The covariance of two random variables measures the average joint deviations from their respective means. We generalise this well-known measure by replacing the means with other statistical functionals such as quantiles, expectiles, or thresholds. Deviations from these functionals are defined via generalised errors, often induced by identification or moment functions. As a normalised measure of dependence, a generalised correlation is constructed. Replacing the common Cauchy-Schwarz normalisation by a novel Fr\'echet-Hoeffding normalisation, we obtain attainability of the entire interval $[-1, 1]$ for any given marginals. We uncover favourable properties of these new dependence measures. The families of quantile and threshold correlations give rise to function-valued distributional correlations, exhibiting the entire dependence structure. They lead to tail correlations, which should arguably supersede the coefficients of tail dependence. Finally, we construct summary covariances (correlations), which arise as (normalised) weighted averages of distributional covariances. We retrieve Pearson covariance and Spearman correlation as special cases. The applicability and usefulness of our new dependence measures is illustrated on demographic data from the Panel Study of Income Dynamics.

翻译：两个随机变量的协方差衡量的是它们各自偏离均值的平均联合偏差。我们将这一广为人知的度量进行推广，用其他统计泛函（如分位数、期望分位数或阈值）替代均值。这些泛函的偏差通过广义误差定义，通常由识别函数或矩函数导出。作为标准化的依赖性度量，我们构建了广义相关性。采用新颖的弗雷歇-赫夫丁归一化替代常见的柯西-施瓦茨归一化，我们能够在任意给定边际分布下实现整个区间$[-1, 1]$的可达性。我们揭示了这些新型依赖性度量的优良性质。分位数相关性和阈值相关性族引出了函数值分布相关性，它们能够呈现完整的依赖结构。这些相关性能导出尾部相关性，这应该可以取代尾部依赖系数。最后，我们构建了综合协方差（相关性），它们作为分布协方差的（归一化）加权平均出现。皮尔逊协方差和斯皮尔曼相关性是其中的特例。我们利用收入动态面板研究的人口统计数据，展示了这些新型依赖性度量的适用性和实用性。