On nonparametric tests of multivariate meta-ellipticity

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.

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 $h \in \ell ^\infty ([0,1]^d)$,

$$\begin{aligned} \varPsi (h) = \left( (\phi _\tau - \phi _\beta )(h_{12}), \ldots , (\phi _\tau - \phi _\beta )(h_{d-1,d}) \right) , \end{aligned}$$

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

$$\begin{aligned} \phi _\tau (h) = 4 \int _{[0,1]^2} h\, {\mathrm{d}}h - 1 ~~\text{ and }~~ \phi _\beta (h) = 4 \, h \left( {1\over 2},{1\over 2} \right) - 1. \end{aligned}$$

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

$$\begin{aligned} \phi '_{\tau ,C}(\varDelta )&= 8 \int _{[0,1]^2} \varDelta \, {\mathrm{d}}C + 4 \varDelta (1,1) - 4 \int _0^1 \left\{ \varDelta (u,1) + \varDelta (1,u) \right\} {\mathrm{d}}u. \end{aligned}$$

(A1)

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

$$\begin{aligned} \left( \phi _\tau - \phi _\beta \right) _C'(\varDelta )= & {} 8 \int _{[0,1]^2} \varDelta \, {\mathrm{d}}C + 4 \varDelta (1,1) - 4 \int _0^1 \left\{ \varDelta (u,1) + \varDelta (1,u) \right\} {\mathrm{d}}u\\&- 4 \varDelta \left( {1\over 2},{1\over 2} \right) . \end{aligned}$$

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

$$\begin{aligned} \varPsi '_C(\varDelta ) = \left( (\phi _\tau - \phi _\beta )_C'(\varDelta _{12}), \ldots , (\phi _\tau - \phi _\beta )_C'(\varDelta _{d-1,d}) \right) . \end{aligned}$$

Note that this derivative can be extended to $\varDelta \in \ell ^\infty ([0,1]^d)$.

1.2 Proof of Proposition 1

First note that

$$\begin{aligned} \tau _{n,jk}= & {} {4 \over n-1} \sum _{i'=1}^n \left\{ {1\over n} \sum _{i=1}^n {\mathbb {I}}\left( {{\widehat{U}}}_{ij} \le {{\widehat{U}}}_{i'j}, {{\widehat{U}}}_{ik} \le {{\widehat{U}}}_{i'k} \right) - {1\over n} \right\} - 1 \\= & {} {4n \over n-1} \times {1\over n} \sum _{i'=1}^n C_{n,jk} \left( {{\widehat{U}}}_{i'j}, {{\widehat{U}}}_{i'k} \right) - {4 \over n-1} - 1 \\= & {} {4n \over n-1} \int _{[0,1]^2} C_{n,jk}(u_j,u_k) \, {\mathrm{d}}C_{n,jk}(u_j,u_k) - {4 \over n-1} - 1 \\= & {} 4 \int _{[0,1]^2} C_{n,jk}(u_j,u_k) \, {\mathrm{d}}C_{n,jk}(u_j,u_k) - 1 \\&+ \, {1\over n-1} \left( 4 \int _{[0,1]^2} C_{n,jk}(u_j,u_k) \, {\mathrm{d}}C_{n,jk}(u_j,u_k) - 4 \right) \\= & {} \phi _\tau (C_{n,jk}) + \left\{ \phi _\tau (C_{n,jk}) - 3 \over n - 1 \right\} . \end{aligned}$$

Since $\beta _{n,jk} = 4 \, C_{n,jk}(1/2,1/2) - 1 = \phi _\beta (C_{n,jk})$, one can write

$$\begin{aligned} {\mathbf{D}}_n = \varPsi (C_n) + \left\{ \varPsi _\tau (C_n) - 3 \, {\mathbf{1}}_L \over n-1 \right\} , \end{aligned}$$

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.

