On an Application of the Student Distribution in the Theory of Probability and Mathematical Statistics

This paper deals with the mathematical basis of the possibility of using a Student distribution in problems of descriptive statistics. We especially separate the case where the Student distribution parameter (``the number of degrees of freedom") is small. We show that the Student distribution with arbitrary ``number of degrees of freedom" can be obtained as the limit when the sample size is random. We emphasize the possibility of using a family of Student distributions as a comfortable model with heavy tails since in this case many relations, in particular, a likelihood function, have the explicit form (unlike stable laws). As an illustration of the possibilities of statistical analysis based on the family of Student distributions, we consider a problem of statistical estimation of the center of the Student distribution under the assumption that the parameter of the form (the number of degrees of freedom) is known. We consider equivariant estimators of the center of the Student distribution based on order statistics, M-estimators, and maximum likelihood estimators, calculate their asymptotic relative efficiency, and study the behavior of the Student distribution when ``the number of degrees of freedom" tends to zero.