Estimation of a nonnegative location parameter with unknown scale

TL;DR

This paper develops minimax estimators for a lower-bounded location parameter in symmetric models, extending classical results to asymmetric loss functions and non-normal distributions, with applications in constrained inference.

Contribution

It introduces new conditions and estimators that dominate existing ones for bounded location parameters across various symmetric models, including non-normal and asymmetric loss cases.

Findings

Generalized Bayes estimators are minimax under scale-invariant loss.

New analytical techniques for risk difference and sign change analysis.

Robustness properties of the proposed estimators under different loss functions.

Abstract

For normal canonical models, and more generally a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-normal models under scale invariant $L^p$ loss with $p>0$. We cover many other situations when the loss is asymmetric, and where other generalized Bayes estimators, obtained with different powers of the scale parameter in the prior measure, are proven to be minimax. We rely on various novel representations, sharp sign change analyses, as well as capitalize on Kubokawa's integral expression for risk difference technique. Several other analytical properties are obtained, including a robustness property of the generalized Bayes estimators above when the loss is either scale invariant $L^p$ or asymmetrized versions. Applications include inference in two-sample normal model with order constraints on the means.