$$\begin{aligned} C_n({\mathbf {u}}) = {1\over n} \sum _{i=1}^n {\mathbb {I}}\left( {\widehat{{\mathbf {U}}}}_i \le {\mathbf {u}}\right) . \end{aligned}$$

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

$$\begin{aligned} {\mathbf{D}}_n&= \left\{ \varPsi \left( C + {{\mathbb {C}}_n({\mathbf {u}}) \over \sqrt{n}} \right) - \varPsi (C) \right\} + \varPsi (C) + \left\{ \varPsi _\tau (C_n) - 3 \, {\mathbf{1}}_L \over n-1 \right\} \nonumber \\&= \varPsi '_C \left( {{\mathbb {C}}_n \over \sqrt{n}} \right) + \varPsi (C) + o_{\mathbb {P}}(n^{-1/2}). \end{aligned}$$

(A2)

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

$$\begin{aligned} {{\mathbb {C}}_{n,jk}(u,v) \over \sqrt{n}}= & {} {1\over n} \sum _{i=1}^n \left\{ {\mathbb {I}}\left( U_{ij} \le u, U_{ik} \le v \right) - {\mathbb {I}}\left( U_{ij} \le u \right) \dot{C}_{jk,1}(u,v) \right. \\&\left. - {\mathbb {I}}\left( U_{ik} \le v \right) \dot{C}_{jk,2}(u,v) \right\} \\&- \left\{ C_{jk}(u,v) - u \, \dot{C}_{jk,1}(u,v) - v \, \dot{C}_{jk,2}(u,v) \right\} + o_{\mathbb {P}}(n^{-1/2}). \end{aligned}$$

Since

$$\begin{aligned} \int _0^1 \int _a^1 \dot{C}_1(u,v) \, {\mathrm{d}}C(u,v) = \int _a^1 \int _0^1 \dot{C}_2(u,v) \, {\mathrm{d}}C(u,v) = {1-a \over 2} \,, \end{aligned}$$

one has from the definition of $\phi '_{\tau ,C}$ in Eq. (A1) that

$$\begin{aligned}&\phi '_{\tau ,C} \left\{ {\mathbb {I}}\left( U_{ij} \le u, U_{ik} \le v \right) - {\mathbb {I}}\left( U_{ij} \le u \right) \dot{C}_{jk,1}(u,v) - {\mathbb {I}}\left( U_{ik} \le v \right) \dot{C}_{jk,2}(u,v) \right\} \\&\quad =8 \left\{ C_{jk}(U_{ij},U_{ik}) - U_{ij} - U_{ik} + 1 \right\} - 4 \left( 1 - U_{ij} \right) - 4 \left( 1 - U_{ik} \right) \\&\quad = 4 \left\{ 2 C_{jk}(U_{ij},U_{ik}) - U_{ij} - U_{ik} \right\} . \end{aligned}$$

Also, using the fact that

$$\begin{aligned} \int _0^1 \int _0^1 u \, \dot{C}_1(u,v) \, {\mathrm{d}}C(u,v) = \int _0^1 \int _0^1 v \, \dot{C}_2(u,v) \, {\mathrm{d}}C(u,v) = {1\over 4} \,, \end{aligned}$$

the Hadamard derivative $\phi '_{\tau ,C}$ applied to the remaining expression yields

$$\begin{aligned} \phi '_{\tau ,C} \left\{ C_{jk}(u,v) - u \, \dot{C}_{jk,1}(u,v) - v \, \dot{C}_{jk,2}(u,v) \right\} = 2(\tau _{jk} + 1) - 2 - 2 = 2(\tau _{jk} - 1). \end{aligned}$$

Since Hadamard derivatives are linear,

$$\begin{aligned} \phi '_{\tau ,C} \left( {{\mathbb {C}}_{n,jk} \over \sqrt{n}} \right) = {4\over n} \sum _{i=1}^n \left\{ 2 C_{jk}(U_{ij},U_{ik}) - U_{ij} - U_{ik} \right\} - 2(\tau _{jk} - 1) + o_{\mathbb {P}}(n^{-1/2}). \end{aligned}$$

