Abstract
A statistical procedure to determine if the dependence structure of a multivariate random vector belongs or not to the general class of elliptical copulas has been proposed by Jaser et al. (Depend Model 5:330–353, 2017). Their test exploits the fact that when the copula of a multivariate population is elliptical, the theoretical Kendall and Blomqvist dependence measures of each pair are the same. Under a setup where the marginal distributions are known, they based their decision rule on the asymptotic distribution of the proposed test statistic, which is chi-squared. In this paper, the restrictive assumption of known marginals is relaxed by the use of ranks. In addition, new test statistics are proposed and their p-values are computed from suitably adapted bootstrap replicates based on the form of their limit under the null hypothesis. Unlike Jaser et al.’s test, the proposed procedures keep their nominal level well when the dimension exceeds two. It is also shown that the new tests have good power properties against several types of alternatives to copula ellipticity.
Similar content being viewed by others
References
Abdous B, Genest C, Rémillard B (2005) Dependence properties of meta-elliptical distributions. In: Statistical modeling and analysis for complex data problems, vol 1 of GERAD 25th Anniv. Ser. Springer, New York, pp 1–15
Bahraoui T, Bouezmarni T, Quessy J-F (2018) A family of goodness-of-fit tests for copulas based on characteristic functions. Scand J Stat 45:301–323
Bahraoui T, Quessy J-F (2017) Tests of radial symmetry for multivariate copulas based on the copula characteristic function. Electron J Stat 11:2066–2096
Blomqvist N (1950) On a measure of dependence between two random variables. Ann Math Stat 21:593–600
Bàrdossy A (2006) Copula-based geostatistical models for groundwater quality parameters. Water Resour Res 42:1–12
Cambanis S, Huang S, Simons G (1981) On the theory of elliptically contoured distributions. J Multivar Anal 11:368–385
Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions, vol 36 of Monographs on statistics and applied probability. Chapman and Hall Ltd, London
Favre A-C, Quessy J-F, Toupin M-H (2018) The new family of Fischer copulas to model upper tail dependence and radial asymmetry: properties and application to high-dimensional rainfall data. Environmetrics 29(e2494):17
Genest C, Carabarín-Aguirre A, Harvey F (2013) Copula parameter estimation using Blomqvist’s beta. J SFdS 154:5–24
Genest C, Nešlehová J, Quessy J-F (2012) Tests of symmetry for bivariate copulas. Ann Inst Stat Math 64:811–834
Genest C, Nešlehová JG (2014) On tests of radial symmetry for bivariate copulas. Stat Pap 55:1107–1119
Jaser M, Haug S, Min A (2017) A simple non-parametric goodness-of-fit test for elliptical copulas. Depend Model 5:330–353
Joe H (2015) Dependence modeling with copulas vol 134 of Monographs on statistics and applied probability. CRC Press, Boca Raton
Kendall MG (1938) A new measure of rank correlation. Biometrika 30:81–93
Klüppelberg C, Kuhn G, Peng L (2007) Estimating the tail dependence function of an elliptical distribution. Bernoulli 13:229–251
Kojadinovic I, Yan J (2010) Nonparametric rank-based tests of bivariate extreme-value dependence. J Multivar Anal 101:2234–2249
Kosorok MR (2008) Introduction to empirical processes and semiparametric inference. Springer series in statistics. Springer, New York
Lee AJ (1990) \(U\)-statistics, vol 110 of statistics: textbooks and monographs. Marcel Dekker Inc., New York (Theory and practice)
Mai J-F, Scherer M (2012) Simulating copulas, vol 4 of series in quantitative finance. Imperial College Press, London (Stochastic models, sampling algorithms, and applications)
