关于参数型copula函数的拟合检验-

1、J.Sys.Sci.&Math.Scis.27(1(2007,2,93101copula(200062(200062;999077copulacopula copulacopulacopulaMR(200062G101copulacopula(the Curse of Dimensionality.14 Sklars(copula copulacopula5,6.(7,8.tcopula*2006-10-05.9427copulaH 0:C (;0C =C (;:,(1Ccopula9Archimedean copula10Kendalls taucopula11(the Probality

2、Integral Transform,12Anderson Darling type13“”;14copula15copula=0T n =nC n (u,v C (u,v ;0S n =T 2n (u,v d u d vC n (u,v 2copula 16(P.389T n17T nS nS n =T 2n (u,v d u d v121nn312342copula(X,Y H (x,y ,F (x =H (x,G (y =H (,y .SklarcopulaH (x,y =C (F (x ,G (y .(2F (u =inf x R |F (x u G (v =inf y R |G (y

3、 v (0u,v 1F (G (R(,+.(2C (u,v =H F (u ,G (v ,0u,v 1.(31copula95(x 1,y 1,(x 2,y 2,(x n ,y n (X,Y nH n (x,y =1nn i =1I x i x,y i y ,F n (x =H n (x,+G n (y =H n (+,y .copulaC n (u,v =H n (F n (u ,G n (v ,F n (u =inf s R |F n (s u G n (v =inf t R |G n (t v F n (x G n (y copula18,19,C n (u,v =1n ni =1I F

4、 n (x i u,G n (y i v .(4nF n (x i x ix 1,x 2,x nnG n (y i y iy 1,y 2,y nC n (17sup0u,v 1 C n (u,v C n(u,v 2n .(5(1S n =T 2n (u,v d u d v,T n =n (C n (u,v C (u,v ;.T nS ncopulaT nS nAA ,u vuvU C (u,v =B C (u,v C 1(u,v B C (u,1C 2(u,v B C (1,v ,B CE B C (u,v B C (u ,v=C (u u ,v v C (u,v C (u ,v .C 1(u

5、,v C 2(u,v (A2(A1f (g (A2copulaC (u,v ;C 1(u,v =C (u,v ;uC 2(u,v =C (u,v ;v,c (u,v ;(A3R q=1n ni =1L (F (X i ,G (Y i ;+o p (1,L (F (X ,G (Y ;q=E (L (F (X ,G (Y ;L T (F (X ,G (Y ;.(A1(A2(A313(A3:=1n A (1n i =1ln c (F (x i ,G (x i ;+O p ln(ln n n,A (=lim n E2Q n (Tq qQ n (=1nn i =1ln c (F n (x i ,G n

6、(x i ;.96272.1(A1(A3(1copulaT n(0,12G C =U C CT V,V.S nS =(G C (u,v 2d u d v .G CE (G C (u,v G C (u ,v =E (U C (u,v U C (u ,v +C (u,v T C (u ,v C (u,v T E (U C(u ,v V C (u ,vTE (U C (u,v V ,E (U C (u,v V =0,u 0,v L (s,t ;d C (s,t ;+C 1(u,v 0,u 0,1L (s,v ;d C (s,v ;+C 2(u,v 0,10,v L (u,t ;d C (u,t ;.

7、2.1,(i.i.dT nT n =n (C n (u,v C (u,v ;=n (C (u,v ;C (u,v +n (C n (u,v C (u,v =J 1+J 2,C (u,v copula2.1J 1=1n C (u,v ;Tni =1LF (X i ,G (Y i ; +o P (1.(6J 2.copula(3(4,n (C n (u,v C (u,v =1n n i =1I F n (X i u,G n (Y i v I F (X i u,G (Y i v +I F (X i u,G (Y i v C (u,v =1n ni =1I F n (X i u,G n (Y i v

