Moment-Generating Operators for Determinants of Product Moments in Samples from a Normal System

A considerable number of commonly used statistical functions, when generalized for samples from a normal multivariate population, can be expressed in terms of the determinants and their principal minors of matrices of second order moments in the sample. In a recent paper the author2 considered generalizations of several statistical functions and found their moments from Wishart's3 generalized product moment distribution by a method of variation of the sample number. Sampling distributions of such functions were found by solving the resulting moment equations, and the uniqueness of the solutions established by the theory of closure. In the present paper operational methods will be developed for finding the moments of a more general class of statistical functions directly from the probability law of the sample. The generality lies in the fact that some of the variates are treated as independent or fixed, that is, they are not subject to sampling variations. Hence the results will extend to least-square regression problems in which values of the independent variates are known without error. They are also applicable, as will be seen in the case of the multiple correlation coefficient, to problems in which an arbitrary number of the independent variates are subject to sampling variation. We shall be primarily interested in deriving a method of obtaining general moment formulas by establishing the validity of certain operations on the population parameters involved in the probability law of the sample. However, several examples will be given in the section on Applications to show how the operators provide an effective analytical method of solving some of the more complicated sampling problems; e.g. that of the multiple correlation coefficient, some of the generalized Neyman-Pearson X criteria, and the generalized Student ratio. All of the results of the author's paper cited above can be obtained at once by means of the operators.