Perlman Michael D, Wellner Jon A
Department of Statistics, University of Washington, Box 354322, Seattle, WA 98195-4322, USA.
Symmetry (Basel). 2011 Aug 23;3(3):574-599. doi: 10.3390/sym3030574.
Do there exist circular and spherical copulas in [Formula: see text]? That is, do there exist circularly symmetric distributions on the unit disk in [Formula: see text] and spherically symmetric distributions on the unit ball in [Formula: see text], d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in [Formula: see text] by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in [Formula: see text] are also described, and determined explicitly for d = 2.
在(\mathbb{R}^d)中是否存在圆形和球形的连接函数？也就是说，在(\mathbb{R}^2)中的单位圆盘上是否存在圆对称分布，以及在(\mathbb{R}^d)（(d\geq3)）中的单位球上是否存在球对称分布，其一维边缘分布是均匀的？对于(d = 2)和(3)，答案是肯定的，其中圆形和球形连接函数是唯一的并且可以明确确定，但对于(d\geq4)，答案是否定的。通过斜坐标变换从(\mathbb{R}^2)中唯一的圆形连接函数得到了一个单参数族的椭圆二元连接函数。还描述了通过对(\mathbb{R}^d)中单位球上的均匀分布进行非线性变换得到的连接函数，并针对(d = 2)明确确定了它们。
