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.

Efficient computation of the stochastic behavior of partial sum processes

“…(2011)). On the other hand, if the probability distribution is a multivariate standard normal or t-distribution with some special correlation structure, an efficient numerical integration may be constructed to compute a rectangular event, say (Dunnett & Sobel (1955), Soong & Hsu (1997)), or an event based on a complete ordering (Kiatsupaibul et al(2017)).…”

“…This is an application of the general discussion of recursive integration given in Hayter (2006) with d = 2. Recursive computational techniques similar to the ones developed in this paper have been applied to the problem of confidence band construction for a distribution function in Kiatsupaibul & Hayter (2015), and to ranked constrained computations in Kiatsupaibul et al (2017). Of course, the probability in equation (1) can be put in this form for continuous random variables with the sets I i equal to the sets A i and the functions h i (x i ) equal to the probability density functions f i (x i ).…”

“…This is an application of the general discussion of recursive integration given in Hayter (2006) with d = 2. Recursive computational techniques similar to the ones developed in this paper have been applied to the problem of confidence band construction for a distribution function in Kiatsupaibul & Hayter (2015), and to ranked constrained computations in Kiatsupaibul et al (2017).…”

A Statistical Model with Qualitative Input

“…This specific problem is proved to be quite challenging for an MCMC. However, Kiatsupaibul et al (2017) proposes a solution method based on the recursive integration technique (Hayter, 2006) that can be adapted to solve this specific problem in O(n 2 ). In this paper, we propose an adapted solution method based on that of Kiatsupaibul et al (2017) to solve the problem.…”

“…In Section 3, the finite dimensional properties of the model are explored through the proposed recursive integration technique. In this section, the recursive integration technique adapted from Kiatsupaibul et al (2017) is also described. In Section 4, the properties of the model when applied to a portfolio selection problem is investigated.…”