Testing the constancy of Spearman's rho in multivariate time series
aa r X i v : . [ s t a t . M E ] F e b Testing the constancy of Spearman’s rho inmultivariate time series
Ivan Kojadinovic
Laboratoire de math´ematiques et applications, UMR CNRS 5142Universit´e de Pau et des Pays de l’AdourB.P. 1155, 64013 Pau Cedex, France [email protected]
Jean-Fran¸cois Quessy
D´epartement de math´ematiques et d’informatiqueUniversit´e du Qu´ebec `a Trois-Rivi`eresTrois-Rivi`eres, Qu´ebec, C.P. 500, G9A 5H7 Canada [email protected]
Tom Rohmer
Laboratoire de math´ematiques Jean LerayUniversit´e de NantesB.P. 92208, 44322 Nantes Cedex 3, France [email protected]
February 27, 2015
Abstract
A class of tests for change-point detection designed to be particularly sensi-tive to changes in the cross-sectional rank correlation of multivariate time seriesis proposed. The derived procedures are based on several multivariate extensionsof Spearman’s rho. Two approaches to carry out the tests are studied: the firstone is based on resampling, the second one consists of estimating the asymptoticnull distribution. The asymptotic validity of both techniques is proved under thenull for strongly mixing observations. A procedure for estimating a key band-width parameter involved in both approaches is proposed, making the derived testsparameter-free. Their finite-sample behavior is investigated through Monte Carloexperiments. Practical recommendations are made and an illustration on trivariatefinancial data is finally presented.
Keywords: change-point detection; empirical copula; HAC kernel variance estima-tor; multiplier central limit theorems; partial-sum processes; ranks; Spearman’srho; strong mixing. Introduction
Let X , . . . , X n be a multivariate times series of d -dimensional observations and, for any i ∈ { , . . . , n } , let F ( i ) denote the cumulative distribution function (c.d.f.) of X i . We areinterested in procedures for testing H : F (1) = · · · = F ( n ) against ¬ H . Notice that theaforementioned null hypothesis can be simply rewritten as H : ∃ F such that X , . . . , X n have c.d.f. F. (1.1)Such statistical procedures are commonly referred to as tests for change-point detection (see, e.g., Cs¨org˝o and Horv´ath, 1997, for an overview of possible approaches). The ma-jority of tests for H developed in the literature deal with the case d = 1. We aim atdeveloping nonparametric tests for multivariate time series that are particularly sensitiveto changes in the dependence among the components of the d -dimensional observations.The availability of such tests seems to be of great practical importance for the analysisof economic data, among others. In particular, assessing whether the dependence amongfinancial assets can be considered constant or not over a given time period appears crucialfor risk management, portfolio optimization and related statistical modeling (see, e.g.,Wied et al., 2014; Dehling et al., 2014, and the references therein for a more detaileddiscussion about the motivation for such statistical procedures).The above context, rather naturally, suggests to address the informal notion of depen-dence through that of copula (see, e.g., Nelsen, 2006). Assume that H in (1.1) holds andthat, additionally, the common marginal c.d.f.s F , . . . , F d of X , . . . , X n are continuous.Then, from the work of Sklar (1959), the common multivariate c.d.f. F of the observationscan be written as F ( x ) = C { F ( x ) , . . . , F d ( x d ) } , x ∈ R d , where the function C : [0 , d → [0 ,
1] is the unique copula associated with F . It followsthat H can be rewritten as H ,m ∩ H ,c , where H ,m : ∃ F , . . . , F d such that X , . . . , X n have marginal c.d.f.s F , . . . , F d , (1.2) H ,c : ∃ C such that X , . . . , X n have copula C. (1.3)Several nonparametric tests designed to be particularly sensitive to certain alterna-tives under H ,m ∩ ¬ H ,c were proposed in the literature. Tests for the constancy ofKendall’s tau (which is a functional of C ) were investigated by Gombay and Horv´ath(1999) (see also Gombay and Horv´ath, 2002) and Quessy et al. (2013) in the case of seri-ally independent observations. A version of the previous tests adapted to a very generalclass of bivariate time series was proposed by Dehling et al. (2014). Recent multivariatealternatives are the tests studied in B¨ucher et al. (2014, see also the references therein)based on Cram´er–von Mises functionals of the sequential empirical copula process .The aim of this work is to derive tests for the constancy of several multivariate exten-sions of Spearman’s rho (which are also functionals of C ) in multivariate strongly mixingtime series. A similar problem was recently tackled by Wied et al. (2014). However, asthe functional they considered does not exactly correspond to a multivariate extension ofSpearman’s rho (because of the way ranks are calculated), the corresponding test turn2ut to have a rather low power. We remedy to that situation by computing ranks withrespect to the relevant subsamples. From a theoretical perspective, as in Wied et al.(2014), no assumptions on the first order partial derivatives of the copula are made. Thelatter is actually an advantage of the studied tests over that investigated in B¨ucher et al.(2014). An inconvenience with respect to the aforementioned approach is however that,as all tests based on moments of copulas (such as Spearman’s rho or Kendall’s tau), thederived tests will have no power, by construction, against alternatives involving changesin the copula at a constant value of Spearman’s rho.To carry out the tests, we propose two approaches for computing approximate p-values: the first one is based on resampling while the second one consists of estimatingthe asymptotic null distribution. In addition, a procedure for estimating a key bandwidthparameter involved in both approaches is proposed, making the derived tests fully data-driven. The versions of the studied tests based on the estimation of the asymptoticnull distribution can be seen as alternatives to the test based on Kendall’s tau recentlyproposed by Dehling et al. (2014).The paper is organized as follows. The test statistics are defined in the second sectionand their limiting null distribution is established under strong mixing. Section 3 presentstwo approaches for computing approximate p-values based, respectively, on bootstrappingand on the estimation of an asymptotic variance. The fourth section partially reports theresults of Monte Carlo experiments involving bivariate and fourvariate time series gen-erated from autoregressive and GARCH-like models. The fifth section contains practicalrecommendations and an illustration on trivariate financial data, while the last sectionconcludes.In the rest of the paper, the arrow ‘ ’ denotes weak convergence in the sense of Def-inition 1.3.3 in van der Vaart and Wellner (2000). Also, given a set T , ℓ ∞ ( T ; R ) denotesthe space of all bounded real-valued functions on T equipped with the uniform metric.The proofs of the stated theoretical results are available in the online supplementary ma-terial and the studied tests for change-point detection are implemented in the package npcp (Kojadinovic, 2014) for the R statistical system (R Development Core Team, 2014). Spearman’s rho is a very well-known measure of bivariate dependence (see, e.g., Nelsen,2006, Section 5.1 and the references therein). For a bivariate random vector with contin-uous margins and copula C , it can be expressed as ρ ( C ) = 12 Z [0 , C ( u )d u − Z [0 , u u d C ( u ) − . d -dimensional with d >
2, the following threepossible extensions were proposed by Schmid and Schmidt (2007): ρ ( C ) = d + 12 d − d − (cid:26) d Z [0 , d C ( u )d u − (cid:27) ,ρ ( C ) = ρ ( ¯ C ) ,ρ ( C ) = (cid:18) d (cid:19) − X ≤ i
1. Indeed, it is easy to verify that, under H defined in (1.1), T n,A ( s ) = λ n ( s, S n,A (0 , s ) − λ n (0 , s ) S n,A ( s, , s ∈ [0 , . (2.9)As we shall see below, the limiting null distribution of T n is then a mere consequence ofthe fact that the empirical processes S n,A , A ⊆ D , | A | ≥
1, are asymptotically equivalentto continuous functionals of the sequential empirical process B n ( s, t, u ) = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 { ( U i ≤ u ) − C ( u ) } , ( s, t, u ) ∈ ∆ × [0 , d , (2.10)where U , . . . , U n is the unobservable sample obtained from X , . . . , X n by the probabil-ity integral transforms U ij = F j ( X ij ), i ∈ { , . . . , n } , j ∈ D .If U , . . . , U n is drawn from a strictly stationary sequence ( U i ) i ∈ Z whose strong mixingcoefficients satisfy α r = O ( r − a ) with a >
1, we have from B¨ucher (2014) that B n (0 , · , · )converges weakly in ℓ ∞ ([0 , d +1 ; R ) to a tight centered Gaussian process B ◦ C with covari-ance function cov { B ◦ C ( s, u ) , B ◦ C ( t, v ) } = ( s ∧ t ) κ C ( u , v ), ( s, u ) , ( t, v ) ∈ [0 , d +1 , where κ C ( u , v ) = cov { B ◦ C (1 , u ) , B ◦ C (1 , v ) } = X k ∈ Z cov { ( U ≤ u ) , ( U k ≤ v ) } . (2.11)As a consequence of the continuous mapping theorem, B n B C in ℓ ∞ (∆ × [0 , d ; R ),where B C ( s, t, u ) = B ◦ C ( t, u ) − B ◦ C ( s, u ) , ( s, t, u ) ∈ ∆ × [0 , d . (2.12)The following proposition, proved in Section A of the supplementary material, is thekey step for obtaining the limiting null distribution of the vector-valued process T n definedin (2.5). Proposition 1.
Assume that X , . . . , X n is drawn from a strictly stationary sequence ( X i ) i ∈ Z with continuous margins and whose strong mixing coefficients satisfy α r = O ( r − a ) , a > . Then, for any A ⊆ D , | A | ≥ , sup ( s,t ) ∈ ∆ | S n,A ( s, t ) − ψ C,A { B n ( s, t, · ) }| = o P (1) , (2.13) where ψ C,A is a linear map from ℓ ∞ ([0 , d ; R ) to R defined by ψ C,A ( g ) = φ A ( g ) − Z [0 , d X j ∈ A Y l ∈ A \{ j } (1 − v l ) g ( v { j } )d C ( v ) , g ∈ ℓ ∞ ([0 , d ; R ) , (2.14) with φ A given in (2.4) . X , . . . , X n is drawn froma stationary vector ARMA process with absolutely continuous innovations. A similarconclusion holds for a large class of GARCH processes (see Lindner, 2009, Section 5, andthe references therein).The next result, proved in Section B of the supplementary material, is a consequenceof the previous proposition and establishes the limiting null distribution of the genericstatistic S n,f defined in (2.6) under strong mixing. Corollary 2.
Under the conditions of Proposition 1, T n s T C ( s ) = (cid:0) T C, { } ( s ) , T C, { } ( s ) , . . . , T C,D ( s ) (cid:1) (2.15) in ℓ ∞ ([0 , R d − ) , where T C ( s ) = ψ C { B C (0 , s, · ) − s B C (0 , , · ) } , s ∈ [0 , , (2.16) with B C defined in (2.12) and ψ C a map from ℓ ∞ ([0 , d ; R ) to R d − defined by ψ C ( g ) = (cid:0) ψ C, { } ( g ) , ψ C, { } ( g ) , . . . , ψ C,D ( g ) (cid:1) , g ∈ ℓ ∞ ([0 , d ; R ) . (2.17) As a consequence, for any f : R d − → R continuous, S n,f = sup s ∈ [0 , | f { T n ( s ) }| S C,f = sup s ∈ [0 , | f { T C ( s ) }| , and, if f is additionally linear and σ C,f = var[ f ◦ ψ C { B C (0 , , · ) } ] > , the weak limitof σ − C,f S n,f is equal in distribution to sup s ∈ [0 , | U ( s ) | , where U is a standard Brownianbridge on [0 , . Corollary 2 suggests two related ways to compute p-values for the generic test statistic S n,f defined in (2.6). The first approach, based on resampling, consists of exploitingthe fact that, under H , T n defined in (2.5) is asymptotically equivalent to a continuousfunctional of the sequential empirical process B n defined in (2.10) and can be applied assoon as f : R d − → R is continuous. The second approach, restricted to the situationwhen f is linear, is motivated by the last claim of Corollary 2. It consists of estimating σ C,f and thus the asymptotic null distribution of S n,f . The first approach that we consider consists of bootstrapping the vector-valued empiricalprocess T n defined in (2.5) using a bootstrap for the sequential empirical process B n .This way of proceeding actually allows us to consider not only linear but also continuous f in (2.6). More specifically, we consider a multiplier bootstrap for B n in thespirit of van der Vaart and Wellner (2000, Chapter 2.9) when observations are seriallyindependent, or B¨uhlmann (1993, Section 3.3) when they are serially dependent. In thelatter case, we rely on the recent work of B¨ucher and Kojadinovic (2014).The notion of multiplier sequence is central to this resampling technique. We say thata sequence of random variables ( ξ i,n ) i ∈ Z is an i.i.d. multiplier sequence if:(M0) ( ξ i,n ) i ∈ Z is i.i.d., independent of X , . . . , X n , with distribution not changing with n ,having mean 0, variance 1, and being such that R ∞ { P( | ξ ,n | > x ) } / d x < ∞ .We say that a sequence of random variables ( ξ i,n ) i ∈ Z is a dependent multiplier sequence if:(M1) The sequence ( ξ i,n ) i ∈ Z is strictly stationary with E( ξ ,n ) = 0, E( ξ ,n ) = 1 andsup n ≥ E( | ξ ,n | ν ) < ∞ for all ν ≥
1, and is independent of the available sample X , . . . , X n .(M2) There exists a sequence ℓ n → ∞ of strictly positive constants such that ℓ n = o ( n )and the sequence ( ξ i,n ) i ∈ Z is ℓ n -dependent, i.e., ξ i,n is independent of ξ i + h,n for all h > ℓ n and i ∈ N .(M3) There exists a function ϕ : R → [0 , ϕ (0) = 1 and ϕ ( x ) = 0 for all | x | > ξ ,n ξ h,n ) = ϕ ( h/ℓ n ) forall h ∈ Z .The choice of the function ϕ and an approach to generate dependent multiplier sequencesis briefly discussed in Section 4. More details can be found in B¨ucher and Kojadinovic(2014, Section 5.2).Let M be a large integer and let ( ξ (1) i,n ) i ∈ Z , . . . , ( ξ ( M ) i,n ) i ∈ Z be M independent copiesof the same multiplier sequence. Then, following B¨ucher and Kojadinovic (2014) andB¨ucher et al. (2014), for any m ∈ { , . . . , M } and ( s, t, u ) ∈ ∆ × [0 , d , letˆ B ( m ) n ( s, t, u ) = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ξ ( m ) i,n { ( ˆ U ni ≤ u ) − C n ( u ) } , ˇ B ( m ) n ( s, t, u ) = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( ξ ( m ) i,n − ¯ ξ ( m ) ⌊ ns ⌋ +1: ⌊ nt ⌋ ) ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ i ≤ u ) , (3.1)where ¯ ξ ( m ) k : l is the arithmetic mean of ξ ( m ) i,n for i ∈ { k, . . . , l } .The following proposition is a consequence of Theorem 1 in Holmes et al. (2013),Theorem 2.1 and the proof of Proposition 4.2 in B¨ucher and Kojadinovic (2014), as well asthe proof of Proposition 4.3 in B¨ucher et al. (2014). It suggests interpreting the multiplierreplicates ˆ B (1) n , . . . , ˆ B ( M ) n (resp. ˇ B (1) n , . . . , ˇ B ( M ) n ) as “almost” independent copies of B n as n increases. Proposition 3.
