Tests of radial symmetry for multivariate copulas based on the copula characteristic function

: A new class of rank statistics is proposed to assess that the copula of a multivariate population is radially symmetric. The proposed test statistics are weighted L 2 functional distances between a nonparametric estimator of the characteristic function that one can associate to a copula and its complex conjugate. It will be shown that these statistics behave asymptotically as degenerate V-statistics of order four and that the limit distributions have expressions in terms of weighted sums of independent chi-square random variables. A suitably adapted and asymptotically valid multiplier bootstrap procedure is proposed for the computation of p -values. One advantage of the proposed approach is that unlike methods based on the empirical copula, the partial derivatives of the copula need not be estimated. The good properties of the tests in ﬁnite samples are shown via simulations. In particular, the superiority of the proposed tests over competing ones based on the empirical copula investigated by [6] in the bivariate case is clearly demonstrated.


Introduction
A random vector X = (X 1 , . . . , X d ) is said to be symmetric about a point μ = (μ 1 , . . . , μ d ) ∈ R d if X − μ and μ − X have the same distribution. In particular when μ = (0, . . . , 0), this is called central symmetry in standard books like [5]. Of interest in this work is the relationship that exists between this notion of multivariate symmetry and the copula that can be extracted from the distribution of a random vector. The starting point is that when the marginal distributions of X are continuous, Sklar's Theorem ensures that there exists a unique copula C : [0, 1] d → [0, 1] such that for all x = (x 1 , . . . , x d ) ∈ R d , Of course, the symmetry of X around μ entails the symmetry of X around μ for each ∈ {1, . . . , d}. In addition, it also entails the so-called radial symmetry of C. In other words, the radial symmetry of C means that U ∼ C and 1 d − U have the same distribution, where 1 d = (1, . . . , 1) ∈ R d . Hence, X is symmetric about μ if and only if X 1 , . . . , X d are marginally symmetric and the copula C of X is radially symmetric; for more details, see [8].
From a model-building perspective using copulas, it may be of interest to check if the dependence structure of a random vector is radially symmetric. In other words, it is a good idea to test the radial symmetry hypothesis before trying to fit a specific copula model to multivariate observations. It is only in the case of its non rejection that the use of a family of radially symmetric copulas would be justified, e.g. the well-known Normal and Student copulas, or more generally the models in the elliptical class. However, if the null hypothesis of radial symmetry is rejected, one would have to seek for radially asymmetric models, e.g. the skew-elliptical, extreme-value or chi-square copulas.
Letting U j = F j (X j ) for j ∈ {1, . . . , d}, the radial symmetry of U is equivalent to the equality in distribution of U and 1 d − U. This can be conveniently expressed in terms of the random vector that takes value in [−1/2, 1/2] d . Namely, the interest in this work is to test for the central symmetry of W, i.e.
Tests of radial symmetry based on empirical copulas have recently been proposed by [2] and [6] in the special case when d = 2; see also [15] for the definition of measures of bivariate radial asymmetry. Essentially, these authors adopt a distribution-oriented perspective based on the comparison of nonparametric estimators of the copulas of (U 1 , U 2 ) and (1 − U 1 , 1 − U 2 ), respectively. As noted by [13], one can derive powerful and easy-to-implement tests by using the characteristic function associated to C. The latter arises as a natural version of the usual multivariate characteristic function. To be more specific, let ψ C be the characteristic function of W, i.e. for i 2 = −1 and t ∈ R d , ψ C (t) = E(e i tW ).
Under the null hypothesis of radial symmetry described in (3), it is clear that ψ C (t) = ψ C (−t) for all t ∈ R d . Using the identity e ix = cos x + i sin x, it can be seen to be equivalent to L C (t) = 0 for all t ∈ R d , where In fact, as noted e.g. by [7], H 0 holds if and only if L C (t) = 0 for each t ∈ R d , so that the null and alternative hypotheses of interest can be reformulated as H 0 : L C (t) = 0 ∀t ∈ R d and H 1 : L C (t) = 0 for some t ∈ R d .
From a practical point-of-view, it will be seen in this work that this characterization of radial symmetry has many advantages: (i) nice and easy-to-implement formulas are available for test statistics based on weighted L 2 -functionals of L C ; (ii) p-values can be computed from the multiplier bootstrap adapted to Vstatistics and avoids the task of estimating the partial derivatives of the copula usually necessary when dealing with empirical copulas; (iii) in the bivariate case, the tests based on these statistics are more powerful than the only available tests yet, namely those three investigated by [6].
The paper is organized as follows. Section 2 introduces the new class of test statistics and provides explicit formulas for their computation. Section 3 derives the asymptotic behavior of these test statistics under the null hypothesis of radial symmetry. Section 4 describes and validates a resampling procedure based on the multiplier bootstrap adapted to the context of rank-based V-statistics; the consistency of the tests against general radially asymmetric alternatives is established as well. In Section 2, the efficiency of the tests in terms of size and power is investigated with the help of Monte-Carlo simulations and the results are compared to the test statistics studied by [6] in the bivariate case. All the proofs are relegated to Appendix A and some complementary computations are to be found in Appendix B.

