Some Unified Results on Isotonic Regression Estimators of Order Restricted Parameters of a General Bivariate Location/Scale Model

TL;DR

This paper investigates isotonic regression estimators for order-restricted parameters in a bivariate model, providing unified theoretical results, dominance properties over classical estimators, and simulation comparisons.

Contribution

It unifies and extends existing results on isotonic regression estimators for order-restricted parameters, characterizing admissibility and dominance in a general bivariate setting.

Findings

Identifies estimators that dominate BLEE/BSEE under squared error loss.

Characterizes admissible isotonic regression estimators within specific classes.

Provides simulation results comparing risk performances of various estimators.

Abstract

We consider component-wise estimation of order restricted location/scale parameters $\theta_1$ and $\theta_2$ ($\theta_1\leq \theta_2$) of a general bivariate distribution under the squared error loss function. To find improvements over the best location/scale equivariant estimators (BLEE/BSEE) of $\theta_1$ and $\theta_2$, we study isotonic regression of suitable location/scale equivariant estimators (LEE/SEE) of $\theta_1$ and $\theta_2$ with general weights. Let $\mathcal{D}_{1,\nu}$ and $\mathcal{D}_{2,\beta}$ denote suitable classes of isotonic regression estimators of $\theta_1$ and $\theta_2$, respectively. Under the squared error loss function, we characterize admissible estimators within classes $\mathcal{D}_{1,\nu}$ and $\mathcal{D}_{2,\beta}$, and identify estimators that dominate the BLEE/BSEE of $\theta_1$ and $\theta_2$. Our study unifies and extends several studies reported in the literature for specific probability distributions having independent marginals. Additionally, some new and interesting results are obtained. A simulation study is also considered to compare the risk performances of various estimators.