重叠样本的顺序统计量：二元密度和回归性质
Order statistics from overlapping samples: bivariate densities and regression properties
摘要：
本文研究重叠样本下两个顺序统计量的联合分布。在父分布绝对连续（关于实直线上的勒贝格测度）的假设下，给出了这类随机变量对的分布的一个显式公式.该分布是通过密度的形式来确定的，该密度是$\R^2$上的二元勒贝格测度和对角$\{(x,x):\,x\in\R\}$上的一元勒贝格测度之和。我们还感兴趣的问题，在多大程度上的条件期望的一个这样的顺序统计量给另一个决定父分布。特别是，我们提供了一个新的表征线性回归的顺序统计量从扩展的样本给定的一个从原始样本，特殊情况下解决了一个问题，在文献中明确指出。看来，为了描述正确的母体分布，使用分位数密度函数是方便的。在其他几种情况下回归的顺序统计量，我们提供了新的结果，在样本中的分布的唯一性。然而，一般问题的可识别性的父分布的回归顺序统计量重叠样本仍然开放。
英文摘要：
In this paper we are interested in the joint distribution of two order statistics from overlapping samples. We give an explicit formula for the distribution of such a pair of random variables under the assumption that the parent distribution is absolutely continuous (with respect to the Lebesgue measure on the real line). The distribution is identified through the form of the density with respect to a measure which is a sum of the bivariate Lebesgue measure on $\R^2$ and the univariate Lebesgue measure on the diagonal $\{(x,x):\,x\in\R\}$. We are also interested in the question to what extent conditional expectation of one of such order statistic given another determines the parent distribution. In particular, we provide a new characterization by linearity of regression of an order statistic from the extended sample given the one from the original sample, special case of which solves a problem explicitly stated in the literature. It appears that to describe the correct parent distribution it is convenient to use quantile density functions. In several other cases of regressions of order statistics we provide new results regarding uniqueness of the distribution in the sample. Nevertheless the general question of identifiability of the parent distribution by regression of order statistics from overlapping samples remains open.
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