Nelsen R B (2006) An introduction to copulas, 2nd edn. Springer series in statistics. Springer, New York
Peng L (2008) Estimating the probability of a rare event via elliptical copulas. N Am Actuar J 12:116–128
Quessy J-F, Durocher M (2019) The class of copulas arising from squared distributions: properties and inference. Econ Stat 12:148–166
Quessy J-F, Rivest L-P, Toupin M-H (2016) On the family of multivariate chi-square copulas. J Multivar Anal 152:40–60
Scaillet O (2005) A Kolmogorov-Smirnov type test for positive quadrant dependence. Can J Stat 33:415–427
Schmid F, Schmidt R (2007) Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence. Metrika 66:323–354
Segers J (2012) Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18:764–782
Shorack GR, Wellner JA (1986) Empirical processes with applications to statistics. Wiley, New York
Telford RD, Cunningham RB (1991) Sex, sport, and body-size dependency of hematology in highly trained athletes. Med Sci Sports Exerc 23:788–794
Valdez EA, Chernih A (2003) Wang’s capital allocation formula for elliptically contoured distributions. Insur Math Econ 33:517–532
van der Vaart A W, Wellner J A (1996) Weak convergence and empirical processes. Springer series in statistics. Springer, New York
Veraverbeke N, Omelka M, Gijbels I (2011) Estimation of a conditional copula and association measures. Scand J Stat 38:766–780
Zhao Y, Genest C (2019) Inference for elliptical copula multivariate response regression models. Electron J Stat 13:911–984
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
I state that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Proofs
Proofs
1.1 Preliminaries
1.1.1 Notation
Let \(\ell ^\infty ([0,1]^d)\) be the space of bounded functions on \([0,1]^d\) and for \(h \in \ell ^\infty ([0,1]^d)\), define \(h_{jk} \in \ell ^\infty ([0,1]^2)\) as the function \(h(u_1, \ldots , u_d)\) evaluated at \(u_j := u\), \(u_k := v\), while letting all other components equal to 1. Then, define the functional \(\varPsi : \ell ^\infty ([0,1]^d) \rightarrow {\mathbb {R}}^L\) such that for \(h \in \ell ^\infty ([0,1]^d)\),
where \(\phi _\tau , \phi _\beta : \ell ^\infty ([0,1]^2) \rightarrow {\mathbb {R}}\) are the Kendall and Blomqvist functionals given respectively for \(h \in \ell ^\infty ([0,1]^2)\) by
1.1.2 Hadamard differentiability of \(\varPsi \)
Lemma 1 in Veraverbeke et al. (2011) ensures that \(\phi _\tau \) is Hadamard-differentiable tangentially to the set \({\mathcal {D}}_2\) of continuous functions on \([0,1]^2\) with derivative at C given for \(\varDelta \in {\mathcal {D}}_2\) by
It can be shown easily that \(\phi _\beta \) is Hadamard-differentiable with derivative at C given for \(\varDelta \in \ell ^\infty ([0,1]^2)\) by \(\phi '_{\beta ,C}(\varDelta ) = 4 \varDelta (1/2,1/2)\); in particular, \(\phi _\beta \) is Hadamard-differentiable tangentially to \({\mathcal {D}}_2\). Hence, \(\phi _\tau - \phi _\beta \) is Hadamard-differentiable tangentially to \({\mathcal {D}}_2\) with
By straightforward arguments, one can conclude that the vector of functionals \(\varPsi \) is Hadamard-differentiable tangentially to the set \({\mathcal {D}}_d\) of continuous functions on \([0,1]^d\) with derivative at C given for \(\varDelta \in {\mathcal {D}}_d\) by
Note that this derivative can be extended to \(\varDelta \in \ell ^\infty ([0,1]^d)\).
1.2 Proof of Proposition 1
First note that
Since \(\beta _{n,jk} = 4 \, C_{n,jk}(1/2,1/2) - 1 = \phi _\beta (C_{n,jk})\), one can write
where \(\varPsi _\tau (h) = ( \phi _\tau (h_{12}), \ldots , \phi _\tau (h_{d-1,d}) )\), \({\mathbf{1}}_L\) is a vector of ones of length L and \(C_n\) is the multivariate empirical copula, i.e.