Also,

$$\begin{aligned} \phi '_{\beta ,C} \left( {{\mathbb {C}}_{n,jk} \over \sqrt{n}} \right)= & {} {4\over \sqrt{n}} \, {\mathbb {C}}_{n,jk} \left( {1\over 2},{1\over 2} \right) \\= & {} {4\over n} \sum _{i=1}^n \left\{ {\mathbb {I}}\left( U_{ij} \le {1\over 2}, U_{ik} \le {1\over 2} \right) - {\mathbb {I}}\left( U_{ij} \le {1\over 2} \right) \dot{C}_{jk,1} \left( {1\over 2},{1\over 2} \right) \right. \\&\left. - \, {\mathbb {I}}\left( U_{ik} \le {1\over 2} \right) \dot{C}_{jk,2} \left( {1\over 2},{1\over 2} \right) \right\} \\&- 4 \left\{ C_{jk} \left( {1\over 2},{1\over 2} \right) - {1\over 2} \, \dot{C}_{jk,1} \left( {1\over 2},{1\over 2} \right) - {1\over 2} \, \dot{C}_{jk,2}\left( {1\over 2},{1\over 2} \right) \right\} \\&+ o_{\mathbb {P}}(n^{-1/2}). \end{aligned}$$

As a consequence, one has

$$\begin{aligned} (\phi '_{\tau ,C} - \phi '_{\beta ,C}) \left( {{\mathbb {C}}_{n,jk} \over \sqrt{n}} \right)= & {} \phi '_{\tau ,C} \left( {{\mathbb {C}}_{n,jk} \over \sqrt{n}} \right) - \phi '_{\beta ,C} \left( {{\mathbb {C}}_{n,jk} \over \sqrt{n}} \right) \\= & {} {4\over n} \sum _{i=1}^n \left\{ 2 C_{jk}(U_{ij},U_{ik}) - U_{ij} - U_{ik} - \, {\mathbb {I}}\left( U_{ij} \le {1\over 2}, U_{ik} \le {1\over 2} \right) \right. \\&\left. + \, {\mathbb {I}}\left( U_{ij} \le {1\over 2} \right) \dot{C}_{jk,1} \left( {1\over 2},{1\over 2} \right) + {\mathbb {I}}\left( U_{ik} \le {1\over 2} \right) \dot{C}_{jk,2} \left( {1\over 2},{1\over 2} \right) \right\} \\&- 2 \left( \tau _{jk} - 1 \right) + 4 \, C_{jk} \left( {1\over 2},{1\over 2} \right) - 2 \, \dot{C}_{jk,1} \left( {1\over 2},{1\over 2} \right) \\&- \, 2 \, \dot{C}_{jk,2} \left( {1\over 2},{1\over 2} \right) + o_{\mathbb {P}}(n^{-1/2}) \\= & {} {4\over n} \sum _{i=1}^n \left\{ \delta _{C_{jk}}^\tau (U_{ij},U_{ik}) - \delta _{C_{jk}}^\beta (U_{ij},U_{ik}) \right\} + 2 - 2 \tau _{jk} + \beta _{jk} \\&+ \, 4 \left\{ \dot{C}_{jk,1} \left( {1\over 2},{1\over 2} \right) - {1\over 2} \right\} \left\{ \dot{C}_{jk,2} \left( {1\over 2},{1\over 2} \right) - {1\over 2} \right\} + o_{\mathbb {P}}(n^{-1/2}). \end{aligned}$$

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

$$\begin{aligned} D_{n\ell }= & {} {4\over n} \sum _{i=1}^n \left\{ \delta _{C_{jk}}^\tau (U_{ij},U_{ik}) - \delta _{C_{jk}}^\beta (U_{ij},U_{ik}) \right\} + 2 - \tau _{jk} \\&+ \, 4 \left\{ \dot{C}_{jk,1} \left( {1\over 2},{1\over 2} \right) - {1\over 2} \right\} \left\{ \dot{C}_{jk,2} \left( {1\over 2},{1\over 2} \right) - {1\over 2} \right\} + o_{\mathbb {P}}(n^{-1/2}). \end{aligned}$$

