A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent samples. In this paper, we study this invariance property on the Pearson correlation coefficient and its applications. A multivariate random vector is said to have an invariant correlation if its pairwise correlation coefficients remain unchanged under any common marginal transforms. For a bivariate case, we characterize all models of such a random vector via a certain combination of comonotonicity -- the strongest form of positive dependence -- and independence. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fr\'echet copulas. In the general multivariate case, we characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model that admits any prescribed invariant correlation matrix. This model turns out to be the joint distribution of samples with duplicate records. In this context, we provide an application of invariant correlation to the statistical inference in the presence of sample duplication. Finally, we show that all our characterization results of invariant correlation, except one special case, remain the same if the common marginal transforms are confined to the set of increasing ones.

翻译：独立样本的一个有用性质是，在应用边际变换后其相关性保持不变。这一不变性在统计推断中具有基础性作用，但对于相依样本通常不成立。本文研究皮尔逊相关系数的这一不变性及其应用。若多元随机向量的成对相关系数在任何常见边际变换下均保持不变，则称该向量具有不变相关性。在二元情形下，我们通过共单调性（正相关的最强形式）与独立性的特定组合刻画了此类随机向量的所有模型。特别地，我们证明了具有不变相关性的可交换联结函数类可由所谓的正弗雷歇联结函数精确描述。在一般多元情形中，我们通过团划分多面体刻画了所有不变相关矩阵的集合。同时提出了一类正回归依赖模型，该模型可容纳任意指定的不变相关矩阵，且恰好对应具有重复记录的样本联合分布。在此背景下，我们给出了不变相关性在存在样本重复时的统计推断应用。最后证明，除一个特例外，若将常见边际变换限制为单调递增变换，所有关于不变相关性的刻画结论保持不变。