AI 中文总结
该研究刻画了至多二阶的二元一致性度量，证明其与满足平方群等变性等性质的copula索引测度场存在双射，并构造了具有平方群等变性的二阶一致性度量，推导了对应加权平均算子的充要条件。
AI 中文摘要
我们在“基于copula混合段计算的度量为至多二次的多项式”的意义下，刻画了至多二阶的二元一致性度量。我们的主要工具是典范极化，它将泛函的二次性转化为分离的亲和性，从而可逐段应用于仿射泛函的积分表示。利用该技术，我们证明至多二阶的度量与满足包括平方群等变性在内的若干性质的copula索引测度场之间存在双射。独立性对应的场决定仿射线性化，其位移决定纯二次部分；此外，精确二阶等价于场的非恒定性。利用我们的刻画结果，我们基于仿射copula算子构造了二阶一致性度量，这些算子具有平方群等变性。作为更具体的例子，我们推导了由平方群变换后的copula的加权平均算子生成二阶一致性度量的充要条件。
英文摘要
We characterize bivariate measures of concordance of degree at most two in the sense that the measures evaluated at copula-mixture segments are polynomials of degree at most two. Our main device is the canonical polarization, which converts quadraticity of a functional into separate affinity, thereby allowing an integral representation for affine functionals to be applied sectionwise. Using this technique, we prove that measures of degree at most two are in bijection with copula-indexed measure fields satisfying several properties including square-group equivariance. The field at independence determines the affine linearization, while its displacement determines the purely quadratic part. Moreover, exact degree two is equivalent to field non-constancy. Leveraging our characterization result, we provide a construction of degree-two measures of concordance based on affine copula operators that are square-group equivariant. As a more concrete example, we derive a necessary and sufficient condition for a weighted averaging operator of square-group transformed copulas to generate a measure of concordance of degree two.