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 $i \in \{ 1, \ldots , n \}$,

$$\begin{aligned} {\mathbf {U}}_i = \left( U_{i1}, \ldots , U_{id} \right) = \left( F_1(Y_{i1}), \ldots , F_d(Y_{id} \right) . \end{aligned}$$

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

$$\begin{aligned} \sqrt{n} {{\widehat{D}}}_{n\ell }= & {} {4\over \sqrt{n}} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ {\widehat{\delta }}_{C_{jk}}^\tau \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) - \delta ^\beta \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) \right\} \\= & {} {4\over \sqrt{n}} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ \delta _{C_{jk}}^\tau \left( U_{ij}, U_{ik} \right) - \delta ^\beta \left( U_{ij}, U_{ik} \right) \right\} \\&+ \, {{\widehat{A}}}_{n\ell }^{(1)} + {{\widehat{A}}}_{n\ell }^{(2)} + {{\widehat{A}}}_{n\ell }^{(3)} + {{\widehat{A}}}_{n\ell }^{(4)}, \end{aligned}$$

where for ${\mathbb {C}}_{jk}$ that is the weak limit of ${\mathbb {C}}_{n,jk} = \sqrt{n}(C_{n,jk} - C_{jk})$,

$$\begin{aligned} {{\widehat{A}}}_{n\ell }^{(1)}= & {} {8\over \sqrt{n}} \left\{ {1\over \sqrt{n} } \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) {\mathbb {C}}_{jk} \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) \right\} , \nonumber \\ {{\widehat{A}}}_{n\ell }^{(2)}= & {} {8\over n} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ {\mathbb {C}}_{n,jk} \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) - {\mathbb {C}}_{jk} \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) \right\} , \nonumber \\ {{\widehat{A}}}_{n\ell }^{(3)}= & {} {4\over \sqrt{n}} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ \delta ^\beta \left( U_{ij}, U_{ik} \right) - \delta ^\beta \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) \right\} , \nonumber \\ \text{ and } ~~ {{\widehat{A}}}_{n\ell }^{(4)}= & {} {4\over \sqrt{n}} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ \delta _{C_{jk}}^\tau \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) - \delta _{C_{jk}}^\tau \left( U_{ij}, U_{ik} \right) \right\} . \end{aligned}$$

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

$$\begin{aligned} {1\over \sqrt{n} } \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) {\mathbb {C}}_{jk} \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) ~~\text{ and }~~ {1\over \sqrt{n} } \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) {\mathbb {C}}_{jk}(U_{ij},U_{ik}) \end{aligned}$$

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,

$$\begin{aligned} \left| {{\widehat{A}}}_{n\ell }^{(2)} \right|&\le {8\over n} \sqrt{ \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) ^2 } \sqrt{ \sum _{i=1}^n \left( {\mathbb {C}}_{n,jk} - {\mathbb {C}}_{jk} \right) ^2( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik}) } \nonumber \\&\le 8 \sqrt{ {1\over n} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) ^2 } \sqrt{ \sup _{(u,v) \in [0,1]^2} \left( {\mathbb {C}}_{n,jk} - {\mathbb {C}}_{jk} \right) ^2(u,v) }. \end{aligned}$$

(A3)