Letting \({\mathbb {C}}_n = \sqrt{n} ( C_n - C )\) be the multivariate empirical copula process, Theorem 3.9.4 of van der Vaart and Wellner (1996), as well as the fact that \(\varPsi _\tau \) is bounded, entails
From Segers (2012), one can deduce an asymptotic representation for \({\mathbb {C}}_n\) in terms of \({\mathbf {U}}_1, \ldots , {\mathbf {U}}_n\). The bivariate marginals \({\mathbb {C}}_{n,12}, \ldots , {\mathbb {C}}_{n,d-1,d}\) of this large-sample representation are such that
Since
one has from the definition of \(\phi '_{\tau ,C}\) in Eq. (A1) that
Also, using the fact that
the Hadamard derivative \(\phi '_{\tau ,C}\) applied to the remaining expression yields
Since Hadamard derivatives are linear,
Also,
As a consequence, one has
Finally, since \((\phi _\tau - \phi _\beta )(C_{jk}) = \tau _{jk} - \beta _{jk}\), one has from Eq. (A2) that the components of \({\mathbf{D}}_n\) are given for \(\ell := (j-1) d + k - {j+1\atopwithdelims ()2}\) by
1.3 Proof of Corollary 1
By construction, elliptical copulas are radially symmetric. In particular, their bivariate marginals are radially symmetric, which means that \(C_{jk}(u,v) = u + v - 1 + C_{jk}(1-u,1-v)\). Differentiating on both sides yields \(\dot{C}_{jk,1}(u,v) = 1 - \dot{C}_{jk,1}(1-u,1-v)\) and \(\dot{C}_{jk,2}(u,v) = 1 - \dot{C}_{jk,2}(1-u,1-v)\), which allows to conclude that \(\dot{C}_{jk,1}(1/2,1/2) = \dot{C}_{jk,2}(1/2,1/2) = 1/2\). As a consequence, \(\delta _{C_{jk}}^\beta (u,v) = \delta ^\beta (u,v)\) and \(\{ \dot{C}_{jk,1}(1/2,1/2) - 1/2 \} \{ \dot{C}_{jk,2}(1/2,1/2) - 1/2 \} = 0\).
1.4 Proof of Proposition 2
Let \({\mathbf {U}}_1, \ldots , {\mathbf {U}}_n\) i.i.d. C be the unobservable random vectors such that for each \(i \in \{ 1, \ldots , n \}\),
From the definition of the components of \(\widehat{\mathbf{D}}_n = ({{\widehat{D}}}_{n1}, \ldots , {{\widehat{D}}}_{nL})\) given in (7), one can write for \(\ell := (j-1) d + k - {j+1\atopwithdelims ()2} \in \{ 1, \ldots , L \}\) that
where for \({\mathbb {C}}_{jk}\) that is the weak limit of \({\mathbb {C}}_{n,jk} = \sqrt{n}(C_{n,jk} - C_{jk})\),
Because of the asymptotic behavior of the univariate empirical distribution function as stated for instance by Shorack and Wellner (1986), \(| {{\widehat{U}}}_{ij} - U_{ij} | = | F_{nj}(Y_{ij}) - F_j(Y_{ij}) | \rightarrow 0\) almost surely as \(n \rightarrow \infty \). Hence, since the trajectories of \({\mathbb {C}}\), and therefore of \({\mathbb {C}}_{jk}\), are continuous, it follows that the difference between
is \(o_{\mathbb {P}}(1)\). From the multiplier central limit Theorem for Euclidean variables that is stated for instance in Lemma 10.5 of Kosorok (2008), the second expression above converges in distribution to the centered Normal with variance \({\mathrm{var}}\{ {\mathbb {C}}_{jk}(U_j,U_k) \}\), where \((U_j,U_k) \sim C_{jk}\). One can then conclude that \({{\widehat{A}}}_{n\ell }^{(1)} = o_{\mathbb {P}}(1)\). Next, from the Cauchy–Schwarz inequality,
The first expression in (A3) converges in probability to \({\mathrm{var}}(\xi _i) = 1\), while the second one is \(o_{\mathbb {P}}(1)\), due to the weak convergence of the empirical copula process \({\mathbb {C}}_n\) to \({\mathbb {C}}\) in the space \(\ell ^\infty ([0,1]^d)\) established by Segers (2012) under a regular copula; thus, \({{\widehat{A}}}_{n\ell }^{(2)} = o_{\mathbb {P}}(1)\). To deal with \({{\widehat{A}}}_{n\ell }^{(3)}\), define
