Séminaire de statistique IRMA Strasbourg

Transcription

1 Test non paramétrique de détection de rupture dans la copule d observations multivariées avec changement dans les lois marginales -Illustration des méthodes et simulations de Monte Carlo- Séminaire de statistique IRMA Strasbourg Tom Rohmer, Université de Bordeaux2, ISPED with collaboration from: Ivan Kojadinovic (Université de Pau) & Jean-François Quessy (Université de Trois-Rivières) and Axel Bücher (Université d Heidelberg) & Johan Segers (Université catholique de Louvain) 24 novembre / 43

2 Summary 1 Introduction 2 Measure of the multivariate dependence The Sklar s theorem The role of copulas to test for breaks detection 3 A Cramér-von Mises test statistic Empirical copula The test statistic 4 Computation of approximate p-values The sequential empirical copula process A resampling scheme for the sequential empirical copula process 5 Monte Carlo Simulations 2 / 43

3 Introduction Summary 1 Introduction 2 Measure of the multivariate dependence The Sklar s theorem The role of copulas to test for breaks detection 3 A Cramér-von Mises test statistic Empirical copula The test statistic 4 Computation of approximate p-values The sequential empirical copula process A resampling scheme for the sequential empirical copula process 5 Monte Carlo Simulations 3 / 43

4 Introduction Test for break detection Let X 1,..., X n be d-dimensional random vectors, where for i = 1,..., n, X i = (X i1,..., X id ). We aim at testing the null hypothesis H 0 : F such that X 1,..., X n have c.d.f. F. 4 / 43

5 Introduction Test for break detection Let X 1,..., X n be d-dimensional random vectors, where for i = 1,..., n, X i = (X i1,..., X id ). We aim at testing the null hypothesis H 0 : F such that X 1,..., X n have c.d.f. F. Example of alternative hypotheses : An abrupt change at the instant k [[1, n 1]] : F (1), F (2) such that X 1,..., X k F (1), X k +1,..., X n F (2) 5 / 43

6 Introduction Test for break detection Let X 1,..., X n be d-dimensional random vectors, where for i = 1,..., n, X i = (X i1,..., X id ). We aim at testing the null hypothesis H 0 : F such that X 1,..., X n have c.d.f. F. Example of alternative hypotheses : A gradual change : 1 k 1 k 2 n 1 X1,..., X k1 F (1) X k2,..., Xn F (2) For i = k 1 + 1,..., k 2 the law of X i will gradually go from F (1) to F (2). 6 / 43

7 Introduction Cumulative Sum test for H 0, serially independent data I Example 1 : Test for change in the mean, d=1 { T n µ k(n k) 1 = max k=1,...,n 1 n 3/2 k ns = sup 1 (X i X n) s [0,1] n. i=1 k i=1 X i 1 n k } n X i i=k+1 Under H 0, as soon as X 1,..., X n are i.i.d., T µ n σ sup U(s), s [0,1] where σ 2 is the unknown variance of X i and U is a standard Brownian bridge, i.e. a centered Gaussian process with covariance function : cov{u(s), U(t)} = min(s, t) st, s, t [0, 1]. 7 / 43

8 Introduction Cumulative Sum test for H 0, serially independent data II Example 2 : test à la [Csörgő et Horváth(1997)] T n # k(n k) = max sup F k=1,...,n 1 n 3/2 1:k (x) F k+1:n (x) x R d ns = sup sup 1 {1(X i x) F 1:n(x)} s [0,1] x R d n, i=1 where for 1 k n, F 1:k (resp. F k+1:n ) is the empirical c.d.f. relative to the subsample X 1,..., X k (resp. X k+1,..., X n) : et F 1:k (x) = 1 k k i=1 1(X i x), F k+1:n (x) = 1 n k F 1:n(x) = 1 n n 1(X i x). i=1 n 1(X i x), i=k+1 8 / 43

9 Introduction Strong mixing conditions In this work, we do not necessarily assume the observations to be serially independent. The asymptotic validity of techniques is proved for strongly mixing observation : [Rosenblatt(1956)] Consider a sequence of d-dimensional random vectors (Y i ) i Z. For s, t Z {, + }, let F t s the σ-algebra generated by Y i, s i t, i.e., F t s = σ(y i, s i t). The strong mixing coefficient is defined by α r = sup t Z sup A F t,b F+ t+r P(A B) P(A)P(B). The sequence (Y i ) i Z is said α-mixing if α r 0 as r +. 9 / 43

