A Note On Simultaneous Estimation of Order Restricted Location Parameters of a General Bivariate Symmetric Distribution Under a General Loss Function

TL;DR

This paper develops a unified approach for estimating order-restricted location parameters in bivariate symmetric distributions using a general loss function, introducing improved estimators that are robust and validated through simulations and real data.

Contribution

It unifies previous results by considering a general bivariate symmetric model and loss function, deriving robust improved estimators using Stein and Kubokawa techniques.

Findings

Improved estimators outperform traditional ones under general conditions

The Stein type estimator is robust across various distributions and loss functions

Simulation and real data validate the effectiveness of the proposed estimators

Abstract

The problem of simultaneous estimation of order restricted location parameters $\theta_1$ and $\theta_2$ ($-\infty<\theta_1\leq \theta_2<\infty$) of a bivariate location symmetric distribution, under a general loss function, is being considered. In the literature, many authors have studied this problem for specific probability models and specific loss functions. In this paper, we unify these results by considering a general bivariate symmetric model and a quite general loss function. We use the Stein and the Kubokawa (or IERD) techniques to derive improved estimators over any location equivariant estimator under a general loss function. We see that the improved Stein type estimator is robust with respect to the choice of a bivariate symmetric distribution and the loss function, as it only requires the loss function to satisfy some generic conditions. A simulation study is carried out to validate the findings of the paper. A real-life data analysis is also provided.