Test statistics
Let X 1 , . . . , X n , where X j = (X j1 , . . . , X jd ), be independent copies of a random vector X = (X 1 , . . . , X d ) ∼ F . In the sequel, it is assumed that the marginal distributions F 1 , . . . , F d are continuous, so that there is a unique copula C that satisfies (1). Usually, the marginal distributions are unknown, in which case the vector W defined in (2) is not observable. For that reason, one has to rely on the vectors of pseudo-observations W 1 , . . . , W n , where for each j ∈ {1, . . . , n}, with F being the -th re-scaled marginal empirical distribution function, i.e.
An empirical version of L C based on its definition in Equation (4) is given by For some weight function ω : R d → R, a test statistic for radial symmetry is As will be seen later, it is usually assumed that ω is strictly positive, except maybe on a subset of R d of Lebesgue measure zero. This requirement ensures that a test based on R n,ω is consistent against all alternatives to H 0 . An explicit and useful formula for R n,ω arises easily upon defining for a, b ∈ R d the function It is then a routine exercise to show that It is usual in characteristic-function testing to assume that ω is a probability density. The next lemma provides a formula for B ω in that case.

Lemma 1.
If ω is a probability density on R d , then where A ω is the real part of the characteristic function of ω, i.e.

Example 1.
A special case of Lemma 1 occurs when ω is a product of densities that are symmetric around zero, i.e.
for all x ∈ R. Since in this situation, the characteristic function of ω factorizes into the product of the marginal characteristic functions, one has for a = (a 1 , . . . , a d ) and where α (a) = R cos(x a) g (x) dx. In order to accomplish some sort of smoothing, one can substitute g (t) with g (t/σ) for some σ > 0, so that Example 2. Suppose that ω is the density of a standardized bivariate elliptical distribution. In that case, for some ϕ ω : R + → R + and some positive-definite correlation matrix R, the characteristic function of ω is real and of the form A ω (a) = ϕ ω (a R a ), where a = (a 1 , . . . , a d ). By considering ω(t/σ) instead of ω(t), where σ > 0 is a real-valued smoothing parameter, the characteristic function becomes A ω (a) = σ 2 ϕ ω (σ 2 a R a ) and then One recovers the standard Normal distribution when ϕ ω (x) = e −x/2 , and then where φ R is the d-variate standard Normal density with correlation matrix R.

Asymptotic behavior of R n,ω under radial symmetry
The large-sample behavior of R n,ω under the null hypothesis of radial symmetry is derived in this section. It will first be shown that R n,ω is asymptotically equivalent to a V-statistic of degree four. The reader is referred to the excellent monograph by [10] for further details on the theory of U-and V-statistics. Before stating the result, define for Also, let Φ ω be such that

Proposition 1.
Suppose that X 1 , . . . , X n are i.i.d. from a multivariate distribution function having continuous marginal distributions and whose unique copula C is radially symmetric. Then as long as the weight function ω is integrable and Tests of radial symmetry

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One can now invoke results in the theory of V-statistics to obtain an asymptotic representation for R n,ω . Before stating it, define the bivariate degenerate kernel Under the null hypothesis of radial symmetry, one has W d = −W and then one can show from the definition of Λ t in Equation (10) that

Proposition 2. Under the conditions of Proposition 1,
As a consequence, R n,ω converges in distribution to a random variable having representation where {Z j } ∞ j=1 is a sequence of i.i.d. N(0, 1) random variables and {κ j } ∞ j=1 are the eigenvalues of η → E W {Ψ ω (w, W) η(W)}.

