The structure of isotonic regression class for LAD problems with quasi-order constraints

Abstract

In this paper, weighted least absolute deviation problems (LAD) with order restrictions on a quasi-ordered finite set are considered. A well known example of these problems is to find maximum likelihood estimates for the location parameters of bilateral exponential distributions under order restrictions. It is shown that the class of all isotonic regressions for a LAD problem is a complete lattice. All these isotonic regressions are bounded between two functions, termed the least and the greatest isotonic regressions. Furthermore, there exists a partition of the index set such that all these isotonic regressions are constants on each set of the partition. A technique to find the partition is developed. It is also proved in this paper that the class of all these isotonic regressions is a compact convex set and the extreme points of this set are identified. A numerical example is given to illustrate the results in this paper.