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 $$\theta _1$$ θ 1 and $$\theta _2$$ θ 2 of a general bivariate location/scale model, when the ordering between the parameters is known a priori (say, $$\theta _1\le \theta _2$$ θ 1 ≤ θ 2 ), has been considered. We consider isotonic regression estimators based on the best location/scale equivariant estimators (BLEEs/BSEEs) of $$\theta _1$$ θ 1 and $$\theta _2$$ θ 2 with general weight functions. Let $$\mathcal {D}$$ D denote the corresponding class of isotonic regression estimators of $$(\theta _1,\theta _2)$$ ( θ 1 , θ 2 ) . Under the sum of the weighted squared error loss function, we characterize admissible estimators within the class $$\mathcal {D}$$ D , and identify estimators that dominate the BLEE/BSEE of ( $$\theta _1$$ θ 1 , $$\theta _2$$ 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 (Ann Math Stat 34:967–972, 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 carried out to validate the findings of the paper, and to compare different competing estimators.