Componentwise Equivariant Estimation of Order Restricted Location and Scale Parameters In Bivariate Models: A Unified Study

TL;DR

This paper provides a unified framework for estimating ordered location and scale parameters in bivariate models, extending previous results and comparing estimator performances through simulations.

Contribution

It unifies existing results on order-restricted estimation for bivariate models under general loss functions and applies these to specific distributions like bivariate normal and gamma.

Findings

Unified estimation approach for bivariate models with order restrictions.

Simulation results show estimator performance varies with distribution and loss function.

Applications demonstrate the practical relevance of the theoretical results.

Abstract

The problem of estimating location (scale) parameters $\theta_1$ and $\theta_2$ of two distributions when the ordering between them is known apriori (say, $\theta_1\leq \theta_2$) has been extensively studied in the literature. Many of these studies are centered around deriving estimators that dominate the best location (scale) equivariant estimators, for the unrestricted case, by exploiting the prior information that $\theta_1 \leq \theta_2$. Several of these studies consider specific distributions such that the associated random variables are statistically independent. This paper considers a general bivariate model and general loss function and unifies various results proved in the literature. We also consider applications of these results to a bivariate normal and a Cheriyan and Ramabhadran's bivariate gamma model. A simulation study is also considered to compare the risk performances of various estimators under bivariate normal and Cheriyan and Ramabhadran's bivariate gamma models.