Multiplier versions of the test statistics
The asymptotic representation of the test statistic R n,ω under the null hypothesis, as described in Equation (13) of Proposition 2, can hardly be used for the computation of p-values. On one part, this representation depends on eigenvalues that are difficult to compute, and on another part, the latter depend on a radially symmetric copula C that is not specified under the null hypothesis. For these reasons, it will rather be representation (12) that will be exploited in conjunction with a nonparametric approach based on the multiplier bootstrap. This resampling method is described in details in a general empirical process context by [19] and [9]. Versions suitably adapted to U-and V-statistics are considered by [3] in the i.i.d. case and by [11] under serial dependence.
Proposition 2 has established that R n,ω is asymptotically equivalent to a first-order degenerate V-statistic. One can therefore, at least in principle, adapt results on the multiplier bootstrap of degenerate U and V-statistics that one can find in [3]. To this end, start with independent multiplier random variables ξ 1 , . . . , ξ n , where for each j ∈ {1, . . . , n}, E(ξ j ) = Var(ξ j ) = 1. In view of the asymptotic representation in Equation (12) and recalling that Ψ ω (w 1 , (11), a multiplier version of R n,ω would be given by where Δ j = (ξ j /ξ) − 1 andξ = (ξ 1 + · · · + ξ n )/n. However, since the random vectors W 1 , . . . , W n are unobservable, the latter will be replaced, in a rather natural way, by W 1 , . . . , W n in the above expression. Moreover, the computation of λ t , and at the same time of Ψ ω , involves an expectation with respect to the unspecified distribution of W under the null hypothesis. For that reason, Ψ ω will be estimated by The proposed multiplier version of R n,ω is then

Asymptotic validity of the multiplier bootstrap and consistency of the tests
The following result characterizes the asymptotic behavior of R n,ω conditional on the data, both under H 0 and under fixed alternatives.

Proposition 3.
Let P be the probability measure conditional on where R ω has the same limit as the first-order degenerate V-statistic

Tests of radial symmetry
Note that under H 0 , λ t = λ t . Hence, a consequence of Proposition 3 is that the asymptotic distribution of R n,ω matches that of R n,ω stated in Proposition 2.
In other words, R n,ω replicates R n,ω properly under the null hypothesis as n goes to infinity. Another consequence of the result is the consistency of the test based on R n,ω . On one hand, since R ω has a non-degenerate distribution, R n,ω = O P (1), while under the assumption that the set {t ∈ R d : ω(t) = 0} has Lebesgue measure zero and since L C vanishes on R d if and only if H 0 holds true, R n,ω /n converges in probability to a positive constant. Thus, R n,ω goes to infinity under general alternatives to H 0 , so the test based on R n,ω is consistent. In practice, one proceeds a large number M of times to obtain asymptotically valid replicates R where the entries of D ω ∈ R n×n are (D ω ) jj = Ψ ω ( W j , W j ). Since D ω needs to be computed only once from the data, these multiplier bootstrap replicates obtain very quickly. An approximate and asymptotically valid p-value for the test of radial symmetry based on R n,ω is then

Implementation issues
An easy-to-implement procedure for the computation of D ω will be derived in this section. To this end, the following lemma will prove useful as it provides an expression for Ψ ω in terms of B ω and some of its partial derivatives.
Lemma 2 can now be exploited in order to derive a compact formula for the computation of D ω based on products of matrices. To this end, define D 0 ∈ R n×n whose entries are Explicit formulas for the computation of the partial derivatives of B ω when the weight functions are those considered in Examples 1-2 are given in Appendix B.