Assume that either i) the random vectors X , . . . , X n are i.i.d. with continuous margins and the se-quences ( ξ (1) i,n ) i ∈ Z , . . . , ( ξ ( M ) i,n ) i ∈ Z are independent copies of a multiplier sequence sat-isfying (M0), (ii) or the random vectors X , . . . , X n are drawn from a strictly stationary sequence ( X i ) i ∈ Z with continuous margins whose strong mixing coefficients satisfy α r = O ( r − a ) for some a > d/ , and ( ξ (1) i,n ) i ∈ Z , . . . , ( ξ ( M ) i,n ) i ∈ Z are independent copiesof a dependent multiplier sequence satisfying (M1)–(M3) with ℓ n = O ( n / − ε ) forsome < ε < / .Then, (cid:16) B n , ˆ B (1) n , . . . , ˆ B ( M ) n (cid:17) (cid:16) B C , B (1) C , . . . , B ( M ) C (cid:17) , (cid:0) B n , ˇ B (1) n , . . . , ˇ B ( M ) n (cid:1) (cid:16) B C , B (1) C , . . . , B ( M ) C (cid:17) in { ℓ ∞ (∆ × [0 , d ; R ) } M +1 , where B C is given in (2.12) and B (1) C , . . . , B ( M ) C are independentcopies of B C . Starting from the quantities defined above, we shall now define appropriate multiplierreplicates under H of T n defined in (2.5). From (2.9), we see that to do so, we firstneed to define multiplier replicates of the processes S n,A , A ⊆ D , | A | ≥
1, definedin (2.8). From (2.13) and Proposition 3, natural candidates would be the processes( s, t ) ψ C,A { ˆ B ( m ) n ( s, t, · ) } or the processes ( s, t ) ψ C,A { ˇ B ( m ) n ( s, t, · ) } , m ∈ { , . . . , M } ,where the map ψ C,A is defined in (2.14). These however still depend on the unknowncopula C . The latter could be estimated either by C n or by C ⌊ ns ⌋ +1: ⌊ nt ⌋ , which led us toconsider the following two computable versions instead:ˆ S ( m ) n,A ( s, t ) = ψ C n ,A { ˆ B ( m ) n ( s, t, · ) } , ˇ S ( m ) n,A ( s, t ) = ψ C ⌊ ns ⌋ +1: ⌊ nt ⌋ ,A { ˇ B ( m ) n ( s, t, · ) } , for ( s, t ) ∈ ∆. The processes ˇ S ( m ) n,A were found to lead to better behaved tests than theˆ S ( m ) n,A in our Monte Carlo experiments, which is why, from now on, we focus solely on theformer. It is easy to verify that the ˇ S ( m ) n,A can be rewritten asˇ S ( m ) n,A ( s, t ) = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( ξ ( m ) i,n − ¯ ξ ( m ) ⌊ ns ⌋ +1: ⌊ nt ⌋ ) I C ⌊ ns ⌋ +1: ⌊ nt ⌋ ,A ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ i ) , where, for any u ∈ [0 , d , I C,A ( u ) = ψ C,A { ( u ≤ · ) } = Y l ∈ A (1 − u l ) − Z [0 , d X j ∈ A Y l ∈ A \{ j } (1 − v l ) ( u j ≤ v j )d C ( v ) . (3.2)Next, by analogy with (2.9), for any m ∈ { , . . . , M } , A ⊆ D , | A | ≥
1, letˇ T ( m ) n,A ( s ) = λ n ( s, S ( m ) n,A (0 , s ) − λ n (0 , s )ˇ S ( m ) n,A ( s, , s ∈ [0 , , T ( m ) n be the corresponding version of T n in (2.5). Finally, for some continuousfunction f : R d − → R , let ˇ S ( m ) n,f = sup s ∈ [0 , | f { ˇ T ( m ) n ( s ) }| by analogy with (2.6). Inter-preting the ˇ S ( m ) n,f as multiplier replicates of S n,f under H , it is natural to compute anapproximate p-value for the test as1 M M X m =1 (cid:16) ˇ S ( m ) n,f ≥ S n,f (cid:17) . (3.3)The null hypothesis is rejected if the estimated p-value is smaller than the desired signif-icance level.The following result, proved in Section C of the supplementary material, can becombined with Proposition F.1 in B¨ucher and Kojadinovic (2014) to show that a testbased on S n,f whose p-value is computed as in (3.3) will hold its level asymptotically as n → ∞ followed by M → ∞ . Proposition 4.
Under the conditions of Proposition 3, for any A ⊆ D , | A | ≥ , (cid:16) S n,A , ˇ S (1) n,A , . . . , ˇ S ( M ) n,A (cid:17) (cid:16) S C,A , S (1) C,A , . . . , S ( M ) C,A (cid:17) in { ℓ ∞ (∆; R ) } M +1 , where, for any ( s, t ) ∈ ∆ , S C,A ( s, t ) = ψ C,A { B C ( s, t, · ) } and S (1) C,A , . . . , S ( M ) C,A are independent copies of S C,A . As a consequence, (cid:0) T n , ˇ T (1) n , . . . , ˇ T ( M ) n (cid:1) (cid:16) T C , T (1) C , . . . , T ( M ) C (cid:17) in { ℓ ∞ ([0 , R d − ) } M +1 , where T C is given in (2.16) and T (1) C , . . . , T ( M ) C are independentcopies of T C , and, for any continuous function f : R d − → R , (cid:16) S n,f , ˇ S (1) n,f , . . . , ˇ S ( M ) n,f (cid:17) (cid:16) S C,f , S (1)
C,f , . . . , S ( M ) C,f (cid:17) in R M +1 , where S C,f = sup s ∈ [0 , | f { T C ( s ) }| and S (1) C,f , . . . , S ( M ) C,f are independent copies of S C,f . The finite-sample behavior of the tests under consideration based on the processesˇ S ( m ) n,A is not however completely satisfactory: the tests appear too liberal for multivariatetime series with strong cross sectional dependence. This prompted us to try other asymp-totically equivalent versions of the ˇ S ( m ) n,A . Under an additional assumption on the partialderivatives of the copula, the generic test statistic S n,f defined in (2.6) can be writtenunder H as a functional of the two-sided sequential empirical copula process studied inB¨ucher and Kojadinovic (2014), and could therefore be bootstrapped via the multiplierprocesses defined in (4.4) of B¨ucher et al. (2014). Without imposing any condition onthe partial derivatives of the copula, the latter remark led us to consider, instead of theprocessesˇ S ( m ) n,A ( s, t ) = φ A { ˇ B ( m ) n ( s, t, · ) }− Z [0 , d X j ∈ A Y l ∈ A \{ j } (1 − v l ) ˇ B ( m ) n ( s, t, v { j } )d C ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) , (3.4)11he processes˜ S ( m ) n,b n ,A ( s, t ) = φ A { ˇ B ( m ) n ( s, t, · ) }− Z [0 , d X j ∈ A Y l ∈ A \{ j } (1 − v l ) ˜ B ( m ) n,b n ,j ( s, t, v j )d C ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) , (3.5)where, for any j ∈ D , ˜ B ( m ) n,b n ,j is a linearly smoothed version of ( s, t, u ) ˇ B ( m ) n ( s, t, u j )with u j the vector of [0 , d whose components are all equal to 1 except the j th one whichis equal to u , and b n a strictly positive sequence of constants converging to 0. Specifically,for ( s, t, v ) ∈ ∆ × [0 , B ( m ) n,b n ,j ( s, t, v ) = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( ξ ( m ) i,n − ¯ ξ ( m ) ⌊ ns ⌋ +1: ⌊ nt ⌋ ) L b n ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij , v ) , where L b n ( u, v ) = u + ∧ v − u − ∧ vu + − u − , u, v ∈ [0 , , with u + = ( u + b n ) ∧ u − = ( u − b n ) ∨
0. It is easy to verify that, for any u ∈ [0 , L b n ( u, · ) differs from ( u ≤ · ) only on the interval ( u − , u + ) on which it linearly increasesfrom 0 to 1.Notice that (3.5) can be rewritten as˜ S ( m ) n,b n ,A ( s, t ) = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( ξ ( m ) i,n − ¯ ξ ( m ) ⌊ ns ⌋ +1: ⌊ nt ⌋ ) I b n ,C ⌊ ns ⌋ +1: ⌊ nt ⌋ ,A ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ i ) , where, for any u ∈ [0 , d , I b n ,C,A ( u ) = Y l ∈ A (1 − u l ) − Z [0 , d X j ∈ A Y l ∈ A \{ j } (1 − v l ) L b n ( u j , v j )d C ( v ) . (3.6)For any m ∈ { , . . . , M } , let ˜ T ( m ) n,b n and ˜ S ( m ) n,b n ,f be the analogues of ˇ T ( m ) n and ˇ S ( m ) n,f ,respectively, defined from the processes ˜ S ( m ) n,b n ,A in (3.5). The following result, proved inSection C of the supplementary material, is then the analogue of Proposition 4 above. Proposition 5. If b n = o ( n − / ) , Proposition 4 holds with ˇ S ( m ) n,A replaced by ˜ S ( m ) n,b n ,A , ˇ T ( m ) n replaced by ˜ T ( m ) n,b n and ˇ S ( m ) n,f replaced by ˜ S ( m ) n,b n ,f . Finally, notice that it is possible to consider a version of the above construction inwhich the smoothing sequence is b ⌊ nt ⌋−⌊ ns ⌋ instead of b n . We focused above only on thelatter approach as it led to better behaved tests in our Monte Carlo experiments.12 .2 Estimating the asymptotic null distribution When the function f used in the definition of S n,f in (2.6) is linear, Corollary 2 givesconditions under which, provided σ C,f = var[ f ◦ ψ C { B C (0 , , · ) } ] >
0, the weak limit of σ − C,f S n,f under H is equal in distribution to sup s ∈ [0 , | U ( s ) | . The distribution of the latterrandom variable can be approximated very well (this aspect is discussed in more detailin Section 4). To be able to estimate an asymptotic p-value for S n,f , it thus remains toestimate the unknown variance σ C,f .Let E ξ and var ξ denote the expectation and variance, respectively, conditional on thedata. By analogy with the classical way of proceeding when estimating variances usingresampling procedures (see, e.g., K¨unsch, 1989; Shao, 2010), in our context, a first naturalestimator of the unknown variance under H is of the formˇ σ n,C,f = var ξ [ f ◦ ψ C { ˇ B ( m ) n (0 , , · ) } ] , (3.7)where ˇ B ( m ) n is defined in (3.1). To simplify the notation, we shall drop the superscript ( m )in the rest of this section. The previous estimator is not computable as C is unknown,which is why we will eventually consider the estimator ˇ σ n,C n ,f instead.To obtain a more explicit expression of ˇ σ n,C,f , first, let I C ( u ) = (cid:0) I C, { } ( u ) , I C, { } ( u ) , . . . , I C,D ( u ) (cid:1) , u ∈ [0 , d , (3.8)where I C,A , A ⊆ D , | A | ≥
1, is defined in (3.2). From the linearity of f ◦ ψ C , we thenobtain thatˇ σ n,C,f = var ξ ( √ n n X i =1 ( ξ i,n − ¯ ξ n ) f ◦ I C ( ˆ U ni ) ) = var ξ " √ n n X i =1 ξ i,n ( f ◦ I C ( ˆ U ni ) − n n X j =1 f ◦ I C ( ˆ U nj ) ) . Using the fact that, from (3.2) and (3.8),1 n n X i =1 f ◦ I C ( ˆ U ni ) = 1 n n X i =1 f ◦ ψ C { ( ˆ U ni ≤ · ) } = f ◦ ψ C ( C n ) , we obtain thatˇ σ n,C,f = 1 n n X i,j =1 E ξ ( ξ i,n ξ j,n ) f n I C ( ˆ U ni ) − ψ C ( C n ) o × f n I C ( ˆ U nj ) − ψ C ( C n ) o . On one hand, should the sequence ( ξ i,n ) i ∈ Z be an i.i.d. multiplier sequence, that is, shouldit satisfy (M0), unsurprisingly, the above estimator simplifies toˇ σ n,C,f = 1 n n X i =1 h f n I C ( ˆ U ni ) − ψ C ( C n ) oi . (3.9)13n the other hand, if the multiplier sequence satisfies (M1)–(M3), one obtainsˇ σ n,C,f = 1 n n X i,j =1 ϕ (cid:18) i − jℓ n (cid:19) f n I C ( ˆ U ni ) − ψ C ( C n ) o × f n I C ( ˆ U nj ) − ψ C ( C n ) o , (3.10)which has the form of the HAC kernel estimator of de Jong and Davidson (2000).Very naturally, once C has been replaced by C n , we use the form in (3.9) (resp. (3.10))for serially independent (resp. weakly dependent) observations. The following result,proved in Section D of the supplementary material, establishes the consistency of ˇ σ n,C n ,f under H . Proposition 6.
