Improved estimation of medians subject to order restrictions in unimodal symmetric families

Abstract

Suppose mutually independent observations are drawn from absolutely continuous, unimodal, symmetric distributions with an order restriction on the medians, μ₀ ≤ min{μ₁,μ₂,...,μ_m}. An isotonic regression estimator is shown to stochastically dominate the marginal sample median when estimating μ₀, under some regularity conditions. These conditions allow the tails of the first population (i.e., the population with median μ₀) to be quite heavy, whereas the tails of the remaining distributions are required to be relatively light. Examples involving the Cauchy and Laplace distributions are shown to satisfy these regularity conditions. Counterexamples illustrate the importance of these regularity conditions for proving stochastic domination. The results expressed herein are theoretical advancements in order restricted inference.