Abstract
Abstract Given a d-dimensional random vector X = (X 1, . . ., X d ), if the standard uniform vector U obtained by the component-wise probability integral transform (PIT) of X has the same distribution of its point reflection through the center of the unit hypercube, then X is said to have copula radial symmetry. We generalize to higher dimensions the bivariate test introduced in [11], using three different possibilities for estimating copula derivatives under the null. In a comprehensive simulation study, we assess the finite-sample properties of the resulting tests, comparing them with the finite-sample performance of the multivariate competitors introduced in [17] and [1].
Highlights
Let X = (X, . . . , Xd) be a continuous random vector with marginal cumulative distribution functions (CDFs) Fi (x), i =, . . . , d
Given a d-dimensional random vector X = (X, . . . , Xd), if the standard uniform vector U obtained by the component-wise probability integral transform (PIT) of X has the same distribution of its point re ection through the center of the unit hypercube, X is said to have copula radial symmetry
Let U be the standard uniform random vector obtained by using the component-wise probability integral transform on X: U = (U, . . . , Ud) = (F (X ), . . . , Fd (Xd))
Summary
The survival copula Cis the CDF of the random vector d − U With these de nitions, equation (1.1) is equivalent to the following identity:. The purpose of this paper is to ll this gap by investigating the nite sample properties of the test statistic Sn in equation (1.3) using the multiplier Bootstrap under three di erent speci cations of the multiplier empirical process, comparing its performance to the procedures proposed in [17] and [1]. [23] obtains the weak convergence of the empirical process based on the copula of the transformed vector: Proposition 1 ([23]). All the proposed derivatives estimators are consistent, implying the convergence of the di erent multiplier bootstrap processes to independent copies of the empirical copula process: Cn , C [n ],mid , . Similar results were obtained under other speci cations of copula models
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