Componentwise Equivariant Estimation of Order Restricted Location and Scale Parameters In Bivariate Models: A Unified Study
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The problem of estimating location (scale) parameters θ_1 and θ_2 of two distributions when the ordering between them is known apriori (say, θ_1≤θ_2) has been extensively studied in the literature. Many of these studies are centered around deriving estimators that dominate the maximum likelihood estimators and/or best location (scale) equivariant estimators for the unrestricted case, by exploiting the prior information θ_1 ≤θ_2. Several of these studies consider specific distributions such that the associated random variables are statistically independent. In this paper, we consider a general bivariate model and general loss function and unify various results proved in the literature. We also consider applications of these results to various dependent bivariate models (bivariate normal, a bivariate exponential model based on a Morgenstern family copula, a bivariate model due to Cheriyan and Ramabhadran's and Mckay's bivariate gamma model) not studied in the literature. We also apply our results to two bivariate models having independent marginals (exponential-location and power-law distributions) that are already studied in the literature, and obtain the results proved in the literature for these models as a special cases of our study.
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