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

Steven T. Garren

Abstract

Suppose mutually independent observations are drawn from absolutely continuous, unimodal, symmetric distributions with an order restriction on the medians, μ0 ≤ min{μ1,μ2,...,μm}. 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.

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Statistics & Risk Modeling publishes articles that discuss modern methods of statistics and probabilistic modeling and their applications to risk management in finance, insurance, and related areas. It also welcomes papers that present methodological innovations in statistical theory as well as papers on innovative statistical modeling applications and inference in risk management.

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