10 Introduction Nasdaq, Dow Jones and the black Monday ( ) : ˆp T # n,α=0.05 val = : we conclude in favour of H / 43

11 Measure of the multivariate dependence Summary 1 Introduction 2 Measure of the multivariate dependence The Sklar s theorem The role of copulas to test for breaks detection 3 A Cramér-von Mises test statistic Empirical copula The test statistic 4 Computation of approximate p-values The sequential empirical copula process A resampling scheme for the sequential empirical copula process 5 Monte Carlo Simulations 11 / 43

12 Measure of the multivariate dependence The Sklar s theorem The Sklar s theorem Theorem of [Sklar(1959)] Let X = (X 1,..., X d ) be a d-dimensional random vector with c.d.f. F and let F 1,... F d be the marginals c.d.f. of X assuming continuous. Then it exists an unique function C : [0, 1] d [0, 1] such that : F (x) = C{F 1(x 1),..., F d (x d )}, x = (x 1,..., x d ) R d. The copula C characterizes the dependence structure of vector X. The copula C can be expressed as follows : C(u) = F {F 1 1 (u 1),..., F 1 d (u d )}, u = (u 1,..., u d ) [0, 1] d. A. Sklar. Fonctions de répartition à n dimensions et leurs marges. Publications de l Institut de Statistique de l Université de Paris, 8 : , / 43

13 Measure of the multivariate dependence The Sklar s theorem Classical copulas Independence copula : Normal copulas d C Π (u) = u j ; j=1 Gumbel Hougaard copulas : ( Clayton copulas : C N Σ (u) = Φ d,σ {Φ 1 (u 1),..., Φ 1 (u d )}; Cθ GH (u) = exp C Cl θ (u) = ( d j=1 d j=1 ) [{ log(u j )} θ] 1/θ, θ 1; u θ j d + 1) 1/θ, θ > 0; 13 / 43

14 Measure of the multivariate dependence The role of copulas to test for breaks detection The role of copulas to test for breaks detection H 0 : F such that X 1,..., X n have c.d.f. F. Sklar s theorem allows to rewrite H 0 as H 0,m H 0,c where H 0,m H 0,c : H 0,c : C, such that X 1,..., X n have copula C H 0,m : F 1,..., F d such thar X 1,..., X n have m.c.d.f. F 1,..., F d. Construction of a test for H 0 more powerful than its predecessors against alternatives involving a change in the copula, based on the CUSUM approach. F, F 1,..., F d and C are unknown. 14 / 43

15 A Cramér-von Mises test statistic Summary 1 Introduction 2 Measure of the multivariate dependence The Sklar s theorem The role of copulas to test for breaks detection 3 A Cramér-von Mises test statistic Empirical copula The test statistic 4 Computation of approximate p-values The sequential empirical copula process A resampling scheme for the sequential empirical copula process 5 Monte Carlo Simulations 15 / 43

16 A Cramér-von Mises test statistic Empirical copula Empirical copula I Let X 1,..., X n be d-dimensional random vectors with continuous m.c.d.f. F 1,..., F d and copula C. For i = 1,..., n, the random vectors U i = (F 1(X i1 ),..., F d (X id )) C. When F 1,..., F d are supposed known, a natural estimator of C is given by the empirical c.d.f. of U 1,..., U n. Here F 1,..., F d are unknown, an estimator of C is given by the empirical c.d.f. of pseudo-observations of copula C : For j = 1,..., d let F 1:n,j be the empirical c.d.f. of sample X 1j,..., X nj. For i = 1,..., n, consider the vectors Û 1:n i = (F 1:n,1(X i1 ),..., F 1:n,d (X id )) = 1 n (R1:n i1,..., Rid 1:n ), where for j = 1,..., d, R 1:n ij is the maximal rank of X ij among X 1j,..., X nj. 16 / 43

17 A Cramér-von Mises test statistic Empirical copula Empirical copula II [Rüschendorf(1976)], [Deheuvels(1979)] C 1:n(u) = 1 n n i=1 1(Û 1:n i u), u [0, 1] d. Let C 1:k (resp. C k+1:n ) the empirical copula evaluates on X 1,..., X k (resp. X k+1,..., X n) : C 1:k (u) = 1 k k i=1 1(Û 1:k i u), C k+1:n (u) = 1 n k n i=k+1 1(Û k+1:n i u) u [0, 1] d. X 1 = X k X n = (X 11,, X 1d) (X n1,, X nd) (R 1:k 11,, R 1:k 1d ) (R 1:k k1,, R 1:k kd ) (R k+1:n k+11,, Rk+1:n k+1d ) (R k+1:n k+11,, Rk+1:n k+1d ) C 1:k. C k+1:n. 17 / 43

