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

We consider component-wise estimation of order restricted location/scale parameters $θ_1$ and $θ_2$ ($θ_1\leq θ_2$) of a general bivariate distribution under the squared error loss function. To find improvements over the best location/scale equivariant estimators (BLEE/BSEE) of $θ_1$ and $θ_2$, we study isotonic regression of suitable location/scale equivariant estimators (LEE/SEE) of $θ_1$ and $θ_2$ with general weights. Let $\mathcal{D}_{1,ν}$ and $\mathcal{D}_{2,β}$ denote suitable classes of isotonic regression estimators of $θ_1$ and $θ_2$, respectively. Under the squared error loss function, we characterize admissible estimators within classes $\mathcal{D}_{1,ν}$ and $\mathcal{D}_{2,β}$, and identify estimators that dominate the BLEE/BSEE of $θ_1$ and $θ_2$. 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.

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