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

The problem of simultaneous estimation of location/scale parameters $θ_1$ and $θ_2$ of a general bivariate location/scale model, when the ordering between the parameters is known apriori (say, $θ_1\leq θ_2$), has been considered. We consider isotonic regression estimators based on the best location/scale equivariant estimators (BLEEs/BSEEs) of $θ_1$ and $θ_2$ with general weight functions. Let $\mathcal{D}$ denote the corresponding class of isotonic regression estimators of $(θ_1,θ_2)$. 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 ($θ_1$,$θ_2$). 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.

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