8、I F (X i u,G (Y i v (H (F n (u ,G n (v H (F (u ,G (v +1n ni =1(I F (X i u,G (Y i v C (u,v +n (H (F n (u ,G n (v H (F (u ,G (v =V 1+V 2+V 3.n (H n (H (5V 1=o P (1.(71copula97V 2i.i.dV 3.n (H (F n (u ,G n (v H (F (u ,G (v =n (C (F (F n (u ,G (G n (v C (u,v =nC 1(u,v (F (F n (u u +nC 2(u,v (G (G n (v v

9、 +o p (1=nC 1(u,v (u F n (F (u +nC 2(u,v (v G n (G (v +o p (1=C 1(u,v 1n n i =1(I F n (X i u u C 2(u,v 1n ni =1(I G n (Y i v v +o p (1.(8(6(820(P.157VII.21,2.1.copulaC (;W (;,W (1,0;=0=W (0,1;,W (1,u ;=0=W (v,1;.C (=C (;+W (;n ,0.(9nC (C (u 2,v 2C (u 2,v 1C (u 1,v 2+C (u 1,v 1(u 1u 2,v 2v 1C (copula

10、012=121n2.2(A1(A3(9012T n;=12,T n(0,12G C +W ,S n(G C (u,v +W (u,v ;2d u d v .2.1(9T n =n C n (u,v C (u,v ; W (u,v ; n +W (u,v ;n 12=J 3+J 4.J 3(0,12G C ,012=12J 4W (.copulaH 0S n2.1321S n222498 27 23 Rademacher Rademacher 1 1 , 2, , n i.i.d Rademacher (x1 , y1 , (x2 , y2 , , (xn , yn 2.1 Tn Tn (n ,

11、 u, v 1 = n n i (IFn (Xi u,Gn (Yi v i=1 n i=1 n i=1 Cn (u, v n i (IGn (Yi v i=1 1 C1n (u, v n C(u, v 1 T n 1 i (IFn (Xi u u C2n (u, v n i L(F (Xi , G(Yi ; |= , v 0 u, v 1. n =( 1 , 2 , , n T , C1n C2n C1 C2 copula H(F (u,G (v H(F (u,G (v (3, C1 (u, v = x C2 (u, v = y . f (F (v g(G (u H (F (u,G (v C1

12、n C2n C1n (u, v = x f (F (v H (F (u,G (v C2n (u, v = y . Hn (, fn ( gn ( H( g (G (u f ( g( ( 25. n n n n n n n n n n Sn (n = 2 Tn (n , u, vdudv. Step 1 Step 2 (i Sn (n , m . Rademacher (i = ( (i , n 1 ppn = k (m+1 , (i (i 2 , , n , i = 1, 2, , m. i = 1, 2, , m. Step 3 Sn . (0 0 k = #Sn (i Sn , i = 0

13、, 1, , m(Sn = n 4 13, C (u, v, = uv copula (expu 1(expv 1 1 ln 1 + , exp 1 = 0, 0, 1. (10 copula copula Franks copula copula ( = 0, 6 13. , (A3. C1 C2 (Gaussian product kernel, 1 ( n copula 9 100. 99 5%, 3 (rejecting rate, 1 n = 100 m 1000, 1000 p 1 p 0.05 5 10 0 15 20 25 5 10 (S 0.00 0.00 0.00 0.00

14、 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.03 0.17 0.31 0.03 0.13 0.18 0.57 0.84 0.07 0.19 0.58 0.89 0.95 0.02 0.03 0.06 0.02 0.37 (1000 . (Sn 0.02 0.00 0.00 0.00 0.04 0.04 0.05 0.07 0.23 0.38 0.05 0.10 0.30 0.50 0.66 0.10 0.25 0.57 0.80 0.91 0.11 0.34 0.63 0.84 0.96 0.11 0.13 0.14 0.16 0.13 (T 0.02

15、 0.00 0.01 0.01 0.08 0.00 0.00 0.07 0.22 0.60 0.01 0.05 0.36 0.80 0.95 0.03 0.21 0.67 0.95 1.00 0.12 0.33 0.71 0.98 1.00 0.01 0.00 0.00 0.02 0.03 0.1 15 20 25 5 10 0.2 15 20 25 5 10 0.3 15 20 25 5 10 0.5 15 20 25 5 10 0.9 15 20 25 1 = 0, 13 = 0.5 S T, 100 27 = 25 = 0.5 1 X Y S T; X Y T Sn X Y copula

