On improved estimation of the larger location parameter

TL;DR

This paper develops improved estimators for the larger location parameter of two distributions, demonstrating their superiority over natural estimators through theoretical proofs, simulations, and real data applications.

Contribution

It introduces new dominating estimators for the larger location parameter, extending existing methods using decision theory and the Kubokawa's IERD approach.

Findings

The natural estimator is inadmissible under various criteria.

Proposed estimators outperform the natural estimator in risk.

Explicit formulas for improved estimators are provided.

Abstract

This paper investigates the problem of estimating the larger location parameter of two general location families from a decision-theoretic perspective. In this estimation problem, we use the criteria of minimizing the risk function and the Pitman closeness under a general bowl-shaped loss function. Inadmissibility of a general location and equivariant estimators is provided. We prove that a natural estimator (analogue of the BLEE of unordered location parameters) is inadmissible, under certain conditions on underlying densities, and propose a dominating estimator. We also derive a class of improved estimators using the Kubokawa's IERD approach and observe that the boundary estimator of this class is the Brewster-Zidek type estimator. Additionally, under the generalized Pitman criterion, we show that the natural estimator is inadmissible and obtain improved estimators. The results are implemented for different loss functions, and explicit expressions for the dominating estimators are provided. We explore the applications of these results to for exponential and normal distribution under specified loss functions. A simulation is also conducted to compare the risk performance of the proposed estimators. Finally, we present a real-life data analysis to illustrate the practical applications of the paper's findings.