigration in a Markovian random environment is investigated. We obtain a limit theorem in the near-critical case. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

On stochastic processes with linear conditional moments

In the paper, we show that a stochastic process with linear regression and linear conditional variance must be either a Gaussian process or a Poisson-type process. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Planar random evolutions governed by hyperbolic equations

Planar random evolutions governed by hyperbolic equations are considered. Two cases are presented in detail: motion with three directions and the model with four perpendicular directions. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part I.     

Multiparameter Shimizu equation

An analog of the Shimizu equation is considered, and the set of solutions is described for the case of diagonal matrices. The result is applied to one characterization problem. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995. Part III     

On reconstruction of distributions by the translated moments of linear statistics

The problem of reconstruction of a multivariate distribution by means of translated moments is considered. Supported in part by the Russian Foundation for Fundamental Research (project code # 95-011-01260) and the International Science Foundation. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

Maximal size of the Student and the ANOVA tests under exactly one contaminant

In the paper, the problem of how much a single contaminant can increase the size of the Student and theANOVA test is discussed. Exact upper bounds for the size of the tests are given. Numerical results are presented. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Stability ofL 1-approximations and robustness of least module estimates

The paper is concerned with the robustness properties of least module (LM) estimates of regression parameters depending on the stability properties of L₁-norm approximations. The conditions of stability for L₁-approximations are obtained in linear, nonlinear, continuous, and discrete cases. The breakdown points of LM estimates and their upper bounds are derived in the linear case of approximation by algebraic and trigonometric polynomials, and in the nonlinear case of approximation by exponentials. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.