16、 (10 copula uv. copula (A3, Franks copula 13 S T Sn X Y 13 copula ( = 0.1 ,Sn 1 Bouy E, Surrleman V, Nikeghmali A, Riboulet G and RoncalliO T. Copulas for nance: A reading e guide and some applications. Paris: Groupe de Recherche Operationnelle, Credit Lyonnais, 2000. 2 Denuit M, Purcaru O and Van K

17、eilegom I. Bivariate Archimedean copulamodelling for censored data in non-life insurance. Journal of Actuarial Practice, 2006 (13: 532. 3 Embrechts P, McNeil A and Straumann D. Correlation and Dependence Risk Management: Properties and Pitfalls. Risk Management: Value at Risk and Beyond, ed. M. A. H

18、. Dempster, Cambridge University Press, 2002, 176223. 4 Frees E W and Valdezk E A. Understanding relationships using copulas. North American Actuarial Journal, 1998, (2: 125. 5 Joe H. Multivariate and dependence concepts. Chapman & Hall, London, 1997. 6 Nelsen R B. An Introduction to Copulas. Spring

19、er, New York, 1999. 7 11981. 8 2004. 9 Wang W, & Wells M T. Models selection and semiparametric inference for bivariate failure-time data. Journal of The American Statistical Association, 2000, 95: 6272. 10 Genest C, Quessy J-F and Rmillard B. Goodness-of-t procedures for copula models based on the

20、e probability integral transformation. Scandinavian Journal of Statistics, 2006, 33: 337366. 11 Rosenblatt M. Remarks on a multivariate transformation. Annals of Mathematical Statistics, 1952, 23: 470472. 12 Breyman W, Dias A and Embrechts P. Dependence structures for multivariate high-frequency dat

21、a in nance. Quantitative Finance, 2003, (3: 116. 13 Fermanian J D. Goodness-of-t tests for copulas. Journal of Multivariate Analysis, 2005, 95: 119152. 14 Chen X, Fan Y and Patton A. Simple tests for models of dependence between multiple nancial time series, with applications to US equity returns an

22、d exchange rates. London Economics Financial 1 copula 101 15 16 17 18 19 20 21 22 23 24 25 Markets Group, Working paper, No. 483. Available at SSRN: 2004. Chen X, Fan Y. Pseudo-likelihood ratio tests for semiparametric multivariate copula model selection. The Canadian Journal of Statistics, 2005, 33

23、: 389414. Van der Varrt, A. & Wellner J A. Weak Convergence and Empirical Processes. Springer, New York, 1996. Fermanian J D. Radulovic D and Wegkamp M J. Weak convergence of empirical copula processes. Bernoulli, 2004, 10: 847860. Deheuvels P. La fonction de dpendance empirique et ses proprits: Un

24、test non paramtrique e ee e dindpendence. Bulletin de lAcadmie royale de Belgique, Class des dcience, 1979, 65: 274292. e e Genest C, Ghoudi K and Rivest L P. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 1995, 82: 543552. Polla

25、rd D. Convergence of Stochastic Process. Springer, New York, 1984. Zhu L X, and Neuhaus G. Nonparametric Monte Carlo tests for multivariate distributions. Biometrika, 2001, 87: 919928. Zhu L X. Model checking of dimension-reduction type for regression. Statistica Sinica, 2003, 13: 283296. Zhu L X. N

26、onparametric Monte Carlo Cests and Applications. Springer, New York, 2005. Zhu L X, Fujikoshi Y and Naito K. Heteroscedasticity checks for regression models. Science in China, 2001, 10: 12361252. Fermanian J D and Scaillet, O. Nonparametric estimation of copulas for time series. Journal of Risk, 2003, (5: 25-54. CHECKING THE ADEQUACY OF COPULAS WITH PARAMETRIC STRUCTURE Wu Ping Yu Zhou Zhu Liping (Department of Statistics, East China Normal Universi