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 $$\theta _1$$ θ 1 and $$\theta _2$$ θ 2 ( $$\theta _1\le \theta _2$$ θ 1 ≤ θ 2 ) of a general bivariate distribution under the squared error loss function. Motivated by the fact that the isotonic regression of the unrestricted best location/scale equivariant estimators (BLEE/BSEE) or the isotonic regression of the unrestricted maximum likelihood estimators (MLE), of $$\theta _1$$ θ 1 and $$\theta _2$$ θ 2 may not necessarily dominate the unrestricted BLEE/BSEE or the unrestricted MLE of $$\theta _1$$ θ 1 and $$\theta _2$$ θ 2 , to find improvements over the best location/scale equivariant estimators (BLEE/BSEE) of $$\theta _1$$ θ 1 and $$\theta _2$$ θ 2 , we study isotonic regression of suitably chosen location/scale equivariant estimators (LEE/SEE) of $$\theta _1$$ θ 1 and $$\theta _2$$ θ 2 with general weights. Let $${\mathcal {D}}_{1,\nu }$$ D 1 , ν and $${\mathcal {D}}_{2,\beta }$$ D 2 , β denote suitable classes of isotonic regression estimators of $$\theta _1$$ θ 1 and $$\theta _2$$ θ 2 , respectively. Under the squared error loss function, we characterize admissible estimators within classes $${\mathcal {D}}_{1,\nu }$$ D 1 , ν and $${\mathcal {D}}_{2,\beta }$$ D 2 , β , and identify estimators that dominate the BLEE/BSEE of $$\theta _1$$ θ 1 and $$\theta _2$$ 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 considered to compare the risk performances of various estimators. A real-life data analysis is also provided to demonstrate the utility of the findings.