18 A Cramér-von Mises test statistic The test statistic Break detection in copula We consider the process D n(s, u) = n ns n (n ns ) {C 1: ns (u) C ns +1:n (u)}, (s, u) [0, 1] d+1, n A Cramér-von Mises statistic : S n,k = D 2 n(k/n, u)dc 1:n(u) = 1 [0,1] d n and S n = max k {1,...,n 1} S n,k. n i=1 D 2 n(k/n, Û 1:n i ), Under H 0, for all k {1,..., n 1}, C 1:k and C k+1:n estimate the same copula C, thus S n tends to be relatively weak. For an abrupt change in copula, the unknown break time can be estimate by the integer k which maximise S n,k. 18 / 43

19 A Cramér-von Mises test statistic The test statistic Asymptotic behaviour of S n under H 0 Condition : [Segers(2012)] For any j {1,..., d}, the partial derivatives C j = C/ u j exist and are continuous on V j = {u [0, 1] d, u j (0, 1)}. Proposition Under H 0 S n = 1 max k {1,...,n 1} n n i=1 D n(k/n, Ûi 1:n ) 2 S C = sup {D C (s, u)} 2 dc(u). s [0,1] [0,1] d D C (s, u) = (1 s)c C (0, s, u) sc C (s, 1, u), (s, u) [0, 1] d+1. and for s t [0, 1] and u [0, 1] d d C C (s, t, u) = {Z C (t, u) Z C (s, u)} C j (u){z C (t, u {j} ) Z C (s, u {j} )} j=1 with Z C a C-Kiefer Müller process and u {j} = (1,..., 1, u j, 1,..., 1). 19 / 43

20 Computation of approximate p-values Summary 1 Introduction 2 Measure of the multivariate dependence The Sklar s theorem The role of copulas to test for breaks detection 3 A Cramér-von Mises test statistic Empirical copula The test statistic 4 Computation of approximate p-values The sequential empirical copula process A resampling scheme for the sequential empirical copula process 5 Monte Carlo Simulations 20 / 43

21 Computation of approximate p-values The sequential empirical copula process A decomposition for the process D n Consider the process D n(s, u) = n ns n ( 1 ns ) {C1: ns (u) C } ns +1:n(u), n (s, u) [0, 1] d+1. D n can be written as function of C n the sequential empirical copula process : ( D n(s, u) = 1 ns ) C n(0, s, u) ns Cn(s, 1, u) n n with for s t [0, 1] and u [0, 1] d and λ n(s, t) = ( nt ns )/n Sequential empirical copula process C n(s, t, u) = nλ n(s, t){c ns +1: nt (u) C(u)} = 1 n nt i= ns +1 { } 1(Û ns +1: nt i u) C(u). 21 / 43

22 Computation of approximate p-values The sequential empirical copula process Asymptotic behaviour of the sequential empirical copula process Theorem Let X 1,..., X n be 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 > 1. Under the previous condition, sup C n(s, t, u) C P n(s, t, u) 0, u [0,1] d where for u [0, 1] d, and u {j} = (1,..., 1, u j, 1,..., 1), C n(s, t, u) = B n(s, t, u) d C j (u)b n(s, t, u {j} ). j=1 B n(s, u) = 1 n nt i= ns +1 {1(U i u) C(u)}, (s, u) [0, 1] d / 43

23 Computation of approximate p-values The sequential empirical copula process A resampling scheme of the sequential empirical copula process From the previous theorem, the process C n is asymptotically equivalent to process C n C n(s, t, u) = {Z n(t, u) Z n(s, u)} with d C j (u){z n(t, u {j} ) Z n(s, u {j} )}, Z n(s, u) = 1 ns {1(U i u) C(u)} (s, u) [0, 1] d+1. n i=1 Z n Z C in l ([0, 1] d+1 ), Z C the C-Kiefer-Müller process and C n C C in l ( [0, 1]) j=1 To resample C n, we construct a resampling of Z n. 23 / 43