The bivariate case
The aim of the section is to study the sampling properties of the tests of bivariate radial symmetry based on R n,ω when ω is the product of standard Normal, double-exponential and double-gamma densities. The computation of B ω is then based on formula (8) in the special case when α 1 = α 2 = α. For the three abovementioned densities, one has respectively The case when ω is the standard bivariate Normal density with R 12 = R 21 = .5 has also been considered; the formula for the computation of B ω is given in equation (9). It is clear that these weight functions are integrable and satisfy the requirement The formulas for the computation of their multiplier versions are given in Appendix B.1 and Appendix B.2. The multiplier variables ξ 1 , . . . , ξ n are taken to be i.i.d. exponential with mean one. The number of bootstrap replicates has been set to M = 1 000 and the estimated probabilities of rejection are based on 1 000 Monte-Carlo repetitions.
A first step is to investigate how well the multiplier method succeeds in the replication of the distribution of the test statistics under the null hypothesis for small and moderate sample sizes. To this end, the bivariate one-parameter Table 1 Families of radially symmetric bivariate copulas Frank, Normal, Plackett and Student (with ν = 4 degrees of freedom) families of radially symmetric copulas have been considered. Their expressions are given in Table 1; more details on these models can be found in the monographs by [12] and [16]. The value of the parameter for each model has been chosen in such a way that Kendall's measure of association takes values in {.25, .50, .75}.
Recall that for a given copula C, Kendall's tau is defined by The simulation results are reported in Table 2 for sample sizes n ∈ {125, 250} and where the value of the smoothing parameter σ lies in {.5, 1, 3, 5, 7}. Generally speaking, all the tests are quite good at keeping their 5% nominal level when n = 250. The only notable exception is for σ = .5 when C is either the Normal or the Student copula with a high level of dependence, i.e. τ (C) = .75. A similar phenomenon, though less marked, occurs when σ = 7. Experiments not shown here indicate that this issue is resolved when the sample size increases to n = 500. When n = 125, the tests are conservative, especially for high values of τ (C). This behavior is typical of methods based on the multiplier bootstrap.
In order to study the power of the tests based on R N n , R DE n , R DG n and R BN n , the radially asymmetric Gumbel, Clayton, chi-square and skew-Student copulas have been considered. The Gumbel and Clayton models are described in standard books like [12] and [16]. A special case of the bivariate skew-Student copula with ν degrees of freedom as defined by [4] is the dependence structure of (X 1 , where (Z 1 , Z 2 ) is standard bivariate Normal with correlation θ and Y is a chi-square random variable with ν degrees of freedom that is independent of (Z 1 , Z 2 ). The radially symmetric Student copula occurs when γ 1 = γ 2 = 0, while otherwise, the construction yields a radially asymmetric copula. In the current simulation study, (γ 1 , γ 2 ) = (1, 1) and ν = 4. The multivariate chisquare copula family was introduced by [1] in the context of spatial statistics and formally investigated by [14]. A particular case is the centered bivariate chi-square copula defined as the dependence structure of (Z 2 1 , Z 2 2 ). Scatterplots of n = 10 000 simulated pairs from these four copulas are provided in Figure 1; the corresponding plots of L C (t 1 , t 2 ) as a function of t 1 ∈ [−40, 40] when t 2 ∈ {10, 20, 30} are given in Figure 2. An overall look at these curves leads to the conclusion that for the tests based on R N n , R DE n and R DG n , the best choice seems to be σ = 1 under the four alternatives, whatever the value of τ (C). Things are a little less clear for the test based on R BN n , since in that case the influence of σ on the power is less obvious. Nevertheless, σ = 5 seems to be an appropriate choice.

Comparisons with the tests of bivariate radial symmetry of [6]
Now the power of R N n , R DE n and R DG n when σ = 1, and of R BN n when σ = 5, will be compared to the procedures based on the empirical copula investigated by [6]. Letting U j1 = F 1 (X j1 ) and U j2 = F 2 (X j2 ), define the empirical copulas Three statistics based on functional distances between C n and D n were investigated by [6]. In the sequel, only the statistic that has been identified by [6] as the most powerful will be considered, namely the Cramér-von Mises statistic  Table 3, where the power obtained by [6] under the Gumbel, Clayton and Skew-Student when n ∈ {250, 500} is given as well (see line S n ). It can first be seen that the results of [6], obtained with M = 250 multiplier bootstrap replicates, are accurately reproduced here with M = 1 000. Otherwise, some expected conclusions can be made, namely that the power of each test increases with the sample size, as a consequence of their consistency. Note also that the power of the tests is smaller under Gumbel alternatives, since as noted by [6], the radial asymmetry is quite weak for the members of this family; this conclusion can also be reached upon looking at Figures 1-2. Under the Gumbel, Clayton and chi-square alternatives, the power generally increases with the level of dependence, while an inverse relationship occurs under the skew-Student copula. These results are in accordance with the scatterplots in Figure 1 and  can also be understood in the light of the curves of L C given in Figure 2.
The test based on R BN n is always much less powerful than the three other characteristic function tests based on R N n , R DE n and R DG n , the performance of these three statistics being quite similar. Now comparing the newly introduced tests with that based on S n , the following comments can be made: (i) Under Gumbel, chi-square and skew-Student alternatives, the performance of R N n , R DE n and R DG n is clearly better than that of S n ; (ii) Under Clayton alternatives, the power of R N n , R DE n and R DG n is very similar to that of S n ; The above conclusions hold both for n = 125 and n = 250. Hence, overall, one could warmly recommend the use of the characteristic function tests with a product weight function and a smoothing parameter set to σ = 1.

