Characterizing statistics and their applications

The paper is concerned with the problem of construction of characterizing statistics for some types of distributions within certain classes. A characterizing statistic for multivariate normal distribution is constructed with the use of the theory of random matrices. Properties and characteristics of spherically symmetric distributions are studied. Examples of statistics that can be used for characterization of symmetric distributions are given. Applications of characterizing statistics to the theory of hypothesis testing are given. The problem of discrimination between the uniform and normal types of multivariate distributions within the group of spherically symmetric distributions from small samples is studied on the basis of the results obtained. A lower estimate is obtained for the number of observations that can serve as a basis for discrimination between these distributions.