Randomization Tests of Copula Symmetry

Abstract

New nonparametric tests of copula exchangeability and radial symmetry are proposed. e novel aspect of the tests is a resampling procedure that exploits group invariance conditions associated with the relevant symmetry hypothesis. ey may be viewed as feasible versions of randomization tests of symmetry, the la er being inapplicable due to the unobservability of margins. Our tests are simple to compute, control size asymptotically, consistently detect arbitrary forms of asymmetry, and do not require the specification of a tuning parameter. Simulations indicate excellent small sample properties compared to existing procedures involving the multiplier bootstrap.

Randomization Tests for Equality in Dependence Structure

“…However, the marginal distributions are not known in practice and can only be estimated consistently, and this suggests that we may only rely on asymptotic group invariance conditions in our application of the permutation method. In this respect, our results in the next section complement those of Canay et al (2017) or Beare and Seo (2017), who investigated the behavior of randomization tests under approximate symmetry conditions, albeit with different formalizations of approximate symmetry.…”

“…In our context, the Hoeffding's condition (ξ π N , ξ π N ) (ξ, ξ ), for a random element ξ π N ∈ A, can be replaced by the conditional convergence, ξ π N P π ξ. This property is also used in Beare and Seo (2017) and it turned out to be particularly useful when the test statistic of interest has a nonlinear form and the continuous mapping theorem and functional delta method for the conditional convergence are to be invoked. We provide the details of the continuous mapping theorem and the functional delta method for the conditional convergence in Lemma A.2 and Lemma A.3 in the Appendix.…”

“…Beare and Seo (2017) also explore an asymptotic group invariance condition to develop a quasi-randomization test of copula symmetry. While g ∈ G N in our setting is specified by a permutation π which is uniformly distributed over G N , g ∈ G n in Beare and Seo (2017) is defined to be a transformation from ([0, 1] 2 ) n onto itself defined by g ((x 1 , y 1 ), ..., (x n , y n )) = (π τ 1 (x 1 , y 1 ), ..., π τn (x n , y n )) where (π 0 (x, y), π 1 (x, y)) = ((x, y), (y, x)) or (π 0 (x, y), π 1 (x, y)) = ((x, y), (1 − x, 1 − y)) with τ i being n i.i.d. draws from the Bernoulli distribution.…”

“…draws from the Bernoulli distribution. Unlike our framework, the group invariance conditions in Beare and Seo (2017) hold between paired normalized observations (with n = m) which are completely dependent with each other. Hence, our proofs are considerably different from theirs though some results on the conditional convergence in Beare and Seo (2017) are also useful to us.…”

“…Remark 3.4. Beare and Seo (2017) also explore an asymptotic group invariance condition to develop a quasi-randomization test of copula symmetry. While g ∈ G N in our setting is specified by a permutation π which is uniformly distributed over G N , g ∈ G n…”