24 Computation of approximate p-values A resampling scheme for the sequential empirical copula process A resampling scheme for Z n, case of i.i.d. observations i.i.d. multipliers [van der Vaart and Wellner(2000)] A sequence of i.i.d multipliers (ξ i ) i Z satisfy the following conditions : For all i Z, ξ i are independent of observations X 1,..., X n E(ξ 0) = 0, var(ξ 0) = 1 and 0 {P( ξ 0 > x)} 1/2 dx <. For m = 1,..., M consider the processes Z (m) n (s, u) = 1 ns n i=1 [Holmes, Kojadinovic et Quessy(2013)] ( ) ( Z n, Z (1) n,..., Z (M) n ξ (m) i {1(U i u) C(u)}, (s, u) [0, 1] d+1. Z C, Z (1) C,..., Z(M) C in {l ([0, 1] d+1 )} M+1, where Z C is the C-Kiefer-Müller process and the processes Z (1) C,..., Z(M) C are independent copies of Z C. ) 24 / 43

25 Computation of approximate p-values A resampling scheme for the sequential empirical copula process A resampling scheme for C n, case of serially dependent observations I dependent multipliers [Bühlmann(1993)] A sequence (ξ i,n ) i Z of dependent multipliers satisfy the following conditions : (ξ i,n ) i Z is strictly stationary and for all i Z, ξ i,n are independent of observations X 1,..., X n ; E(ξ 0,n) = 0, var(ξ 0,n) = 1 and sup n 1 E ( ξ 0,n ν ) < for any ν 1 ; 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-dependant ; i.e., such that ξ i,n is independent of ξ i+h,n for all h > l n and all i N ; There exists a function ϕ : R [0, 1], symmetric around 0, continuous at 0 satisfying ϕ(0) = 1 and ϕ(x) = 0 for all x > 1, such that E(ξ 0,nξ h,n ) = ϕ(h/l n) for all h Z. 25 / 43

26 Computation of approximate p-values A resampling scheme for the sequential empirical copula process A resampling scheme for C n, case of serially dependent observations II Using dependent multipliers ξ i,n well-chosen, we can construct a resampling for Z n (or B n) adapted to the case of dependent data. B (m) n (s, u) = 1 n nt i= ns +1 ξ (m) i,n {1(U i u) C(u)}, (s, u) [0, 1] d+1. For m = 1,..., M, consider the estimated processes : ˆB (m) n (s, t, u) = 1 n nt i= ns +1 ξ (m) 1:n i,n {1(Ûi u) C 1:n(u)}, (s, t, u) [0, 1] d, We consider an estimator C j,1:n of C j constructed with finite differencing of the empirical copula at a bandwidth of h n : C j 1:n (u) = C1:n(u + hne j) C 1:n(u h nej ) min(u j + h n, 1) max(u j h n, 0), u [0, 1]d, e j = (0,..., 0, 1, 0,..., 0) and h n = n 1/2. 26 / 43

27 Computation of approximate p-values A resampling scheme for the sequential empirical copula process A resampling scheme for C n, case of serially dependent observations III For the estimated processes Ĉ (m) n d (s, t, u) = ˆB (m) n (s, t, u) C j,1:n (u) ˆB (m) n (s, t, u {j} ), (s, t, u) [0, 1] d, we have the following result : j=1 [Bücher, Kojadinovic, Rohmer et Segers(2014)] Let X 1,..., X n be drawn from a strictly stationary sequence (X i ) i Z. Under mixing conditions : ( ) C n, Ĉ (1) n ) (,..., Ĉ (M) n C C, C (1) C,..., C(M) C in {l ( [0, 1] d )} M+1, where C (1) C,..., C(M) C are independent copies of C C., 27 / 43

28 Computation of approximate p-values A resampling scheme for the sequential empirical copula process A resampling scheme for C n, case of serially dependent observations IV For (s, t, u) [0, 1] d, consider the processes and ˇB (m) n (s, t, u) = 1 n Č (m) n nt i= ns +1 (s, t, u) = ˇB (m) (s, t, u) n ξ (m) ns +1: nt i,n {1(Ûi u) C ns +1: nt (u)}. d C j, ns +1: nt (u) ˇB (m) n (s, t, u {j} ). j=1 [Bücher, Kojadinovic, Rohmer et Segers(2014)] Let X 1,..., X n be drawn from a strictly stationary sequence (X i ) i Z. Under mixing conditions : ( ) C n, Č (1) n in {l ( [0, 1] d )} M+1. ) (,..., Č (M) n C C, C (1) C,..., C(M) C, 28 / 43