Assume that f : R d − → R in the definition of (2.6) is linear and thateither (i) the random vectors X , . . . , X n are i.i.d. with continuous margins, (ii) or the random vectors X , . . . , X n are drawn from a strictly stationary sequence ( X i ) i ∈ Z with continuous margins whose strong mixing coefficients satisfy α r = O ( r − a ) for some a > , and ℓ n = O ( n / − ε ) for some < ε < / such that,additionally, ϕ defined in (M3) is twice continuously differentiable on [ − , with ϕ ′′ (0) = 0 and is Lipschitz continuous on R .Then, ˇ σ n,C n ,f P → σ C,f . As a consequence, the weak limit of ˇ σ − n,C n ,f S n,f is equal indistribution to sup s ∈ [0 , | U ( s ) | . As in the previous subsection, better behaved tests are obtained if (3.6) is used insteadof (3.2) in the above developments. Let I b n ,C ( u ) = (cid:0) I b n ,C, { } ( u ) , I b n ,C, { } ( u ) , . . . , I b n ,C,D ( u ) (cid:1) , u ∈ [0 , d , and let ˜ σ n,b n ,C n ,f be the corresponding estimator of σ C,f . Proceeding as above, for seriallyindependent data, the appropriate form of ˜ σ n,b n ,C n ,f is˜ σ n,b n ,C n ,f = 1 n n X i =1 h f n I b n ,C n ( ˆ U ni ) − ¯ I b n ,C n oi , (3.11)where ¯ I b n ,C n = n − P n =1 I b n ,C n ( ˆ U ni ), while, for weakly dependent observations,˜ σ n,b n ,C n ,f = 1 n n X i,j =1 ϕ (cid:18) i − jℓ n (cid:19) f n I b n ,C n ( ˆ U ni ) − ¯ I b n ,C n o × f n I b n ,C n ( ˆ U nj ) − ¯ I b n ,C n o . (3.12)The following analogue of Proposition 6 is proved in Section D of the supplementarymaterial. Proposition 7. If b n = o ( n − / ) , Proposition 6 holds with ˇ σ n,C n ,f replaced with ˜ σ n,b n ,C n ,f . .3 Estimation of the bandwidth parameter ℓ n When the available observations are weakly dependent, both the approach based onresampling presented in Section 3.1 and the one based on the estimation of the asymptoticnull distribution discussed in Section 3.2 require the choice of the bandwidth parameter ℓ n .The latter quantity appears in the definition of the dependent multiplier sequences and,as mentioned in B¨ucher and Kojadinovic (2014), plays a role somehow analogous to thatof the block length in the block bootstrap. The value of ℓ n is therefore expected to havea crucial influence on the finite-sample performance of the two versions of the test basedon S n,f described previously.The aim of this subsection is to propose an estimator of ℓ n in the spirit of thatinvestigated in Paparoditis and Politis (2001), Politis and White (2004) and Patton et al.(2009), among others, for other resampling schemes. By analogy with (3.7), we start fromthe non computable estimator of σ C,f defined by σ n,C,f = var ξ [ f ◦ ψ C { ¯ B n (0 , , · ) } ] , (3.13)where ¯ B n ( s, t, u ) = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ξ i,n { ( U i ≤ u ) − C ( u ) } , ( s, t, u ) ∈ ∆ × [0 , d , and ( ξ i,n ) i ∈ Z is a dependent multiplier sequence. Proceeding as for (3.7), it is easy toverify that σ n,C,f = 1 n n X i,j =1 ϕ (cid:18) i − jℓ n (cid:19) f {I C ( U i ) − ψ C ( C ) } f {I C ( U j ) − ψ C ( C ) } . (3.14)Under the conditions of Proposition 6 (ii) and from the fact that the random variables | f ◦ I C ( U i ) | are bounded by sup x ∈ [ − , d − | f ( x ) | < ∞ (since sup u ∈ [0 , d |I C,A ( u ) | ≤ A ⊆ D | A | ≥ σ n,C,f ) − σ C,f = Γ ℓ n + o ( ℓ − n ) and var( σ n,C,f ) = ℓ n n ∆ + o ( ℓ n /n ) , where Γ = ϕ ′′ (0) / P ∞ k = −∞ k τ ( k ) with τ ( k ) = cov { f ◦ I C ( U ) , f ◦ I C ( U k ) } , and ∆ =2 σ C,f R − ϕ ( x ) d x . As a consequence, the mean squared error of σ n,C,f isMSE( σ n,C,f ) = Γ ℓ n + ∆ ℓ n n + o ( ℓ − n ) + o ( ℓ n /n ) . (3.15)Differentiating the function x Γ /x + ∆ x/n and equating the derivative to zero, weobtain that the value of ℓ n that minimizes the mean square error of σ n,C,f is, asymptoti-cally, ℓ optn = (cid:18) ∆ (cid:19) / n / .
15o estimate ℓ optn , it is necessary to estimate the infinite sum P k ∈ Z k τ ( k ) as well as σ C,f = P k ∈ Z τ ( k ) through a pilot estimate. To do so, we adapt the approach described inPaparoditis and Politis (2001, page 1111) and Politis and White (2004, Section 3) to thecurrent context (see also Patton et al., 2009). Let ˆ τ n ( k ) be the sample autocovarianceat lag k computed from the sequence f ◦ I b n ,C n ( ˆ U n ) , . . . , f ◦ I b n ,C n ( ˆ U nn ). Then, weestimate Γ and ∆ by ˆΓ n = ϕ ′′ (0) / L X k = − L λ ( k/L ) k ˆ τ n ( k )and ˆ∆ n = 2 ( L X k = − L λ ( k/L )ˆ τ n ( k ) ) (cid:26)Z − ϕ ( x ) d x (cid:27) , respectively, where λ ( x ) = [ { −| x | ) }∨ ∧ x ∈ R , is the “flat top” (trapezoidal) kernelof Politis and Romano (1995) and L is an integer estimated by adapting the procedure de-scribed in Politis and White (2004, Section 3.2). Let ˆ ̺ n ( k ) be the sample autocorrelationat lag k estimated from f ◦ I b n ,C n ( ˆ U n ) , . . . , f ◦ I b n ,C n ( ˆ U nn ). The parameter L is thentaken as the smallest integer k after which ˆ ̺ n ( k ) appears negligible. The latter is deter-mined automatically by means of the algorithm described in detail in Politis and White(2004, Section 3.2). Our implementation is based on Matlab code by A.J. Patton (avail-able on his web page) and its R version by J. Racine and C. Parmeter. In the previous section, two ways to compute approximate p-values for generic change-point tests based on (2.6) were studied under the null. These asymptotic results donot however guarantee that such tests will behave satisfactorily in finite-samples, whichis why additional numerical simulations are needed. In our experiments, we restrictedattention to the three statistics given in (2.3). For each statistic S n,i , i ∈ { , , } ,an approximate p-value was computed using either the resampling approach based onthe processes in (3.5), or the estimated asymptotic null distribution based on varianceestimators of the form (3.11) or (3.12). To distinguish between these two situations, weshall talk about the test ˜ S n,i and the test S an,i , respectively, in the rest of the paper.The experiments were carried out in the R statistical system using the copula package(Hofert et al., 2013). The sequence b n involved in both classes of tests was taken equalto n − . . The only (asymptotically negligible) difference with the theoretical develop-ments presented in the previous sections is that the rescaled maximal ranks in (2.2) werecomputed by dividing the ranks by l − k + 2 instead of l − k + 1. Data generating procedure
Two multivariate time series models were used to gen-erate d -dimensional samples of size n in our Monte Carlo experiments: a simple au-toregressive model of order one and a GARCH(1,1)-like model. Apart from d , n and theparameters of the models, the other inputs of the procedure are a real t ∈ (0 ,
1) determin-ing the location of the possible change-point in the innovations, and two d -dimensional16opulas C and C . The procedure used to generate a d -dimensional sample X , . . . , X n then consists of:1. generating independent random vectors U i , i ∈ {− , . . . , , . . . , n } such that U i , i ∈ {− , . . . , , . . . , ⌊ nt ⌋} are i.i.d. from copula C and U i , i ∈ {⌊ nt ⌋ + 1 , . . . , n } are i.i.d. from copula C ,2. computing ǫ i = (Φ − ( U i ) , . . . , Φ − ( U id )), where Φ is the c.d.f. of the standardnormal distribution,3. setting X − = ǫ − and, for any j ∈ D , computing recursively either X ij = γX i − ,j + ǫ ij , (AR1)or σ ij = ω j + β j σ i − ,j + α j ǫ i − ,j and X ij = σ ij ǫ ij , (GARCH)for i = − , . . . , , . . . , n .If the copulas C and C are chosen equal, the above procedure generates samples un-der H defined in (1.1). Three possible values were considered for the parameter γ controlling the strength of the serial dependence in (AR1): 0 (serial independence),0.25 (mild serial dependence), 0.5 (strong serial dependence). Model (GARCH) wasonly considered in the bivariate case, and following B¨ucher and Ruppert (2013), with( ω , β , α ) = (0 . , . , . ω , β , α ) = (0 . , . , . H ,m ∩ ( ¬ H ,c ), where H ,m and H ,c are defined in (1.2) and (1.3),respectively, were obtained by taking C = C and t ∈ { . , . , . } . Notice that when γ = 0 in (AR1), the latter are samples under H ,m ∩ H ,c , where H ,c : ∃ distinct C and C , and t ∈ (0 ,
1) such that X , . . . , X ⌊ nt ⌋ have copula C and X ⌊ nt ⌋ +1 , . . . , X n have copula C . This is not the case anymore when γ >
Other factors of the experiments
Five copula families were considered (the Clayton,the Gumbel–Hougaard, the Normal, the Frank and the Student), the cross-sectionaldimensional d was taken in { , } , and the values 50, 100, 200, 400 and 500 were usedfor n . To estimate the power of the tests, 1000 samples were generated under eachcombination of factors and all the tests were carried out at the 5% significance level. Computation of the test statistics and of the corresponding p-values
The datagenerating procedure above generates multivariate time series whose component series donot contain ties with probability one. Consequently, as explained in Section 2.2, S n, ismerely S n, computed from the sample − X , . . . , − X n . Furthermore, if d = 2, it is easyto see that S n, = S n, = S n, . However, it can be verified that only the approximate p-values for the tests ˜ S n, and ˜ S n, (resp. S an, and S an, ) will be equal. Indeed, the multiplier17eplicates based on the processes in (3.5) (resp. the variance estimators of the form (3.11)or (3.12)) computed from X , . . . , X n do not coincide in general with those computedfrom − X , . . . , − X n , even in dimension two.From Proposition 7, we see that, to compute an asymptotic p-value for the tests S an,i , itis necessary to be able to compute the c.d.f. of the random variable sup s ∈ [0 , | U ( s ) | . Thedistribution of the latter random variable is known as the Kolmogorov distribution. Asclassically done in other contexts, we approach this distribution by that of the statisticof the classical Kolmogorov–Smirnov goodness-of-fit test for a simple hypothesis. Specif-ically, we use the function pkolmogorov1x given in the code of the R function ks.test .[Table 1 about here.][Table 2 about here.] Empirical levels and power of the tests based on i.i.d. multipliers / a varianceestimator of the form (3.11) Table 1 gives the empirical levels of the tests when theobservations are serially independent. For the sake of brevity, the results are reportedonly for two copula families. Overall, we find that the tests ˜ S n,i with multiplier sequencessatisfying (M0) (here standard normal sequences) hold there level rather well both for d = 2 and d = 4, and all the considered degrees of cross-sectional dependence. This is notthe case for the tests S an,i which frequently appear way too liberal when the cross-sectionaldependence is high.Table 2 partially reports the percentages of rejection of the i.i.d. multiplier tests forserially independent observations generated under H ,m ∩ H ,c resulting from a change ofthe copula parameter within a copula family. The columns CvM give the results of thei.i.d. multiplier test based on the maximally selected Cram´er–von Mises statistic studiedin B¨ucher et al. (2014) (with multiplier replicates of the form (4.6) in the latter reference)and implemented in the R package npcp . Overall, we find that the tests ˜ S n,i are morepowerful than that studied in B¨ucher et al. (2014) for such scenarios, especially when thechange in the copula occurs early or late. Among the tests ˜ S n,i , we observed that thetest ˜ S n, (which coincides with the test ˜ S n, in dimension two) led frequently to slightlyhigher rejection rates, although this conclusion is based on a limited number of simulationscenarios. The rejection rates of the tests S an,i with a variance estimator of the form (3.11)are not reported for the sake of brevity. They were found to be slightly less powerful thanthe tests ˜ S n,i when τ = 0 .