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

$$\begin{aligned} {\widehat{\alpha }}_{n\ell }(u,v) = {4\over \sqrt{n} } \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ {\mathbb {I}}\left( U_{ij} \le u \right) - {1\over 2} \right\} \left\{ {\mathbb {I}}\left( U_{ik} \le v \right) - {1\over 2} \right\} \end{aligned}$$

and note that up to a difference of order 1/n that comes from ${\mathbb {I}}({{\widehat{U}}}_{ij} \le u) = {\mathbb {I}}( F_{nj}(Y_{ij}) \le u) \approx {\mathbb {I}}( Y_{ij} \le F_{nj}^{-1}(u)) = {\mathbb {I}}(U_{ij} \le F_j \circ F_{nj}^{-1}(u))$,

$$\begin{aligned} {{\widehat{A}}}_{n\ell }^{(3)} = {\widehat{\alpha }}_{n\ell } \left( {1\over 2}, {1\over 2} \right) - {\widehat{\alpha }}_{n\ell } \left\{ F_j \circ F_{nj}^{-1} \left( 1\over 2\right) , F_k \circ F_{nk}^{-1} \left( 1\over 2\right) \right\} . \end{aligned}$$

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),

$$\begin{aligned} \sup _{u\in [0,1]} \left| F_{nj}^{-1}(u) - F_j^{-1}(u) \right| {\mathop {\longrightarrow }\limits ^{a.s.}}0, \end{aligned}$$

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 $(U_{ij},U_{ik})$,

$$\begin{aligned} \delta _{C_{jk}}^\tau \left( {{\widehat{U}}}_{ij}, {{\widehat{U}}}_{ik} \right) - \delta _{C_{jk}}^\tau \left( U_{ij}, U_{ik} \right)= & {} \left\{ 2 \, \dot{C}_{jk,1}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} \left( {{\widehat{U}}}_{ij} - U_{ij} \right) \\&+ \, \left\{ 2 \, \dot{C}_{jk,2}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} \left( {{\widehat{U}}}_{ik} - U_{ik} \right) \\= & {} \left\{ 2 \, \dot{C}_{jk,1}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} { {\mathbb {F}}_{nj}(Y_{ij}) \over \sqrt{n} } \\&+ \, \left\{ 2 \, \dot{C}_{jk,2}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} { {\mathbb {F}}_{nk}(Y_{ik}) \over \sqrt{n} } \,, \end{aligned}$$

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

$$\begin{aligned} {{\widehat{A}}}_{n\ell }^{(4)}&= {4\over n} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ 2 \, \dot{C}_{jk,1}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} {\mathbb {F}}_{nj}(Y_{ij}) \nonumber \\&\qquad + \, {4\over n} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ 2 \, \dot{C}_{jk,2}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} {\mathbb {F}}_{nk}(Y_{ik}) \nonumber \\&\quad = {4\over \sqrt{n}} \left\{ {1\over \sqrt{n}} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ 2 \, \dot{C}_{jk,1}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} {\mathbb {F}}_j(Y_{ij}) \right\} \nonumber \\&\qquad + \, {4\over \sqrt{n}} \left\{ {1\over \sqrt{n}} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ 2 \, \dot{C}_{jk,2}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} {\mathbb {F}}_k(Y_{ik}) \right\} \nonumber \\&\qquad + \, {4\over n} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ 2 \, \dot{C}_{jk,2}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} ({\mathbb {F}}_{nj} - {\mathbb {F}}_j)(Y_{ij}) \nonumber \\&\qquad + \, {4\over n} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ 2 \, \dot{C}_{jk,2}(U_{ij}^\star ,U_{ik}^\star ) - 1 \right\} ({\mathbb {F}}_{nk} - {\mathbb {F}}_k)(Y_{ik}). \end{aligned}$$

(A4)

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

$$\begin{aligned} \sqrt{n} \, \widehat{\mathbf{D}}_n = \left( {{\widehat{D}}}^\star _{n1}, \ldots , {{\widehat{D}}}_{nL}^\star \right) + o_{\mathbb {P}}(1), \end{aligned}$$

where

$$\begin{aligned} {{\widehat{D}}}_{n\ell }^\star = {4\over \sqrt{n}} \sum _{i=1}^n \left( \xi _i - {\bar{\xi }} \right) \left\{ \delta _{C_{jk}}^\tau (U_{ij},U_{ik}) - \delta ^\beta (U_{ij},U_{ik}) \right\} . \end{aligned}$$

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).