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

Abstract

We consider component-wise estimation of order restricted location/scale parameters ${\theta }_1$ and ${\theta }_2$ (${\theta }_1\le {\theta }_2$) 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.