4. For τ = 0 .
6, a comparison of the two classes of tests is notnecessarily meaningful as the tests S an,i were often found to be way too liberal understrong cross-sectional dependence.[Table 3 about here.][Table 4 about here.][Table 5 about here.]18 mpirical levels and power of the tests based on dependent multipliers / avariance estimator of the form (3.12) Part of Table 3 reports the empirical lev-els of the test ˜ S n, when dependent multiplier sequences satisfying (M1)–(M3) are used.These sequences were generated using the “moving average approach” proposed initiallyin B¨uhlmann (1993, Section 6.2) and revisited in B¨ucher and Kojadinovic (2014, Sec-tion 5.2). A standard normal sequence was used for the required initial i.i.d. sequence.The kernel function κ in that approach was chosen to be the Parzen kernel defined by κ P ( x ) = (1 − x + 6 | x | ) ( | x | ≤ /
2) + 2(1 − | x | ) (1 / < | x | ≤ x ∈ R , which amountsto choosing the function ϕ in (M3) as x ( κ P ⋆ κ P )(2 x ) / ( κ P ⋆ κ P )(0), where ‘ ⋆ ’ denotesthe convolution operator. The value of the bandwidth parameter ℓ n defined in (M2) wasestimated using the data-driven procedure described in Section 3.3. The same value of ℓ n was used to carry out the test S an, relying on a variance estimator of the form (3.12).From the first three vertical blocks of Table 3, we see that an increase in the degree ofserial dependence in (AR1) (controlled by γ ) appears to result in a small inflation of theempirical levels of the test ˜ S n, . As expected, the situation improves as n increases from100 to 400. For sequences generated using (GARCH), the empirical levels of the test ˜ S n, appear always reasonably close to the 5% nominal level. The test S an, remains overallway too liberal when the cross-sectional dependence is high.The last vertical block of Table 3 reports, for strongly serially dependent observationsgenerated using (AR1), the empirical levels of the test ˜ S n, based on i.i.d. multipliers,as well as those of the test S an, based on an inappropriate variance estimator of theform (3.11). As expected, both tests strongly fail to hold their level.Table 4 partially reports the rejection percentages of the tests based on dependentmultipliers / a variance estimator of the form (3.12) for observations generated under H ,m ∩ ( ¬ H ,c ) resulting from a change of the copula parameter within a copula family.The rejection rates of the test S an, should be considered with care when τ = 0 . S n, appears almost always more powerful than the test S an, . Also,as it could have been expected, the presence of strong serial dependence ( γ = 0 .
5) leadsto lower rejection percentages when compared with serial independence ( γ = 0). Finally,comparing the results for the test ˜ S n, when γ = 0 with the analogue results reportedin Table 2 reveals that, rather naturally, the use of dependent multipliers in the case ofserially independent observations results in a small loss of power.We end this section by a comparison of the tests ˜ S n, and S an, with the similar teststudied in Wied et al. (2014). To do so, we reproduced one of the experiments carriedout in the latter reference. The results are reported in Table 5 and confirm that tests forchange-point detection based on (2.1) are potentially substantially more powerful thantests based on (2.7). Based on the experiments partially reported in the previous section, we recommend,among the tests ˜ S n,i and S an,i , the tests ˜ S n,i . Indeed, the tests S an,i did not hold their level19ell in the case of strong cross-sectional dependence. Furthermore, because of their form,the tests S an,i might suffer from some of the practical issues described in Shao and Zhang(2010), and, in future research, it might be of interest to study a self-normalization versionof these as advocated in the latter reference.The pros and cons of the tests ˜ S n,i compared with the test studied in B¨ucher et al.(2014) are as follows. The tests ˜ S n,i seem more powerful for alternatives involving a changein Spearman’s rho at constant margins; they are also substantially faster to compute.Their main weakness is that, by construction, they have no power against alternativesinvolving a change in the copula at a constant value of Spearman’s rho and constantmargins.Among the tests ˜ S n,i , we recommend the test ˜ S n, , merely because of its slightly betterfinite-sample behavior in our simulations.We end this section by a brief illustration of the studied tests on real financial observa-tions. Specifically, we consider a trivariate version of the data analyzed in Dehling et al.(2014, Section 7). The observations consist of n = 990 daily logreturns computed fromthe DAX, the CAC 40 and the Standard and Poor 500 indices for the years 2006–2009.An approximate p-value of 0.045 was obtained for the test ˜ S n, with dependent multipli-ers, providing some evidence against H . It is however important to bear in mind that itis only under the assumption that H ,m in (1.2) holds that it would be fully justified todecide to reject H ,c in (1.3). Tests for change-point detection based on the generic statistic S n,f defined in (2.6) werefirst studied theoretically. These tests, designed to be particularly sensitive to changes inthe cross-sectional dependence of multivariate time series, can be carried out using eitherresampling based on multipliers, or by estimating the asymptotic null distribution of S n,f .Both approaches were shown to be asymptotically valid under strong mixing and suitableconditions on the underlying function f . In addition, a procedure for estimating a keybandwidth parameter involved in both techniques for computing p-values was suggested,making the tests fully data-driven. Next, their finite-sample behavior was investigatedby means of extensive simulations for three particular choices of the function f resultingin the test statistics defined in (2.3) measuring changes in the cross-sectional dependencein terms of multivariate extensions of Spearman’s rho. Practical recommendations andan illustration were finally given. Acknowledgements
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A Proof of Proposition 1
Let us first introduce some additional notation. For integers 1 ≤ k ≤ l ≤ n , let H k : l denote the empirical c.d.f. of the unobservable sample U k , . . . , U l and let H k : l, , . . . , H k : l,d denote its margins. The corresponding empirical quantile functions are H − k : l,j ( u ) = inf { v ∈ [0 ,
1] : H k : l,j ( v ) ≥ u } , u ∈ [0 , , j ∈ D. Finally, for any u ∈ [0 , d , let h k : l ( u ) = (cid:0) H k : l, ( u ) , . . . , H k : l,d ( u d ) (cid:1) (A.1)and h − k : l ( u ) = (cid:0) H − k : l, ( u ) , . . . , H − k : l,d ( u d ) (cid:1) . (A.2)By convention, all the quantities defined above are taken equal to zero if k > l .23 roof of Proposition 1. Fix A ⊆ D , | A | ≥
1, and ( s, t ) ∈ ∆ such that ⌊ ns ⌋ < ⌊ nt ⌋ .On one hand, from (2.8) and by linearity of φ A defined in (2.4), we have S n,A ( s, t ) = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 Y j ∈ A { − H ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( U ij ) } − √ nλ n ( s, t ) φ A ( C ) , where we have used the fact that ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij = H ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( U ij ) for all j ∈ D and all i ∈ {⌊ ns ⌋ + 1 , . . . , ⌊ nt ⌋} . On the other hand, ψ C,A { B n ( s, t, · ) } = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 Y j ∈ A (1 − U ij ) − √ nλ n ( s, t ) φ A ( C ) − Z [0 , d X j ∈ A Y l ∈ A \{ j } (1 − v l ) B n ( s, t, v { j } )d C ( v ) . Next, let π ( u ) = Q j ∈ A (1 − u j ), u ∈ R d . Then, fix u ∈ [0 , d , and, for any x ∈ [0 , w u ( x ) = u + x { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( u ) − u } and let g ( x ) = π { w u ( x ) } , where h ⌊ ns ⌋ +1: ⌊ nt ⌋ isdefined in (A.1). The function g is clearly continuously differentiable on [0 , x ∗ u ,n,s,t ∈ (0 ,
1) such that g (1) − g (0) = g ′ ( x ∗ u ,n,s,t ), thatis, such that π { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( u ) }− π ( u ) = X j ∈ A ˙ π j [ u + x ∗ u ,n,s,t { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( u ) − u } ] { H ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u j ) − u j } . It follows that S n,A ( s, t ) − ψ C,A { B n ( s, t, · ) } = 1 √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 X j ∈ A ˙ π j [ U i + x ∗ U i ,n,s,t { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( U i ) − U i } ] { H ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( U ij ) − U ij }− Z [0 , d X j ∈ A ˙ π j ( v ) B n ( s, t, v { j } )d C ( v ) . Notice that, by the triangle inequality and the fact that sup u ∈ [0 , d | ˙ π j ( u ) | ≤ j ∈ D ,sup ( s,t ) ∈ ∆ | S n,A ( s, t ) − ψ C,A { B n ( s, t, · ) }| ≤ | A | sup ( s,t, u ) ∈ ∆ × [0 , d | B n ( s, t, u ) | . Next, fix ε, η >
0. Using the previous inequality and the fact that B n vanishes when s = t and is asymptotically uniformly equicontinuous in probability as a consequence ofLemma 2 in B¨ucher (2014), there exists δ ∈ (0 ,
1) such that, for all sufficiently large n ,P sup ( s,t ) ∈ ∆ t − s<δ | S n,A ( s, t ) − ψ C,A { B n ( s, t, · ) }| > ε ≤ P | A | sup ( s,t, u ) ∈ ∆ × [0 , dt − s<δ | B n ( s, t, u ) | > ε < η/ .
24o show (2.13), it remains therefore to prove that, for all sufficiently large n ,P sup ( s,t ) ∈ ∆ t − s ≥ δ | S n,A ( s, t ) − ψ C,A { B n ( s, t, · ) }| > ε < η/ . To show the above, we shall now prove that sup ( s,t ) ∈ ∆ δ | S n,A ( s, t ) − ψ C,A { B n ( s, t, · ) }| con-verges in probability to zero, where ∆ δ = { ( s, t ) ∈ ∆ : t − s ≥ δ } . The latter supremumis smaller than P j ∈ A ( I n,j + II n,j ), where I n,j ≤ sup ( s,t ) ∈ ∆ δ (cid:12)(cid:12)(cid:12) √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 (cid:0) ˙ π j [ U i + x ∗ U i ,n,s,t { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( U i ) − U i } ] − ˙ π j ( U i ) (cid:1) × { H ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( U ij ) − U ij } (cid:12)(cid:12)(cid:12) and II n,j ≤ sup ( s,t ) ∈ ∆ δ (cid:12)(cid:12)(cid:12) Z [0 , d ˙ π j ( v ) B n ( s, t, v { j } )d H ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) − Z [0 , d ˙ π j ( v ) B n ( s, t, v { j } )d C ( v ) (cid:12)(cid:12)(cid:12) . Next, notice thatsup ( s,t, u ) ∈ ∆ δ × [0 , d | H ⌊ ns ⌋ +1: ⌊ nt ⌋ ( u ) − C ( u ) |≤ sup ( s,t, u ) ∈ ∆ δ × [0 , d | B n ( s, t, u ) | × n − / × sup ( s,t ) ∈ ∆ δ { λ n ( s, t ) } − → . (A.3)Fix j ∈ A . Since the function ˙ π j is continuous on [0 , d , by the continuous mappingtheorem, sup ( s,t, u ) ∈ ∆ δ × [0 , d | ˙ π j [ u + x ∗ u ,n,s,t { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( u ) − u } ] − ˙ π j ( u ) | P →
0. Hence, I n,j ≤ sup ( s,t, u ) ∈ ∆ × [0 , d | B n ( s, t, u ) |× sup ( s,t, u ) ∈ ∆ δ ∈ [0 , d | ˙ π j [ u + x ∗ u ,n,s,t { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( u ) − u } ] − ˙ π j ( u ) | P → . It thus remains to show that II n,j P →
0. The latter is mostly a consequence of Lemma 8below. First, notice that (A.3) implies that H ⌊ ns ⌋ +1: ⌊ nt ⌋ P → C in ℓ ∞ (∆ δ × [0 , d ; R ).Hence, ( B n , H ⌊ ns ⌋ +1: ⌊ nt ⌋ ) ( B C , C ) in ℓ ∞ (∆ δ × [0 , d ; R ). Next, combining the previousweak convergence with Lemma 3 in Holmes et al. (2013) and the continuous mappingtheorem, we obtain that the finite-dimensional distributions of ( A n,j , B n ) converge weaklyto those of ( A C,j , B C ), where A n,j and A C,j are defined in Lemma 8. The fact that( A n,j , B n ) ( A C,j , B C ) in { ℓ ∞ (∆ δ × [0 , d ; R ) } then follows from Lemma 8 below andthe fact that marginal asymptotic tightness implies joint asymptotic tightness. The latterweak convergence combined with the continuous mapping theorem finally implies that II n,j P →
0, which completes the proof. (cid:4) emma 8. For any j ∈ D and δ ∈ (0 , , A n,j A C,j in ℓ ∞ (∆ δ ; R ) , where A n,j ( s, t ) = Z [0 , d ˙ π j ( v ) B n ( s, t, v { j } )d H ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) , (A.4) A C,j ( s, t ) = Z [0 , d ˙ π j ( v ) B C ( s, t, v { j } )d C ( v ) . Proof.
