Unique Optimal Partitions of Distributions and Connections to Hazard Rates and Stochastic Ordering

Optimal partitioning of a distribution arises in many contexts, including quantization in information theory, piecewise constant approximation of a function, stratified sampling, goodness-of-fit tests, principal points and clustering, and selective assembly in manufacturing. This article studies the behavior of optimal partitions, develops conditions under which the optimal partitioning of a distribution is unique, and establishes connections to hazard rate and likelihood ratio orderings of the distribution. An earlier proof which gives a slightly weaker condition than the sufficient condition in this article is shown to be incorrect by means of a counter-example. Optimal partitioning is compared with some heuristic partitioning strategies that are commonly used in applications and is shown to lead to substantial improvements in efficiency.