29 Computation of approximate p-values A resampling scheme for the sequential empirical copula process A resampling for S n The process D n can be rewritten as : D n(s, u) = λ n(s, 1)C n(0, s, u) λ n(0, s)c n(s, 1, u). Two possibilities to resample D n : ˆD (m) n Ď (m) n (s, u) = λ n(s, 1)Ĉ (m) n (0, s, u) λ n(0, s)ĉ (m) n (s, 1, u), (s, u) = λ n(s, 1)Č (m) n (0, s, u) λ n(0, s)č (m) n (s, 1, u).... and for S n : n {ˆD (m) n (k/n, Ûi 1:n )} 2 /n, i=1 n {Ď (m) n (k/n, Ûi 1:n )} 2 /n. k {1,...,n 1} i=1 Ŝ n (m) = max k {1,...,n 1} Š (m) n = max 29 / 43

30 Computation of approximate p-values A resampling scheme for the sequential empirical copula process Asymptotic validity of the resampling scheme [Bücher, Kojadinovic, Rohmer et Segers(2014)] Under H 0 and with the same mixing conditions and ξ i,n well-chosen, (S n, Ŝ n (1),..., Ŝ (M) (S n, Š n (1),..., Š (M) n ) (S C, S (1) C n ) (S C, S (1) in R M+1, S (1) C,..., S (M) C independent copies of S C.,..., Ŝ n (M) and Š n (1) independent copies of S n. Ŝ (1) n,..., Š (M) n We compute two approximate p-values for S n as ˆp M,n = 1 M M m=1 ) (m) 1 (Ŝ n S n C,..., S (M) C ),..., S (M) C ) can be interpreted as M almost ou encore ˇp M,n = 1 M M m=1 ) (m) 1 (Š n S n 30 / 43

31 Computation of approximate p-values A resampling scheme for the sequential empirical copula process The pros and cons + : Most powerful test than predecessors for detect a change in copula + : Consistent tests + : Adapted in the case of strong mixing observations - : Less sensitive to detect a change in marginal distribution - : Without the hypothesis than the margins are constant ; we can t conclude in favour of a change in copula. 31 / 43

32 Computation of approximate p-values A resampling scheme for the sequential empirical copula process Break detection in the copula when there exists a change in marginal distribution at time m Consider the following null hypothesis H m 0 = H 1,m H 0,c : H 0,c : C, such that X 1,..., X n have copula C H 1,m : F 1,... F d and F 1,... F d such that X 1,..., X m have m.c.d.f. F 1,... F d and X m+1,..., X n have m.c.d.f. F 1,... F d. Taking the change at time m, an estimator of copula C can be built with the following pseudo-observations : { (F 1:m,1(X i1 ),..., F 1:m,d (X id )) i {1,..., m} Û 1:n i,m = (F m+1:n,1(x i1 ),..., F m+1:n,d (X id )) i {m + 1,..., n}, where in this case j = 1,..., d,f 1:m,j (resp. F m+1:n,j ) is the empirical cumulative distribution function computed with X 1j,..., X mj (resp. X m+1j,..., X nj ) 32 / 43

33 Computation of approximate p-values A resampling scheme for the sequential empirical copula process In the same way, we construct C 1:k,m and C k+1:n,m from sub sample X 1,..., X k and X k+1,..., X n for k in {1,..., n 1}. Figure: Case of m k : X 1 = (X 11,, X 1d) (R 1:m 11,, R 1:m 1d ) (R 1:m m1,, R 1:m md ) C 1:m X m C 1:k,m (R m+1:k m+11,, Rm+1:k m+1d ) (R m+1:k k1,, R m+1:k kd ) C m+1:k X k (R k+1:n k+11,, Rk+1:n k+1d ) X n = (X n1,, X nd) (R k+1:n n1,, R k+1:n nd ) C k+1:n C k+1:n,m 33 / 43

34 Computation of approximate p-values A resampling scheme for the sequential empirical copula process The process The empirical c.d.f. C 1:k,m and C k+1:n,m evaluate from pseudo-observations Û1,m, 1:k..., Ûk,m 1:k and Û k+1:n k+1,m,..., Û n,m k+1:n are given by { m k m C1:m(u) + C C 1:k,m (u) = k k m+1:k (u) m [1, k] C 1:k (u) m / [1, k], and C k+1:n,m (u) = The test statistic is where { m k+1 C n k+1 k+1:m(u) + n m Cm+1:n(u) m [k + 1, n] n k+1 C k+1:n (u) m / [k + 1, n]. S n,m = D n,m(k/n, u) = 1 max k=1,...,n n n i=1 D 2 n,m(k/n, Û 1:n i,m) k(n k) n 3/2 {C 1:k,m (u) C k+1:n,m (u)}. 34 / 43