Fix j ∈ D and δ ∈ (0 , H ⌊ ns ⌋ +1: ⌊ nt ⌋ P → C in ℓ ∞ (∆ δ × [0 , d ; R ). Then, from the fact that B n B C in ℓ ∞ (∆ × [0 , d ; R ), we obtain that, for any ( s , t ) , . . . , ( s k , t k ) ∈ ∆ δ , (cid:0) B n ( s , t , · ) , H ⌊ ns ⌋ +1: ⌊ nt ⌋ , . . . , B n ( s k , t k , · ) , H ⌊ ns k ⌋ +1: ⌊ nt k ⌋ (cid:1) (cid:0) B C ( s , t , · ) , C, . . . , B C ( s k , t k , · ) , C (cid:1) in { ℓ ∞ ([0 , d ; R ) } k . From Lemma 3 in Holmes et al. (2013) and the continuous mappingtheorem, the above implies that (cid:0) A n,j ( s , t ) , . . . , A n,j ( s k , t k ) (cid:1) (cid:0) A C,j ( s , t ) , . . . , A C,j ( s k , t k ) (cid:1) in R k . Hence, we have convergence of the finite-dimensional distributions, that is, condi-tion (i) of Theorem 2.1 in Kosorok (2008) holds.It remains to prove condition (ii) of Theorem 2.1 in Kosorok (2008). Specifically, weshall now show that A n,j is k · k -asymptotically uniformly equicontinuous in probability,which will complete the proof since ∆ δ is totally bounded by k · k . By Problem 2.1.5in van der Vaart and Wellner (2000), we need to show that, for any positive sequence a n ↓
0, sup ( s,t ) , ( s ′ ,t ′ ) ∈ ∆ δ | s − s ′| + | t − t ′|≤ an | A n,j ( s, t ) − A n,j ( s ′ , t ′ ) | P → . (A.5)We bound the supremum on the left of the previous display by I n + II n , where I n = sup ( s,t ) , ( s ′ ,t ′ ) ∈ ∆ δ | s − s ′| + | t − t ′|≤ an (cid:12)(cid:12)(cid:12)(cid:12)Z [0 , d ˙ π j ( v ) B n ( s, t, v { j } )d H ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) − Z [0 , d ˙ π j ( v ) B n ( s ′ , t ′ , v { j } )d H ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) and II n = sup ( s,t ) , ( s ′ ,t ′ ) ∈ ∆ δ | s − s ′| + | t − t ′|≤ an (cid:12)(cid:12)(cid:12)(cid:12)Z [0 , d ˙ π j ( v ) B n ( s ′ , t ′ , v { j } )d H ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) − Z [0 , d ˙ π j ( v ) B n ( s ′ , t ′ , v { j } )d H ⌊ ns ′ ⌋ +1: ⌊ nt ′ ⌋ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) . Now, I n ≤ sup u ∈ [0 , d | ˙ π j ( u ) | × sup ( s,t ) , ( s ′ ,t ′ ) ∈ ∆ δ, u ∈ [0 , d | s − s ′| + | t − t ′|≤ an | B n ( s, t, u ) − B n ( s ′ , t ′ , u ) | P → , B n is asymptotically uniformly equicontinuous in probability as a consequence ofLemma 2 in B¨ucher (2014). Furthermore, II n is smaller than sup ( s,t ) , ( s ′ ,t ′ ) ∈ ∆ δ | s − s ′| + | t − t ′|≤ an (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⌊ nt ⌋ − ⌊ ns ⌋ ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ˙ π j ( U i ) B n ( s ′ , t ′ , U { j } i ) − ⌊ nt ′ ⌋ X i = ⌊ ns ′ ⌋ +1 ˙ π j ( U i ) B n ( s ′ , t ′ , U { j } i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + sup ( s,t ) , ( s ′ ,t ′ ) ∈ ∆ δ | s − s ′| + | t − t ′|≤ an (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ⌊ nt ⌋ − ⌊ ns ⌋ − ⌊ nt ′ ⌋ − ⌊ ns ′ ⌋ (cid:19) ⌊ nt ′ ⌋ X i = ⌊ ns ′ ⌋ +1 ˙ π j ( U i ) B n ( s ′ , t ′ , U { j } i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which is smaller than2 × sup ( s,t ) , ( s ′ ,t ′ ) ∈ ∆ δ | s − s ′| + | t − t ′|≤ an |⌊ nt ⌋ − ⌊ nt ′ ⌋| + |⌊ ns ⌋ − ⌊ ns ′ ⌋|⌊ nt ⌋ − ⌊ ns ⌋× sup u ∈ [0 , d | ˙ π j ( u ) | × sup ( s,t, u ) ∈ ∆ × [0 , d | B n ( s, t, u ) | P → . Hence, II n P → (cid:4) B Proof of Corollary 2
Proof.
Starting from (2.9), using Proposition 1, the linearity of ψ C,A and (2.10), we obtainthat, for any A ⊆ D , | A | ≥ s ∈ [0 , | T n,A ( s ) − ψ C,A { B n (0 , s, · ) − λ (0 , s ) B n (0 , , · ) }| = o P (1) . Hence, T n has the same weak limit as s ψ C { B n (0 , s, · ) − λ (0 , s ) B n (0 , , · ) } and (2.15)follows from the continuous mapping theorem.The second to last claim is a consequence of the continuous mapping theorem. Toprove the last claim, it suffices to show that the Gaussian process σ − C,f f { T C ( · ) } has thesame covariance function as U . For any, s, t ∈ [0 , σ − C,f f { T C ( s ) } , σ − C,f f { T C ( t ) } ]= σ − C,f E[ f ◦ ψ C { B C (0 , s, · ) − s B C (0 , , · ) } f ◦ ψ C { B C (0 , t, · ) − t B C (0 , , · ) } ] . (B.1)By linearity of f ◦ ψ C and Fubini’s theorem, the expectation in the last display is equalto f ◦ ψ C { u f ◦ ψ C ( v E[ { B C (0 , s, u ) − s B C (0 , , u ) }{ B C (0 , t, v ) − t B C (0 , , v ) } ]) } , that is,( s ∧ t − st ) f ◦ ψ C [ u f ◦ ψ C { v κ C ( u , v ) } ] = ( s ∧ t − st ) var[ f ◦ ψ C { B C (0 , , · ) } ] , where κ C is defined in (2.11). Combining the previous display with (B.1), we obtain thatcov[ σ − C,f f { T C ( s ) } , σ − C,f f { T C ( t ) } ] = ( s ∧ t − st ), which completes the proof. (cid:4) Proofs of Propositions 4 and 5
Proof of Proposition 4.
We only show the first claim as the subsequent claims thenmostly follow from the continuous mapping theorem. Also, we only provide the proofunder (ii) in the statement of Proposition 3, the proof being simpler under (i). Fix A ⊆ D , | A | ≥
1. For any ( s, t ) ∈ ∆, let S ( m ) n,A ( s, t ) = ψ C,A { ˇ B ( m ) n ( s, t, · ) } . Using the linearity ofthe map ψ C,A defined in (2.14), Proposition 3 and the continuous mapping theorem, weobtain that (cid:16) S n,A , S (1) n,A , . . . , S ( M ) n,A (cid:17) (cid:16) S C,A , S (1) C,A , . . . , S ( M ) C,A (cid:17) in { ℓ ∞ (∆; R ) } M +1 . The first claim is thus proved if we show that, for any m ∈ { , . . . , M } ,sup ( s,t ) ∈ ∆ | ˇ S ( m ) n,A ( s, t ) − S ( m ) n,A ( s, t ) | is o P (1). Fix m ∈ { , . . . , M } and notice that the lattersupremum is smaller than 2 | A | sup ( s,t, u ) ∈ ∆ × [0 , d | ˇ B ( m ) n ( s, t, u ) | . We can therefore proceedanalogously to the proof of Proposition 1. Fix ε, η >
0. Using the previous inequality aswell as the fact that ˇ B ( m ) n is zero when s = t and is asymptotically uniformly equicontin-uous in probability as a consequence of Lemma A.3 in B¨ucher and Kojadinovic (2014),there exists δ ∈ (0 ,
1) such that, for all sufficiently large n ,P sup ( s,t ) ∈ ∆ t − s<δ | ˇ S ( m ) n,A ( s, t ) − S ( m ) n,A ( s, t ) | > ε < η/ . It remains therefore to prove that sup ( s,t ) ∈ ∆ δ | ˇ S ( m ) n,A ( s, t ) − S ( m ) n,A ( s, t ) | P →
0, where ∆ δ = { ( s, t ) ∈ ∆ : t − s ≥ δ } . The latter supremum is smaller than X j ∈ A sup ( s,t ) ∈ ∆ δ (cid:12)(cid:12)(cid:12) Z [0 , d ˙ π j ( v ) ˇ B ( m ) n ( s, t, v { j } )d C ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) − Z [0 , d ˙ π j ( v ) ˇ B ( m ) n ( s, t, v { j } )d C ( v ) (cid:12)(cid:12)(cid:12) , where ˙ π j is the j th first order partial derivative of the function π ( u ) = Q j ∈ A (1 − u j ), u ∈ R d , introduced in the proof of Proposition 1. Fix j ∈ A . The j th summand in theprevious display is smaller than I n + II n , where I n = sup ( s,t ) ∈ ∆ δ (cid:12)(cid:12)(cid:12) Z [0 , d ˙ π j ( v ) ˇ B ( m ) n ( s, t, v { j } )d C ⌊ ns ⌋ +1: ⌊ nt ⌋ ( v ) − ˇ A ( m ) n,j ( s, t ) (cid:12)(cid:12)(cid:12) ,II n = sup ( s,t ) ∈ ∆ δ (cid:12)(cid:12)(cid:12) ˇ A ( m ) n,j ( s, t ) − Z [0 , d ˙ π j ( v ) ˇ B ( m ) n ( s, t, v { j } )d C ( v ) (cid:12)(cid:12)(cid:12) , and ˇ A ( m ) n,j is defined analogously to the process A n,j in (A.4) with B n replaced by ˇ B ( m ) n . Inaddition, it can be verified that Lemma 8 remains true if B n and B C are replaced by ˇ B ( m ) n and B ( m ) C , respectively, in its statement. It follows that we can proceed as at the end ofproof of Proposition 1 to show that II n above converges to zero in probability.28o show that I n P →
0, we use the fact that I n ≤ I ′ n + I ′′ n , where I ′ n = sup ( s,t ) ∈ ∆ δ (cid:12)(cid:12)(cid:12) ⌊ nt ⌋ − ⌊ ns ⌋ ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 (cid:2) ˙ π j { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( U i ) } − ˙ π j ( U i ) (cid:3) × ˇ B ( m ) n { s, t, h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( U i ) { j } } (cid:12)(cid:12)(cid:12) ,I ′′ n = sup ( s,t ) ∈ ∆ δ (cid:12)(cid:12)(cid:12) ⌊ nt ⌋ − ⌊ ns ⌋ ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ˙ π j ( U i ) h ˇ B ( m ) n { s, t, h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( U i ) { j } } − ˇ B ( m ) n ( s, t, U { j } i ) i (cid:12)(cid:12)(cid:12) . For I ′ n , we have that I ′ n ≤ sup ( s,t, u ) ∈ ∆ × [0 , d (cid:12)(cid:12) ˇ B ( m ) n ( s, t, u ) (cid:12)(cid:12) × sup ( s,t, u ) ∈ ∆ δ × [0 , d (cid:12)(cid:12) ˙ π j { h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( u ) } − ˙ π j ( u ) (cid:12)(cid:12) P → B ( m ) n , (A.3), and the continuous mappingtheorem. For I ′′ n , using the fact that sup u ∈ [0 , d | ˙ π j ( u ) | ≤
1, we obtain that I ′′ n ≤ sup ( s,t, u ) ∈ ∆ δ × [0 , d (cid:12)(cid:12) ˇ B ( m ) n { s, t, h ⌊ ns ⌋ +1: ⌊ nt ⌋ ( u ) { j } } − ˇ B ( m ) n ( s, t, u { j } ) (cid:12)(cid:12) P → . The latter convergence is a consequence of the asymptotic equicontinuity in probabilityof ˇ B ( m ) n and the fact that sup ( s,t,u ) ∈ ∆ δ × [0 , | H ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u ) − u | P → (cid:4) Proof of Proposition 5.
