On the Estimation of a Restricted Location Parameter for Symmetric Distributions

Abstract

For estimating the median θ of a spherically symmetric univariate distribution under squared error loss, when θ is known to be restricted to an interval [−m,m], m known, we derive sufficient conditions for estimators δ to dominate the maximum likelihood estimator δ_mle. Namely: (i) we identify a large class of models where for sufficiently small m, all Bayesian estimators with respect to symmetric about 0 priors supported on [−m,m] dominate δ_mle, and (ii) we provide for Bayesian estimators δ_π sufficient dominance conditions of the form m ≤ c_π, which are applicable to various models and priors π. In terms of the models, applications include Cauchy and Student distributions, densities which are logconvex on (θ,∞) including scale mixtures of Laplace distributions, and logconcave on (θ, ∞) densities with logconvex on (θ,∞) first derivatives such as normal, logistic, Laplace and hyperbolic secant, among others. In terms of priors π which lead to dominating δ_π's in (ii), applications include the uniform density, as well as symmetric densities about 0, which are also absolutely continuous, nondecreasing and logconcave on (0,m).