A deeper investigation on the bivariate Normal weight function
The test statistic R BN n based on the bivariate Normal density with ρ = .5 is systematically less powerful than its competitors. However, this family of weight functions offers an additional flexibility by allowing to select an optimal value of ρ ∈ (−1, 1). Since R BN n = R N n when ρ = 0, the power of the test when using ρ cannot be less than that of the test based on the product of Normal densities.
In order to investigate the influence of ρ on the power of the test based on R BN n , a complementary simulation study has been conducted under the same twelve models of asymmetric copulas. The results are presented in Figure 4 when n = 125 (in blue) and n = 250 (in black); the value of the smoothing parameter has been set to σ = 5. Overall, the influence of ρ seems to de- Test pend heavily on the kind of alternatives at hand, preventing from a general recommendation on a universal value to be chosen. Nevertheless, the advantage of taking a value of ρ different from zero is more obvious when τ (C) takes high values. While a negative value for ρ would clearly be a bad decision in that case, taking ρ > 0 significantly improves the power, especially when n = 125. This little investigation opens the door to a more formal study on the optimal choice of a weight function. Noting that the bivariate Normal weight function can be written in terms of the density c θ of the Normal copula as one could consider a general family of weight functions of the form where c θ is some copula density and G is a cdf on R whose associated density g = G is symmetric around zero. This would allow for a lot of flexibility, as one could choose among many copula families, levels of dependence as controlled by θ, and smoothing parameters σ > 0. However, it raises at the same time the issue of basing this choice on a formal criteria, which even in the domain of standard characteristic function tests, remains an open question.

Performance of the tests in higher dimensions
This section reports the results of an investigation on the size and power of the characteristic function statistic S n,ω when d ∈ {3, 5} and the weight function is based on a product of Normal densities. The copula models that have been considered under H 0 are the Normal and Student with ν = 4 degrees of freedom. Radially asymmetric alternatives have been generated from the multivariate skew-Student copula with ν degrees of freedom. The latter is a straightforward extension of the model already introduced in the bivariate case, namely the dependence structure of where Z = (Z 1 , . . . , Z d ) is standard multivariate Normal with correlation matrix R, Y is a chi-square random variable with ν degrees of freedom that is independent of Z and γ ∈ R d controls the degree of asymmetry. In the sequel, γ = (1, . . . , 1) and ν = 4. In addition, the multivariate centered chi-square copula defined as the dependence structure of (Z 2 1 , . . . , Z 2 d ) has been considered. For each of these four models, the correlation matrix R has been taken equicorrelated, i.e. R jj = θ for each j = j ∈ {1, . . . , d}. The value of θ for each model has been selected in order that the pairwise Kendall's tau τ is in the set {.25, .50, .75}. The results are to be found in Table 4.
One can first say that the tests are generally good at keeping their 5% nominal level under the null hypothesis when n = 250, while they are rather conservative when n = 125; this is particularly true when τ = .75. This behavior was also observed in the bivariate case. Globally, the tests are very good at distinguishing departures from H 0 . In particular, the chi-square alternatives are always well detected, the power of the tests being at its best when τ = .50. Under the skew-Student alternatives, the probabilities of rejection are also good, but this time they are inversely proportional to the value of τ . For a given alternative, the power of the tests tend to be higher when d = 5 compared to d = 3. Finally note that the best choice for the smoothing parameter is σ ∈ {.5, 1}.

A.1. Proof of Lemma 1
Making use of the product-to-sum trigonometric identity, i.e.

sin
one deduces from the definition of B ω in Equation (6) that where A ω is the real part of the characteristic function of ω.

A.2. Proof of Proposition 1
By the mean-value Theorem, one has for each j ∈ {1, . . . , n} that Upon noting that for each ∈ {1, . . . , d}, Hence, in view of the definition of Λ t , One then has At this point, it is worth noting that from the Cauchy-Schwarz inequality, Proposition 2 establishes the convergence in distribution of V n,ω , so that V n,ω = O P (1). It then remains to show that Δ n1,ω = o P (1) in order to conclude that R n,ω = V n,ω + o P (1). To this end, one deduces from Equation (14) that Since for each ∈ {1, . . . , d}, W j − W j = F (X j ) − F (X j ) and √ n{ F (x) − F (x)} converges uniformly in x ∈ R to a brownian bridge (see [18], for instance), One can then conclude that In view of the assumption Finally, simple calculations allow to show that

A.5. Proof of Lemma 2
First define h t (w) = sin(tw ) and h [ ] t (w) = ∂h t (w)/∂w = t cos(tw ) for each ∈ {1, . . . , d}. With this notation, one can write Integrating λ t (w 1 ) λ t (w 2 ) with respect to ω yields The first expression on the righthand side is For the second and third summand, one has for w ∈ [−1/2, 1/2] d that while for the fourth summand, Collecting the four expressions yields the announced formula.
By straightforward computations, one can show that Also, In the case of the standard Normal density, one can show that