On Estimation of Parameters of Distributions within the Scheme of Decomposable Statistics

Abstract

In the present paper, we consider the maximum likelihood estimators of parameters of the distributions of random variables which are dependent in a certain sense. Namely, we assume that the sample forms a scheme of decomposable statistics (DS’s). The latter means that there exist independent random variables whose joint conditional distribution given their sum is fixed at some “point” coincides with the joint distribution of the sample mentioned above. DS’s form a very wide class of dependent random variables appearing both in various sections of mathematics and in other fields of science (see [2, 5, 6, 9, 16, 17] and the references therein). DS’s are most widely used in problems of mathematical statistics: in the construction of goodness-of-fit and homogeneity tests (see, e.g., [7, 8]) and in survey sampling of finite populations. DS’s automatically emerge when Neyman-type tests (see [11]) adjusted for testing hypotheses concerning the informative parameters are constructed in problems with sufficient statistics with respect to the nuisance parameter (see [16, 18–20]). The maximum likelihood method proposed by Fisher is usually used for the estimation of parameters of distributions in the classical situation, where the sample terms are jointly independent and are identically distributed with some distribution function assumed known up to a parameter. There are only a few cases (see, e.g., [15]) in which the sample terms are assumed dependent in some sense. So far as is known to the authors, the problem considered in the present paper is new and the results obtained are general, like those in the classical situation.