We only provide the proof under (ii) in the statement ofProposition 3, the proof being simpler under (i). From Proposition 4, to prove thedesired result it suffices to show that, for any A ⊆ D , | A | ≥ ( s,t ) ∈ ∆ | ˜ S ( m ) n,b n ,A ( s, t ) − ˇ S ( m ) n,A ( s, t ) | P → . Fix A ⊆ D , | A | ≥
1. From (3.4) and (3.5) and the triangle inequality, the latter will holdif, for any j ∈ A , sup ( s,t,u ) ∈ ∆ × [0 , | ˜ B ( m ) n,b n ,j ( s, t, u ) − ˇ B ( m ) n ( s, t, u j ) | P → . The previous supremum can actually be restricted to u ∈ (0 ,
1) as both processes are zeroif u ∈ { , } .Let K > n ≥ i ∈ { , . . . , n } , ξ ( m ) i,n ≥ − K . Also, fix j ∈ A . The supremum on the right of the previous display is thensmaller than I n + II n , where I n = sup ( s,t,u ) ∈ ∆ × (0 , √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( ξ ( m ) i,n + K ) (cid:12)(cid:12)(cid:12) L b n ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij , u ) − ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij ≤ u ) (cid:12)(cid:12)(cid:12) ,II n = sup ( s,t,u ) ∈ ∆ × (0 , K + ¯ ξ ( m ) ⌊ ns ⌋ +1: ⌊ nt ⌋ √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 (cid:12)(cid:12)(cid:12) L b n ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij , u ) − ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij ≤ u ) (cid:12)(cid:12)(cid:12) . u, v ) ∈ [0 , × (0 , |L b n ( u, v ) − ( u ≤ v ) | ≤ ( u − ≤ v ) − ( u + ≤ v ) (C.1)= ( u − b n ≤ v ) − ( u + b n ≤ v )= ( u ≤ v + ) − ( u ≤ v − ) . Then, we write I n ≤ I n, + I n, , where I n, = sup ( s,t,u ) ∈ ∆ × [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( ξ ( m ) i,n − ¯ ξ ( m ) ⌊ ns ⌋ +1: ⌊ nt ⌋ ) ( u − < ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij ≤ u + ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I n, = sup ( s,t,u ) ∈ ∆ × [0 , K + ¯ ξ ( m ) ⌊ ns ⌋ +1: ⌊ nt ⌋ √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( u − < ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij ≤ u + ) . For I n, , we have I n, ≤ sup ( s,t, u , v ) ∈ ∆ × [0 , d k u − v k ≤ bn (cid:12)(cid:12) ˇ B ( m ) n ( s, t, u ) − ˇ B ( m ) n ( s, t, v ) (cid:12)(cid:12) P → B ( m ) n . Before dealing with I n, , let us first show that I n, = sup ( s,t,u ) ∈ ∆ × [0 , √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( u − < ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij ≤ u + ) P → . (C.2)From the proof of Proposition 3.3 of B¨ucher et al. (2014), we have thatsup ( s,t,u ) ∈ ∆ × [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 h { U ij ≤ H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u ) } − ( ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij ≤ u ) i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P → . Consequently, to prove that I n, →
0, it suffices to show thatsup ( s,t,u ) ∈ ∆ × [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 h { U ij ≤ H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u + ) } − { U ij ≤ H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u − ) } i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P → . The supremum on the left of the previous display is smaller than J n, + J n, + J n, , where J n, = sup ( s,t,u ) ∈ ∆ × [0 , (cid:12)(cid:12)(cid:12) B n { s, t, , H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u + ) , } − B n { s, t, , H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u − ) , } (cid:12)(cid:12)(cid:12) ,J n, = sup ( s,t,u ) ∈ ∆ × [0 , √ nλ n ( s, t ) (cid:12)(cid:12)(cid:12) H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u + ) − u + − H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u − ) + u − (cid:12)(cid:12)(cid:12) ,J n, = sup ( s,t,u ) ∈ ∆ × [0 , √ nλ n ( s, t ) | u + − u − | , J n, . We immediately have J n, ≤ √ nb n →
0. The fact J n, → s, t, u )
7→ √ nλ n ( s, t ) { H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u ) − u } , itself following from its weak convergence to( s, t, u )
7→ − B C ( s, t, u j ) in ℓ ∞ (∆ × [0 , R ). The latter is a consequence of the weakconvergence of B n to B C in ℓ ∞ (∆ × [0 , d ; R ), Lemma B.2 of B¨ucher and Kojadinovic(2014) and the extended continuous mapping theorem (van der Vaart and Wellner, 2000,Theorem 1.11.1). The fact that J n, → δ ∈ (0 , ( s,t,u ) ∈ ∆ × [0 , t − s ≥ δ (cid:12)(cid:12)(cid:12) H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u + ) − H − ⌊ ns ⌋ +1: ⌊ nt ⌋ ,j ( u − ) (cid:12)(cid:12)(cid:12) P → . Combined with the asymptotic uniform equicontinuity in probability of B n , the lattercan be used to prove that J n, → I n, → I n, ≤ K × I n, + I n, , where I n, = sup ( s,t,u ) ∈ ∆ × [0 , ¯ ξ ( m ) ⌊ ns ⌋ +1: ⌊ nt ⌋ √ n ⌊ nt ⌋ X i = ⌊ ns ⌋ +1 ( u − < ˆ U ⌊ ns ⌋ +1: ⌊ nt ⌋ ij ≤ u + ) . Hence, to show that I n, →
0, it remains to prove that I n, →
0. The latter can be shownby proceeding as for the term (B.8) in B¨ucher et al. (2014).We therefore have that I n P →
0. The fact that II n P →
0, follows from the fact that II n ≤ I n, →
0. This completes the proof under the condition ξ ( m ) i,n ≥ − K . To show thatthis condition is not necessary, we use the arguments employed at the end of the proofof Proposition 4.3 of B¨ucher et al. (2014). (cid:4) D Proofs of Propositions 6 and 7
Lemma 9.
Assume that U , . . . , U n is drawn from a strictly stationary sequence ( U i ) i ∈ Z whose strong mixing coefficients satisfy α r = O ( r − a ) , a > . Then, for any A ⊆ D , | A | ≥ and j ∈ A , H n,A,j H A,j in ℓ ∞ ([0 , R ) , where, for any t ∈ [0 , , H n,A,j ( t ) = n − / P ni =1 [ Y i,A,j ( t ) − E { Y ,A,j ( t ) } ] , Y i,A,j ( t ) = Q l ∈ A \{ j } (1 − U il ) ( t ≤ U ij ) , and H A,j is atight process.Proof.
Fix A ⊆ D , | A | ≥ j ∈ A . To simplify the notation, we write H n instead of H n,A,j and Y i instead of Y i,A,j as we continue. To prove the desired result, we mostly adaptthe arguments used in the proof of Proposition 2.11 of Dehling and Philipp (2002). FromTheorem 2.1 in Kosorok (2008), two conditions are needed to obtain the desired weakconvergence. The first condition (which is the weak convergence of the finite-dimensionaldistributions) is a consequence of Theorem 3.23 of Dehling and Philipp (2002) as a > Y i ( t ) ∈ [0 ,
1] for all t ∈ [0 , H n is asymptotically | · | -equicontinuous in probability. To do so, we shall first prove that, for31ny ε, δ >
0, there exists a grid 0 = t < t < · · · < t k = 1 such that, for all n sufficientlylarge, P ( max ≤ i ≤ k sup t ∈ [ t i − ,t i ] | H n ( t ) − H n ( t i − ) | ≥ ε ) ≤ δ. (D.1)We first note that there exists constants c ≥ ǫ ∈ (0 ,
1) such that α r ≤ cr − − ǫ .Then, using the fact that, for t, t ′ ∈ [0 , { Y ( t ) − Y ( t ′ ) } ] ≤ E[ | Y ( t ) − Y ( t ′ ) | ] ≤ E { ( t ∧ t ′ ≤ U ij ≤ t ∨ t ′ ) } = | t − t ′ | , we apply Lemma 3.22 of Dehling and Philipp (2002) with ξ i = Y i ( t ) − Y i ( t ′ ) to obtainthatE[ { H n ( t ) − H n ( t ′ ) } ] ≤ cǫ (cid:0) | t − t ′ | η + n − | t − t ′ | η/ (cid:1) = λ (cid:0) | t − t ′ | η + n − | t − t ′ | η/ (cid:1) , where η = 1 + ǫ/ > λ = 10 c/ǫ . It follows that, for any t, t ′ ∈ [0 ,
1] such that | t − t ′ | ≥ n − /η , E[ { H n ( t ) − H n ( t ′ ) } ] ≤ λ | t − t ′ | η . (D.2)Next, consider a grid 0 = t < t < · · · < t k = 1 to be specified later. Furthermore, itcan be verified that the function G : t E { Y ( t ) } is continuous and strictly decreasingon [0 , i ∈ { , . . . , k } , let τ = εn − / /
4, let m = m i = ⌊{ G ( t i − ) − G ( t i ) } /τ ⌋ and define a subgrid t i − = s < s < · · · < s m = t i such that G ( s j ) = G ( s ) − jτ for j ∈ { , . . . , m − } . Notice that this ensures that, for any j ∈ { , . . . , m } , τ ≤ G ( s j − ) − G ( s j ) ≤ τ . Now, fix j ∈ { , . . . , m } . Using the fact that the function t n − P ni =1 Y i ( t ) is also decreasing, it can be verified that, for any t ∈ [ s j − , s j ], H n ( t ) − H n ( t i − ) ≤ | H n ( s j − ) − H n ( t i − ) | + ε/ − ε/ − | H n ( s j ) − H n ( t i − ) | ≤ H n ( t ) − H n ( t i − ) . The above inequalities imply that, for any t ∈ [ t i − , t i ] = S mj =1 [ s j − , s j ], − ε/
2+ min ≤ j ≤ m {−| H n ( s j ) − H n ( t i − ) |} ≤ H n ( t ) − H n ( t i − ) ≤ max ≤ j ≤ m | H n ( s j − ) − H n ( t i − ) | + ε/ , and thus that sup t ∈ [ t i − ,t i ] | H n ( t ) − H n ( t i − ) | ≤ max ≤ j ≤ m | H n ( s j ) − H n ( t i − ) | + ε/ . Hence,P ( sup t ∈ [ t i − ,t i ] | H n ( t ) − H n ( t i − ) | ≥ ε ) ≤ P (cid:26) max ≤ j ≤ m | H n ( s j ) − H n ( t i − ) | ≥ ε/ (cid:27) . (D.3)Now, let ζ l = H n ( s l ) − H n ( s l − ), l ∈ { , . . . , m } with ζ = 0, and let S j = P jl =0 ζ l , j ∈ { , . . . , m } . From (D.2), we then have that, for any 0 ≤ j < j ′ ≤ m and n sufficiently32arge,E { ( S j ′ − S j ) } = E j ′ X l = j +1 ζ l ! = E (cid:2) { H n ( s j ′ ) − H n ( s j ) } (cid:3) ≤ λ ( s j ′ − s j ) η = 2 λ ( X j