and note that up to a difference of order 1/n that comes from \({\mathbb {I}}({{\widehat{U}}}_{ij} \le u) = {\mathbb {I}}( F_{nj}(Y_{ij}) \le u) \approx {\mathbb {I}}( Y_{ij} \le F_{nj}^{-1}(u)) = {\mathbb {I}}(U_{ij} \le F_j \circ F_{nj}^{-1}(u))\),
By the multiplier central limit Theorem for empirical processes, \({\widehat{\alpha }}_{n\ell }\) converges weakly in \(\ell ^\infty ([0,1]^2)\) to a process \(\alpha _\ell \) with continuous trajectories. Therefore, since by Shorack and Wellner (1986),
one can conclude that \({{\widehat{A}}}_{n\ell }^{(3)} = o_{\mathbb {P}}(1)\). Now for \({{\widehat{A}}}_{n\ell }^{(4)}\), a first order Taylor expansion applied to \(\delta _{C_{jk}}^\tau (u,v) = 2 C_{jk}(u,v) - u - v\) yields that for \((U_{ij}^\star ,U_{ik}^\star )\) between \(({{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik})\) and \((U_{ij},U_{ik})\),
where \({\mathbb {F}}_{nj}(y) = \sqrt{n} \{ F_{nj}(y) - F_j(y) \}\) converges weakly in \(\ell ^\infty ({\mathbb {R}})\) to the \(F_j\)-Brownian bridge \({\mathbb {F}}_j\). One can then write
From the multiplier central limit Theorem for Euclidean variables, the expression inside the brackets of the first two summands in (A4) converge in distribution to centered Normal distributions with variance bounded respectively by \({\mathrm{var}}\{ {\mathbb {F}}_j(Y_{1j}) \}\) and \({\mathrm{var}}\{ {\mathbb {F}}_k(Y_{1k}) \}\) since \(|2 \, \dot{C}_{jk,1}(u,v) - 1| \le 1\) and \(|2 \, \dot{C}_{jk,2}(u,v) - 1| \le 1\) for all \((u,v) \in [0,1]^2\); as a consequence, the first two summands in (A4) are \(o_{\mathbb {P}}(1)\). Using again \(|2 \, \dot{C}_{jk,1}(u,v) - 1| \le 1\) and \(|2 \, \dot{C}_{jk,2}(u,v) - 1| \le 1\), one can invoke the same reasoning used for \({{\widehat{A}}}_{n\ell }^{(4)}\) in (A3) and conclude that the third and fourth summands in (A4) are \(o_{\mathbb {P}}(1)\). As a consequence, \({{\widehat{A}}}_{n\ell }^{(4)} = o_{\mathbb {P}}(1)\).
It has been shown that for any \(\ell \in \{ 1, \ldots , L \}\), \({{\widehat{A}}}_{n\ell }^{(1)} + {{\widehat{A}}}_{n\ell }^{(2)} + {{\widehat{A}}}_{n\ell }^{(3)} + {{\widehat{A}}}_{n\ell }^{(4)} = o_{\mathbb {P}}(1)\). This result can be easily extended jointly for all \(\ell \in \{ 1, \ldots , L \}\), so that
where
From an application of the multiplier central limit Theorem for Euclidean variables combined with Slutsky’s theorem, one can conclude that \(( \sqrt{n} ( {\mathbf{D}}_n - {\varvec{\mu }}), \sqrt{n} \, \widehat{\mathbf{D}}_n)\) converges weakly to \(({\mathbb {D}},{{\widetilde{{\mathbb {D}}}}})\), where \({{\widetilde{{\mathbb {D}}}}}\) is an independent copy of \({\mathbb {D}}\), i.e. a L-variate centered Normal with covariance matrix \(\varSigma \) whose components are described in (6).
Rights and permissions
About this article
Cite this article
Quessy, JF. On nonparametric tests of multivariate meta-ellipticity. Stat Papers 62, 2283–2310 (2021). https://doi.org/10.1007/s00362-020-01189-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-020-01189-x