Maximum Likelihood Estimation of the Distributions of Two Stochastically Ordered Random Variables

Abstract

Abstract The maximum likelihood estimates and Ĝ of two distribution functions F and G are found, subject to the restrictions that for all x and that and Ĝ are of the discrete type. Random samples x 1, ···, x m and y 1, ···, y n are taken from the respective distributions and these m+n values are ordered according to magnitude, with x's before y's in cases of equality. These n+m values are divided into a number of “strings” as follows. The first string ends after the y value that is determined so that the ratio of the number n 1 of y values in that string to the number m 1 of x values there is a maximum. If this string does not include all n+m values, then (excluding the first string) the second string and the numbers n 2 and m 2 are determined in a similar manner. This can be continued to construct a (possible) third string, and so on. Let h 1(x i ) and h 2(y i ) be the respective frequencies of x i and yi. If x i and y i are in the k th string, say, then the maximum likelihood estimates assign the following probabilities to x i and and ĝ(y j ) = h 2(y j )(m k +n k )/n k (m+n). These assignments provide consistent estimators of F and G when F ≥ G.