2. The assumption of Theorem 2.12 of Billingsley (1968) beingsatisfied (see also Lemma 2.10 in Dehling and Philipp, 2002), we obtain that there existsa constant K ≥ ν ≥ (cid:18) max ≤ j ≤ m | S j | ≥ ν (cid:19) ≤ ν − K ( s m − s ) η = ν − K ( t i − t i − ) η . Applying the previous inequality to the right-hand side of (D.3), we obtain thatP ( sup t ∈ [ t i − ,t i ] | H n ( t ) − H n ( t i − ) | ≥ ε ) ≤ ε − K ( t i − t i − ) η . It follows thatP ( max ≤ i ≤ k sup t ∈ [ t i − ,t i ] | H n ( t ) − H n ( t i − ) | ≥ ε ) ≤ ε − K k X i =1 ( t i − t i − ) η ≤ ε − K × max ≤ i ≤ k ( t i − t i − ) η − × k X i =1 ( t i − t i − ) . By choosing the initial grid such that max ≤ i ≤ k ( t i − t i − ) ≤ { δε − K − } / ( η − , weobtain (D.1).It remains to verify that H n is asymptotically | · | -equicontinuous in probability. ByProblem 2.1.5 in van der Vaart and Wellner (2000), this amounts to showing that for anypositive sequence a n ↓ ε, δ > sup s,t ∈ [0 , | t − s |≤ an | H n ( s ) − H n ( t ) | > ε ≤ δ (D.4)for n sufficiently large. Fix ε, δ > a n ↓
0, and choose a grid 0 = t < · · · < t k = 1such that (D.1) holds for all n sufficiently large. Furthermore, let µ = min
1, sup u ∈ [0 , d |I C,A ( u ) | ≤ u ∈ [0 , d |I C n ,A ( u ) | ≤ | ψ C,A ( C ) | ≤
1. Hence, by (2.17), (3.8) and linearity of f ,we have that the second term (between square brackets) on the right of inequality (D.5) isbounded by 4 sup x ∈ [ − , d − | f ( x ) | < ∞ . Concerning the first term on the right of (D.5),we have 1 n n X i,j =1 ϕ (cid:18) i − jℓ n (cid:19) = 1 n ℓ n X k = − ℓ n ( n − | k | ) ϕ (cid:18) kℓ n (cid:19) ≤ ℓ n + 1 = O ( n / − ε ) . We will now show that the last supremum on the right of (D.5) is O P ( n − / ), which willcomplete the proof. By the triangle inequality,sup u ∈ [0 , d | f [ I C n { h n ( u ) } − I C ( u ) − ψ C n ( C n ) + ψ C ( C )] |≤ sup u ∈ [0 , d | f [ I C n { h n ( u ) } − I C ( u )] | + | f { ψ C n ( C n ) − ψ C ( C ) }| . By linearity of f , from (3.2) and (3.8), to show that the first term on the right on theprevious inequality is O P ( n − / ), it suffices to show that, for any A ⊆ D , | A | ≥ u ∈ [0 , d |I C n ,A { h n ( u ) } − I C,A ( u ) | = O P ( n − / ) . (D.6)Similarly, for the second term on the right, it suffices to show that, for any A ⊆ D , | A | ≥ | ψ C n ,A ( C n ) − ψ C,A ( C ) | = O P ( n − / ). Now, from Fubini’s theorem, ψ C,A ( C ) = ψ C,A [E { ( U ≤ · ) } ] = E {I C,A ( U ) } . Hence, | ψ C n ,A ( C n ) − ψ C,A ( C ) | is smaller than (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 n I C n ,A ( ˆ U ni ) − I C,A ( U i ) o(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 [ I C,A ( U i ) − E {I C,A ( U ) } ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup u ∈ [0 , d |I C n ,A { h n ( u ) } − I C,A ( u ) | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 [ I C,A ( U i ) − E {I C,A ( U ) } ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . O P ( n − / ). The latter is a consequence of the weak con-vergence of n − / P ni =1 [ I C,A ( U i ) − E {I C,A ( U ) } ] which follows from Theorem 3.23 ofDehling and Philipp (2002) as a consequence of the fact that sup u ∈ [0 , d |I C,A ( u ) | ≤ A ⊆ D , | A | ≥ u ∈ [0 , d |I C,A { h n ( u ) } − I C,A ( u ) | = O P ( n − / ) , (D.7)sup u ∈ [0 , d |I H n ,A ( u ) − I C,A ( u ) | = O P ( n − / ) , (D.8)sup u ∈ [0 , d |I C n ,A ( u ) − I H n ,A ( u ) | = O P ( n − / ) . (D.9)Fix A ⊆ D , | A | ≥ Proof of (D.7) . We havesup u ∈ [0 , d |I C,A { h n ( u ) } − I C,A ( u ) | ≤ sup u ∈ [0 , d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y l ∈ A { − H n,l ( u l ) } − Y l ∈ A (1 − u l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X j ∈ A sup u ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [0 , d Y l ∈ A \{ j } (1 − v l ) [ { H n,j ( u ) ≤ v j } − ( u ≤ v j )] d C ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By an application of the mean value theorem similar to that performed in the proof ofProposition 1, it is easy to verify that the first supremum is O P ( n − / ) since, for any j ∈ D , sup u ∈ [0 , | H n,j ( u ) − u | = O P ( n − / ) as a consequence of the weak convergence of B n defined in (2.10). The second term is smaller than X j ∈ A sup u ∈ [0 , Z [0 , | { H n,j ( u ) ≤ v } − ( u ≤ v ) | d v ≤ X j ∈ A sup u ∈ [0 , Z [0 , { u ∧ H n,j ( u ) ≤ v ≤ u ∨ H n,j ( u ) } d v = X j ∈ A sup u ∈ [0 , | H n,j ( u ) − u | = O P ( n − / ) . Proof of (D.8) : From (3.2) and the triangle inequality, it suffices to show that, forany j ∈ A ,sup u ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 Y l ∈ A \{ j } (1 − U il ) ( u ≤ U ij ) − Z [0 , d Y l ∈ A \{ j } (1 − v l ) ( u ≤ v j )d C ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O P ( n − / ) . The latter is an immediate consequence of the weak convergence result stated in Lemma 9and the continuous mapping theorem. 35 roof of (D.9) : The supremum on the left of (D.9) is smaller than I n + II n + III n ,where I n = sup u ∈ [0 , d |I C n ,A ( u ) − I H n ,A { h − n ( u ) }| ,II n = sup u ∈ [0 , d |I H n ,A { h − n ( u ) } − I C,A { h − n ( u ) } − I H n ,A ( u ) + I C,A ( u ) | , (D.10) III n = sup u ∈ [0 , d |I C,A { h − n ( u ) } − I C,A ( u ) | , (D.11)with h − n is defined in (A.2). The term I n is smallersup u ∈ (0 , d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y l ∈ A (1 − u l ) − Y l ∈ A { − H − n,l ( u l ) } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + sup u ∈ [0 , d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 X j ∈ A Y l ∈ A \{ j } { − H n,l ( U il ) } { u j ≤ H n,j ( U ij ) }− n n X i =1 X j ∈ A Y l ∈ A \{ j } (1 − U il ) { H − n,j ( u j ) ≤ U ij } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since, for any j ∈ D , sup u ∈ [0 , | H − n,j ( u ) − u | = sup u ∈ [0 , | H n,j ( u ) − u | (for instance, bysymmetry arguments on the graphs of H n,j and H − n,j ), and by an application of themean value theorem as above, we obtain that the first supremum is O P ( n − / ). Usingthe fact that, for all u ∈ [0 , u ≤ H n,j ( U ij ) is equivalent to H − n,j ( u ) ≤ U ij , it can beverified that the second supremum is smaller than X j ∈ A sup u ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 Y l ∈ A \{ j } { − H n,l ( U il ) } − Y l ∈ A \{ j } (1 − U il ) { u ≤ H n,j ( U ij ) } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X j ∈ A sup u ∈ [0 , d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y l ∈ A \{ j } { − H n,l ( u l ) } − Y l ∈ A \{ j } (1 − u l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O P ( n − / ) , where the last equality follows again by an application of the mean value theorem asabove. Hence, I n = O P ( n − / ). For II n defined in (D.10), we have II n ≤ n − / X j ∈ A sup u ∈ [0 , (cid:12)(cid:12) H n,A,j { H − n,j ( u ) } − H n,A,j ( u ) (cid:12)(cid:12) = o P ( n − / ) , where H n,A,j is defined in Lemma 9. The last equality is a consequence of the asymp-totic equicontinuity in probability of H n,A,j and the fact that sup u ∈ [0 , | H − n,j ( u ) − u | =sup u ∈ [0 , | H n,j ( u ) − u | a . s . −→
0. The latter convergence follows from the almost sure invari-ance principle established in Berkes and Philipp (1977) and Yoshihara (1979). It impliesa functional law of the iterated logarithm for u H n,j ( u ) − u as soon as a >
3, whichin turn implies the Glivenko–Cantelli lemma under strong mixing.It remains to show that
III n defined in (D.11) is O P ( n − / ). The proof of the latteris similar to that of (D.7). (cid:4) roof of Proposition 7. We only show the result under (ii), the proof being simplerunder (i). To prove the desired result, we shall show that ˜ σ n,b n ,C n ,f − ˇ σ n,C n ,f P → A ⊆ D , | A | ≥ u ∈ [0 , d |I b n ,C n ,A ( u ) − I C n ,A ( u ) | = O P ( n − / ) . Fix A ⊆ D , | A | ≥
1. From (3.2) and (3.6), we have that the supremum on the right ofthe previous display is smaller than P j ∈ A I n,j , where I n,j = sup u ∈ [0 , Z [0 , d |L b n ( u, v j ) − ( u ≤ v j ) | d C n ( v ) . Fix j ∈ A . From (C.1), we have that I n,j ≤ n − / J n,j , where J n,j = sup u ∈ [0 , √ n n X i =1 { ( u − ≤ ˆ U nij ) − ( u + ≤ ˆ U nij ) } = sup u ∈ [0 , √ n n X i =1 { ( ˆ U nij < u + ) − ( ˆ U nij < u − ) }≤ sup u ∈ [0 , √ n n X i =1 { ( ˆ U nij ≤ u + ) − ( ˆ U nij ≤ u − ) } + sup u ∈ [0 , √ n n X i =1 ( ˆ U nij = u ) . Proceeding as for (C.2), we obtain that the first supremum on the right of the previousdisplay converges in probability to zero. The second supremum is smaller thansup u ∈ [0 , √ n n X i =1 { ( ˆ U nij ≤ u ) − ( ˆ U nij ≤ u − /n ) } and can be dealt with along the same lines. Hence, J n,j P →
