The problem of estimating location (scale) parameters $\theta_1$ and $\theta_2$ of two distributions when the ordering between them is known apriori (say, $\theta_1\leq \theta_2$) 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 $\theta_1 \leq \theta_2$. 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.
翻译:文献中广泛研究了在知道两个分布点之间的定序时估算位置( 标度) $theta_ 1美元 和 $theta_ 2美元 的问题。 许多这些研究都围绕在文献中广泛研究的以下问题: 估计位置( 标度) 参数( 标度) $theta_ 1美元 美元 和 $theta_ 2美元 两个分布点 。 当它们之间的定序已知时, 估计位置( 标度( 标度) $theta_ 1美元 美元 美元 和 $theta_ 2美元 美元 美元 。 这些研究中有些研究考虑了具体的分布, 相关的随机变量在统计上是独立的 。 在本文中, 我们考虑的是一般的双变量模型和一般损失函数, 并统一文献中所证明的各种结果。 我们还考虑将这些结果应用到各种依赖的双变量模型( 常数, 以Morgenster 家族立式为基础, 一种双变量指数模型, 是切里扬和 Ramabhadran's 和 Mckay's 特殊变数变量的模型。