Abstract
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.
Highlights
Do there exist spherically symmetric distributions on the closed unit ball Bd in Rd that have uniform one-dimensional marginal distributions on [−1, 1]? A distribution on Bd with this property may be said to “square the circle” when d = 2 and to “cube the sphere” when d ≥ 3.The cumulative distribution function of a multivariate distribution on the unit cube [0, 1]d whose marginal distributions are uniform [0, 1] is commonly called a copula; see Nelsen [1] for an accessible introduction to this topic
If a circular or spherical copula exists on Cd, it is the cdf of a random vector Z ≡ (Z1, . . . , Zd )
For d = 3, the unique spherical copula is generated by the uniform distribution on the unit sphere ∂B3 := {(x1, x2, x3 ) | x21 + x22 + x23 = 1}
Summary
If a circular or spherical copula exists on Cd , it is the cdf of a random vector Z ≡ For d = 3, the unique spherical copula is generated by the uniform distribution on the unit sphere ∂B3 := {(x1 , x2 , x3 ) | x21 + x22 + x23 = 1}. If a spherical copula is to exist for d = 3, it follows from (2) that its generating random vector Z ∈ B3 must satisfy E(R2 ) = 1, R = 1 with probability one.
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