35 Computation of approximate p-values A resampling scheme for the sequential empirical copula process Asymptotic behaviour of S n,m under H 0 Theorem Let X 1,..., X n n independent (or strong mixing) random vectors, with copula C such that for a fixed integer m, X 1,..., X m have marginal c.d.f. F 1,..., F d and X m+1,..., X n have marginal c.d.f. F 1,..., F d. Then with the similar conditions S n,m S C = sup D 2 C (s, u)dc(u). s [0,1] [0,1] d The asymptotic distribution does not depend on m! This result can be generalized at the case of multiple changes in marginal distributions. 35 / 43

36 Monte Carlo Simulations Summary 1 Introduction 2 Measure of the multivariate dependence The Sklar s theorem The role of copulas to test for breaks detection 3 A Cramér-von Mises test statistic Empirical copula The test statistic 4 Computation of approximate p-values The sequential empirical copula process A resampling scheme for the sequential empirical copula process 5 Monte Carlo Simulations 36 / 43

37 Monte Carlo Simulations Cramér-von Mises statistic, case of i.i.d. observations Consider the alternative H 1,c : H 1,c H 1,c : distinct C 1 and C 2, and t (0, 1) such that X 1,..., X nt have copula C 1 and X nt +1,..., X n have copula C 2. Percentages of rejection of hypothesis H 0 computed using 1000 samples of size n {50, 100, 200} generated under : H 0 = H 0,c H 0,m, H 1,c H 0,m With a change in marginal c.d.f. : H m 0 = H 0,c H 1,m H 1,c H 1,m 37 / 43

38 Monte Carlo Simulations Nasdaq, Dow Jones and the black Monday ( ) Suppose there is an unique change in m.c.d.f. at time m = 202 ( ) ˆp Sn val = : no evidence against H 0,c. 38 / 43

39 Monte Carlo Simulations Bibliography I A. Bücher et I. Kojadinovic. A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing. arxiv : , A. Bücher, I. Kojadinovic, T. Rohmer et J. Segers. Detecting changes in cross-sectional dependence in multivariate time series. Journal of Multivariate Analysis, 132 : , M. Csörgő et L. Horváth. Limit theorems in change-point analysis. Wiley Series in Probability and Statistics. John Wiley & Sons, Chichester, UK, P. Bühlmann. The blockwise bootstrap in time series and empirical processes. PhD thesis, ETH Zürich, Diss. ETH No / 43

40 Monte Carlo Simulations Bibliography II P. Deheuvels. La fonction de dépendance empirique et ses propriétés : un test non paramétrique d indépendance. Acad. Roy. Belg. Bull. Cl. Sci. 5th Ser., 65 : , M. Holmes, I. Kojadinovic et J-F. Quessy. Nonparametric tests for change-point detection à la Gombay and Horváth. Journal of Multivariate Analysis, 115 :16 32, I. Kojadinovic, J.-F. Quessy et T. Rohmer. Testing the constancy of Spearman s rho in multivariate time series. ArXiv e-prints, July T. Rohmer. Some results on change-point detection in cross-sectional dependence of multivariate data with changes in marginal distributions. ArXiv : , July / 43

41 Monte Carlo Simulations Bibliography III Murray Rosenblatt. A central limit theorem and a strong mixing condition. Proceedings of the National Academy of Sciences of the United States of America, 42(1) :43, L. Rüschendorf. Asymptotic distributions of multivariate rank order statistics. Annals of Statistics, 4 : , J. Segers. Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions. Bernoulli, 18 : , A. Sklar. Fonctions de répartition à n dimensions et leurs marges. Publications de l Institut de Statistique de l Université de Paris, 8 : , / 43

42 Monte Carlo Simulations Bibliography IV D. Wied, H. Dehling, M. van Kampen et D. Vogel. A fluctuation test for constant Spearman s rho with nuisance-free limit distribution. Computational Statistics and Data Analysis, 76 : , A. Bücher and M. Ruppert. Consistent testing for a constant copula under strong mixing based on the tapered block multiplier technique. Journal of Multivariate Analysis, 116 : , A.W. van der Vaart and J.A. Wellner. Weak convergence and empirical processes. Springer, New York, Second edition. 42 / 43

43 Monte Carlo Simulations Thank you for your attention! 43 / 43