Isotonic Regression Estimators For Simultaneous Estimation of Order Restricted Location/Scale Parameters of a Bivariate Distribution: A Unified Study

TL;DR

This paper develops isotonic regression estimators for jointly estimating ordered bivariate location or scale parameters, characterizes their admissibility, and demonstrates their advantages through theoretical analysis and simulations.

Contribution

It introduces a unified framework for isotonic regression estimators of ordered parameters, extending existing results and providing new admissibility and dominance properties.

Findings

Identifies admissible estimators within the class of isotonic regression estimators.

Shows certain estimators dominate best location/scale equivariant estimators.

Validates theoretical results with simulation studies.

Abstract

The problem of simultaneous estimation of location/scale parameters $\theta_1$ and $\theta_2$ of a general bivariate location/scale model, when the ordering between the parameters is known apriori (say, $\theta_1\leq \theta_2$), has been considered. We consider isotonic regression estimators based on the best location/scale equivariant estimators (BLEEs/BSEEs) of $\theta_1$ and $\theta_2$ with general weight functions. Let $\mathcal{D}$ denote the corresponding class of isotonic regression estimators of $(\theta_1,\theta_2)$. Under the sum of the weighted squared error loss function, we characterize admissible estimators within the class $\mathcal{D}$, and identify estimators that dominate the BLEE/BSEE of ($\theta_1$,$\theta_2$). Our study unifies several studies reported in the literature for specific probability distributions having independent marginals. We also report a generalized version of the Katz (1963) result on the inadmissibility of certain estimators under a loss function that is weighted sum of general loss functions for component problems. A simulation study is also carried out to validate the findings of the paper.