Rank constrained distribution and moment computations

Abstract

Consider a set of independent random variables with specified distributions or a set of multivariate normal random variables with a product correlation structure. This paper shows how the distributions and moments of these random variables can be calculated conditional on a specified ranking of their values. This can be useful when the ordering of the variables can be determined without observing the actual values of the variables, as in ranked set sampling, for example. Thus, prior information on the distributions and moments from their individual specified distributions can be updated to provide improved posterior information using the known ranking. While these calculations ostensibly involve high dimensional integral expressions, it is shown how the previously developed general recursive integration methodology can be applied to this problem so that they can be evaluated in a straightforward manner as a series of one-dimensional or two-dimensional integral calculations. Furthermore, the proposed methodology possesses a self-correction mechanism in the computation that prevents any serious growth of the errors. Examples illustrate how different kinds of ranking information affect the distributions, expectations, variances, and covariances of the variables, and how they can be employed to solve a decision making problem.