0, which implies that I n,j = o ( n − / ) and completes the proof. (cid:4) H computed from 1000 samples of size n ∈{ , , , } generated with γ = 0 in (AR1) and when C = C = C is eitherthe d -dimensional Clayton (Cl) or Gumbel–Hougaard (GH) copula the bivariate marginsof which have a Kendall’s tau of τ . The tests ˜ S n,i are carried out with i.i.d. multipliersequences, while the tests S an,i use variance estimators of the form (3.11). d = 2 d = 4 C n τ ˜ S n, ˜ S n, S an, S an, ˜ S n, ˜ S n, ˜ S n, S an, S an, S an, Cl 50 0.1 6.8 7.4 2.6 3.0 4.6 5.1 4.0 1.2 2.1 0.70.3 4.1 5.2 1.7 4.2 4.9 5.4 3.7 0.5 2.6 0.70.5 3.1 2.7 2.5 8.6 7.1 3.9 4.9 2.8 2.8 1.20.7 3.0 0.5 8.3 23.8 7.4 4.1 3.3 5.4 10.3 3.1100 0.1 3.5 4.3 2.3 2.7 4.1 5.3 4.4 1.6 3.4 2.50.3 4.0 4.4 2.3 3.6 5.7 4.7 4.4 2.0 2.8 1.40.5 4.2 4.0 4.9 8.3 4.3 4.0 3.5 2.2 3.7 1.90.7 5.7 1.6 12.6 23.1 9.1 3.9 7.6 11.3 9.5 7.4200 0.1 4.9 4.7 2.8 3.1 6.1 5.1 5.2 3.1 3.4 3.30.3 4.9 5.3 3.7 4.9 4.1 5.6 4.2 2.3 3.6 1.90.5 4.6 4.3 4.8 6.9 4.6 5.5 4.2 4.1 4.8 3.20.7 5.6 3.1 11.2 15.1 10.5 5.3 11.1 14.1 8.3 9.9400 0.1 4.6 4.9 3.7 3.8 6.3 6.7 6.5 4.5 5.5 4.80.3 4.3 4.6 4.0 4.4 5.8 5.3 5.5 4.1 4.2 3.80.5 4.8 4.6 4.2 4.8 5.8 4.5 5.5 5.5 4.0 4.70.7 5.9 4.0 9.3 10.8 8.5 6.6 8.7 13.5 8.1 8.2GH 50 0.1 6.7 6.3 3.4 2.3 5.8 5.3 4.7 2.4 0.8 2.50.3 4.1 3.9 3.5 2.1 5.9 6.0 5.3 1.8 0.7 3.10.5 3.1 3.4 6.9 3.4 4.6 4.9 4.0 3.0 2.5 6.50.7 2.0 1.8 15.5 10.7 3.4 6.2 2.0 6.2 4.2 10.3100 0.1 5.2 5.1 2.7 2.5 4.3 4.8 4.1 2.5 1.5 2.10.3 5.9 5.3 5.2 3.9 6.1 6.7 6.7 3.1 1.9 4.50.5 3.7 3.7 6.6 5.1 5.3 4.8 5.3 3.6 3.4 6.40.7 1.3 2.3 16.9 13.8 4.5 7.0 2.7 8.6 9.0 14.2200 0.1 5.2 5.2 3.8 3.5 4.8 4.3 4.5 3.3 2.6 3.10.3 5.2 5.1 4.7 3.9 6.0 6.5 5.3 4.7 3.3 4.30.5 4.5 4.5 5.2 4.7 4.2 3.9 4.0 3.2 3.6 3.90.7 2.2 3.7 12.8 10.8 4.6 7.0 4.9 6.6 9.0 10.9400 0.1 6.4 6.1 4.8 4.7 5.1 5.7 4.3 4.0 3.1 3.10.3 4.7 4.6 4.1 3.8 4.6 5.3 5.6 3.7 3.6 4.40.5 3.3 3.3 3.5 3.0 4.3 5.1 4.5 3.9 4.5 4.70.7 4.6 5.8 10.1 9.9 5.3 7.1 5.9 6.3 9.5 10.438able 2: Percentage of rejection of H computed from 1000 samples of size n ∈{ , , } generated with γ = 0 in (AR1), t ∈ { . , . , . } and when C and C are both d -dimensional normal (N) or Frank (F) copulas such that the bivariate mar-gins of C have a Kendall’s tau of 0.2 and those of C a Kendall’s tau of τ . The colunmsCvM give the results for the test studied in B¨ucher et al. (2014). All the tests were carriedout with i.i.d. multiplier sequences. d = 2 d = 4 C n τ t
CvM ˜ S n, ˜ S n, CvM ˜ S n, ˜ S n, ˜ S n, N 50 0.4 0.10 5.6 6.0 5.6 5.9 7.9 7.9 8.30.25 9.1 8.7 8.9 12.2 17.3 18.9 19.50.50 13.4 12.6 12.6 24.3 25.1 27.6 28.20.6 0.10 9.0 8.7 8.9 7.1 20.7 21.7 22.40.25 32.3 34.7 32.6 45.6 66.3 67.0 69.90.50 46.7 42.7 41.6 76.1 78.0 77.5 80.8100 0.4 0.10 5.7 7.8 7.6 7.6 11.2 12.2 12.30.25 14.9 19.7 19.1 27.0 35.3 37.2 43.00.50 25.9 28.9 29.2 54.5 54.6 53.5 59.60.6 0.10 14.6 22.7 23.4 26.1 47.5 51.1 58.80.25 60.0 68.6 69.0 90.3 94.9 94.8 97.60.50 81.9 84.8 84.2 98.8 98.4 99.0 99.5200 0.4 0.10 9.1 11.7 12.3 13.2 18.2 17.9 23.30.25 26.5 36.7 36.9 58.9 64.9 67.1 75.50.50 47.7 54.2 53.7 83.4 83.5 83.3 88.90.6 0.10 34.5 57.7 58.0 63.1 87.3 87.8 93.80.25 92.6 96.5 96.7 100.0 100.0 100.0 100.00.50 99.1 99.5 99.5 100.0 100.0 100.0 100.0F 50 0.4 0.10 6.9 5.7 6.2 4.5 7.8 9.0 8.40.25 10.8 9.7 10.0 12.9 17.9 19.7 19.90.50 15.1 13.6 13.6 24.7 30.2 31.1 29.10.6 0.10 11.1 10.6 11.3 7.3 23.3 29.7 24.80.25 33.1 32.7 31.9 42.3 67.2 70.2 69.50.50 50.9 46.1 46.2 78.3 81.9 82.3 85.5100 0.4 0.10 6.1 7.0 7.4 6.5 9.2 13.6 11.90.25 16.5 18.2 18.7 26.5 38.8 46.8 49.60.50 26.4 28.6 28.3 48.9 52.7 58.3 61.60.6 0.10 17.7 27.3 27.2 22.7 55.3 63.9 68.60.25 66.5 73.6 74.0 91.9 97.7 98.2 99.50.50 86.2 87.3 87.5 99.3 98.8 99.4 99.8200 0.4 0.10 10.2 15.7 15.6 12.5 19.7 25.3 27.10.25 34.3 41.3 41.5 53.6 64.4 76.2 78.80.50 50.7 54.3 54.4 83.2 83.9 90.4 93.20.6 0.10 39.0 64.7 65.6 60.3 88.0 92.2 96.40.25 95.4 98.3 98.3 99.9 100.0 100.0 100.00.50 99.5 99.8 99.8 100.0 100.0 100.0 100.039able 3: Percentage of rejection of H computed from 1000 samples of size n ∈{ , , } when C = C = C is either the bivariate Clayton (Cl), Gumbel–Hougaard(GH) or Frank (F) copula with a Kendall’s tau of τ . In the first four vertical blocks ofthe table, the test ˜ S n, (resp. S an, ) is carried out using dependent multiplier sequences(resp. a variance estimator of the form (3.12)). In the last vertical block, i.i.d. multipliersand a variance estimator of the form (3.11) are used instead. γ = 0 γ = 0 . γ = 0 . γ = 0 . C n τ ˜ S n, S an, ˜ S n, S an, ˜ S n, S an, ˜ S n, S an, ˜ S n, S an, Cl 100 0.10 5.2 2.3 6.6 3.5 8.2 3.3 6.2 2.5 14.5 10.20.30 3.5 1.8 6.7 3.1 7.1 4.7 5.2 3.3 15.0 11.60.50 4.0 3.4 5.0 4.5 5.2 4.7 4.6 4.5 12.0 13.50.70 8.3 12.0 7.5 11.8 7.2 11.2 7.2 13.2 8.9 20.0200 0.10 4.2 2.3 5.1 2.8 6.9 3.6 5.0 3.1 17.2 13.50.30 5.1 2.6 6.2 3.4 7.2 4.4 5.3 3.8 15.7 13.00.50 4.4 4.1 5.0 5.1 4.6 5.1 4.5 4.5 14.1 14.20.70 6.5 12.2 6.6 9.8 7.4 11.2 6.5 10.8 12.4 20.0400 0.10 4.7 3.3 5.6 4.3 6.0 3.5 5.3 3.8 19.4 16.90.30 4.4 3.4 6.3 4.3 6.0 4.2 4.0 3.5 17.3 15.20.50 4.7 4.7 5.9 5.7 5.6 5.0 6.1 5.7 14.6 14.20.70 6.4 8.7 5.7 7.9 5.1 6.8 6.6 9.5 15.7 19.0GH 100 0.10 4.8 2.5 5.1 2.0 7.7 2.7 5.6 2.8 15.3 11.20.30 5.0 3.7 5.9 4.4 7.5 4.5 4.9 2.9 15.0 14.20.50 4.5 6.7 4.3 7.1 6.3 7.9 4.9 6.9 10.7 15.70.70 3.5 16.0 4.3 18.9 5.1 18.9 3.7 16.2 4.5 25.4200 0.10 6.4 3.9 5.6 3.7 7.3 3.9 5.8 3.8 18.2 14.10.30 6.0 5.1 6.4 4.6 6.7 4.6 5.4 4.5 19.1 16.40.50 5.1 4.9 6.0 6.4 6.9 8.0 3.7 4.9 15.6 17.20.70 3.8 14.4 2.8 13.0 4.4 12.4 3.5 12.2 10.0 25.4400 0.10 5.0 4.0 5.8 4.8 6.3 5.1 5.2 3.9 18.5 16.30.30 4.1 3.0 5.1 4.3 6.3 4.6 4.9 4.1 18.5 17.20.50 3.2 3.6 5.0 6.3 7.9 7.5 4.9 4.7 16.7 17.20.70 5.2 9.8 3.8 8.7 5.4 10.6 3.8 8.2 14.5 22.4F 100 0.10 5.5 2.1 5.3 2.3 10.6 4.2 5.0 2.4 15.2 10.20.30 4.4 2.2 5.9 3.9 7.7 4.1 6.4 4.7 13.3 10.30.50 4.0 7.6 4.0 6.0 5.4 7.1 4.2 6.7 12.8 18.00.70 5.2 29.3 4.8 26.5 5.4 18.1 5.4 23.9 5.9 28.5200 0.10 4.0 2.1 6.0 3.9 8.3 4.5 5.1 2.9 17.5 13.40.30 5.0 3.9 5.7 4.1 7.1 3.9 5.3 3.4 17.0 14.50.50 4.8 6.2 4.5 5.7 6.9 7.1 4.4 5.6 15.0 17.30.70 3.2 19.9 4.0 17.5 4.6 13.4 4.9 20.1 8.9 25.1400 0.10 4.1 3.1 6.0 4.4 6.0 4.0 4.5 3.0 18.0 14.80.30 5.5 4.6 6.7 5.6 5.9 4.2 5.2 4.3 14.7 12.50.50 4.6 4.7 4.7 5.0 4.0 3.8 4.8 5.5 15.7 16.50.70 5.3 13.2 4.5 12.3 6.2 9.9 5.7 13.2 14.2 21.740able 4: Percentage of rejection of H computed from 1000 samples of size n ∈ { , } generated with t ∈ { . , . , . } and when C and C are both bivariate Clayton (Cl),Gumbel–Hougaard (GH) or normal (N) copulas with a Kendall’s tau of 0.2 for C anda Kendall’s tau of τ for C . The colunms CvM give the results for the test studied inB¨ucher et al. (2014). The latter test and the test ˜ S n, (resp. the test S an, ) are (resp.is) carried out using dependent multiplier sequences (resp. a variance estimator of theform (3.12)). γ = 0 γ = 0 . C n τ t
CvM ˜ S n, S an, CvM ˜ S n, S an, CvM ˜ S n, S an, Cl 100 0.4 0.10 6.5 6.5 4.3 6.5 8.0 5.0 6.6 6.7 3.80.25 17.9 20.4 13.4 14.0 19.7 10.6 17.2 18.1 11.20.50 23.5 23.2 15.0 18.3 22.4 9.7 28.6 27.6 17.10.6 0.10 12.6 20.6 19.7 9.4 17.1 17.0 13.9 20.1 19.40.25 61.3 65.7 52.7 44.2 53.6 36.4 61.1 64.8 50.70.50 80.0 78.8 61.1 58.4 61.8 34.9 80.3 78.3 59.3200 0.4 0.10 8.2 9.6 7.5 6.9 10.4 7.0 8.3 11.1 8.90.25 26.5 31.8 25.2 19.9 27.7 20.2 27.8 32.0 26.20.50 45.3 47.0 37.0 34.2 40.0 27.9 47.1 48.8 40.10.6 0.10 30.4 42.1 42.3 12.6 28.8 28.6 29.7 43.9 43.40.25 93.2 94.2 87.4 71.1 79.2 65.9 91.1 92.2 83.50.50 98.5 98.3 94.1 89.5 90.5 80.1 98.7 98.2 94.1GH 100 0.4 0.10 5.3 8.0 7.1 5.0 8.2 7.1 6.3 7.6 6.90.25 12.4 17.1 12.1 11.6 18.6 11.1 14.9 18.6 14.90.50 22.5 25.2 16.9 18.2 24.2 14.0 26.0 27.7 19.90.6 0.10 10.4 18.5 26.1 7.7 19.4 25.7 10.2 19.9 26.60.25 53.3 63.1 54.7 41.2 58.0 43.7 55.0 63.8 52.40.50 78.1 80.4 67.4 62.7 69.5 46.1 76.0 76.3 63.1200 0.4 0.10 7.0 10.5 10.0 7.1 11.4 9.9 6.9 10.2 9.00.25 25.2 31.9 27.7 19.1 30.9 22.8 24.6 32.3 26.70.50 43.0 48.3 42.1 31.4 39.3 30.0 43.2 49.1 41.30.6 0.10 25.9 42.7 47.2 13.0 30.1 34.0 23.5 43.4 46.30.25 89.0 92.9 86.3 72.1 83.5 70.0 88.9 94.5 85.00.50 98.3 98.5 95.9 89.6 92.0 83.4 98.4 98.7 93.6N 100 0.4 0.10 6.1 7.8 6.2 6.9 10.2 7.8 6.1 7.0 5.50.25 14.4 19.3 14.7 13.7 19.2 13.2 14.7 17.8 13.30.50 25.6 27.7 19.4 17.5 24.1 12.5 25.2 28.7 19.20.6 0.10 10.6 27.1 32.0 8.2 19.7 23.7 10.2 19.3 24.70.25 61.5 70.1 61.3 46.0 62.3 44.8 58.4 69.2 59.30.50 82.6 85.1 72.3 64.9 71.3 44.9 79.0 82.0 65.7200 0.4 0.10 8.0 10.8 9.2 5.9 12.6 9.2 7.0 9.3 8.90.25 27.7 37.4 33.2 20.4 31.0 24.7 26.8 35.1 30.70.50 47.0 51.5 43.6 33.2 41.7 30.7 43.0 49.5 41.30.6 0.10 27.1 47.3 49.6 14.5 35.6 39.2 28.8 48.3 51.80.25 91.5 96.5 88.4 72.3 85.2 71.0 90.7 96.1 85.70.50 98.8 99.7 96.3 91.7 95.5 83.6 99.1 99.3 94.841able 5: Percentage of rejection of H computed from 1000 samples of size n = 500generated with γ = 0 in (AR1) and when C and C are both either bivariate Studentcopulas with 1 d.f. ( t ), with 3 d.f. ( t ) or with 5 d.f. ( t ) with a Spearman’s rho of 0.4for C and a Spearman’s rho of ρ for C . The test ˜ S n, was carried out with dependentmultiplier sequences, while the test S an, used a variance estimator of the form (3.12). Thecolumns W contain the rejection rates of the similar test studied in Wied et al. (2014).The results are taken from Table 1 in the latter reference. t t t ρ W ˜ S n, S an, W ˜ S n, S an, W ˜ S n, S an,1