关于参数型copula函数的拟合检验-

1、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 Integral Transform,12

2、Anderson 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 v (0u,v 1F (G (R(,+.

3、(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 n (x i u,G n (y i v

4、.(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,v C 2(u,v (A2(A1f (g

5、 (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 (x i ;.96272.1(A1(A3(

6、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 ;.2.1,T nT n =n (C n (u

7、,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 I F (X i u,G (Y i v (H (F n

8、 (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 2V 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 +o p (1=nC 1(u,v (u F n (F (u +

9、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 (copula012=121n2.2(A1(A3(9012T n;=12,T

10、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 n22241 ( 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 0.00 0.00 0.00 0

11、.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 0.00 0.01 0.01 0

12、.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 (10 copula uv. c

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23、e 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 University, Shanghai 200062 Zhu Lixing (Department of Statistics, East China Normal University, Sha