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均值偏离的联合平均程度。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。这些泛函的偏离通过广义误差来定义，此类误差通常由识别函数或矩函数导出。作为依赖关系的归一化度量，我们构建了广义相关。通过采用新颖的弗雷歇-霍夫丁归一化替代常见的柯西-施瓦茨归一化，我们实现了对于任意给定边际分布，整个区间 $[-1, 1]$ 的可达性。我们揭示了这些新依赖度量的优良性质。分位数相关和阈值相关族衍生出函数值的分布相关，能够呈现完整的依赖结构。它们导出了尾部相关，这理应取代尾部依赖系数。最后，我们构建了综合协方差（综合相关），其形式为分布协方差的（归一化）加权平均。皮尔逊协方差和斯皮尔曼相关作为特例被重新获取。我们通过收入动态面板研究的人口统计数据，阐述了新依赖度量的适用性与实用性。