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1、IIIKKK ?3K ?O Copula 8? z ? 11 / ?36 - , ? | 4 - 2013qO?- uuu ?xxx”OOO ?“?n? 2013c5?10F ? IIIKKK ?3K ?O Copula 8? z ? 12 / ?36 - , ? | 4 Outline ?3K a ?O u ?Copula 8? IK ?333KKK ?O Copula 8? z ? 13 / ?36 - , ? | 4 1?333KKK ? 1.37K?x+3? 2.A? ?!?5 !? 3.?A? ?( kJ9 ? ? 1.AzzaliniCapitanio?J?skew-normal(
2、SN), skew-t(ST); 2.2?/CqJscale mixture skew normal(SMSN)? 3.MaGenton?J?fl exible skew symmetric distri- bution. IK ?333KKK ?O Copula 8? z ? 14 / ?36 - , ? | 4 ? 1.?a#? ?A5?O ? 2.u ?copula?E,? ? ?5(?: 3.?3/”xN? ?A IK ?3K ?OOO Copula 8? z ? 15 / ?36 - , ? | 4 2aaa ?OOO ? ? 1 Skew normal(SN)distributio
3、n: p(x,2,) = 2 (x ) ( x ) ()()OIO? PX SN(,2,) 2 Skew student t(ST)distribution: p(x,2,) = 2t(x,2;)T ( x ( +1 +Q )1/2 ; ) t(;)T(;)Ogd?t? Q = (x)2/2aqSTN IK ?3K ?OOO Copula 8? z ? 16 / ?36 - , ? | 4 3 Scale mixture skew normal(SMSN)distribution: CU?L X = +1/2(U)Z Z SN(0,2,),()?Uk H(;)?C PX SMSN(,2,H)
4、4 Flexible skew-symmetric distribution: p(x;,) = 21f0(z)GPK(z) f0()G()Ovf0(x) = f0(x),G(x) = 1 G(x) ?PK(x) = 1x+3x3+2K1x2K1?K? g, z = (x)/ ?f0GO?tFlexible skew-t-normal distributionPFSTN IK ?3K ?OOO Copula 8? z ? 17 / ?36 - , ? | 4 Aa ?3A?x? SMSNaFSSa SNSTSTNFSNFSTN ? 1!?STFSTN? S?5 ?X?Sm?k X lcopul
5、a; K 5?u? ?pd? du?37K?x? 5 IK ?3K ?O Copula 8? z ? 115 / ?36 - , ? | 4 Copula Copula?eCUj, j = 1,.,rl m0,1?!?u = (u1,.,ur)K? C(u1,.,ur) = Pr(U1 u1,.,Ur ur) copula Sklarn?)?S (?(copula)AO/XJS Y?A?copula? IK ?3K ?O Copula 8? z ? 116 / ?36 - , ? | 4 Copula b?X = (X1,.,Xr)k Y?F(x;)zS O?Fj(xj;), j = 1,.,
6、rKL C(u;) = F(F1 1 (u1;),.,F1 r (ur;);) copula Copula XJX?f(x;) Sfj(xj;), j = 1,.,rocopula 30,1r?copula c(u;) = f(x;) rj=1fj(xj;) xj= F1 j (uj;) IK ?3K ?O Copula 8? z ? 117 / ?36 - , ? | 4 ?skew-tCopula ?skew-t b?XQlXe?t ( X Q ) t2r ( 0 0 ) , = ( +D2D DI ) , ) D = diag(1,.,r)?(r r)? ? e?skew-t fST(x
7、;,D,)= 2r det(+D2)1/2 ft ( (+D2)1/2x; ) Pr(V 0;x) ft(x;)?t? V tr(D(+D2)1x, x(+D2)1x+ r+ (ID(+D2)1D),r+) IK ?3K ?O Copula 8? z ? 118 / ?36 - , ? | 4 ?skew-t copula b ? ? ?skew-t ? FST(x;)K? ?t?copula CST(u;) = FST(x;) cST(u;) = fST(x;) rj=1fST,j(xj;j,) IK ?3K ?O Copula 8? z ? 119 / ?36 - , ? | 4 SSSf
8、ff?/x|?N? N?8? ?)?x| ?)xi3?A xi!x?J/?!%n9? ?)n?%nd CAc?n?g A?n? ?U?O N?I?NCD?. 8O?Gr?yx? J IK ?3K ?O Copula 8? z ? 120 / ?36 - , ? | 4 IK ?3K ?O Copula 8? z ? 121 / ?36 - , ? | 4 ? k?/6IU3:5:2? ?p?!?!.|+? +? g3N?g N?5?!?g?5? ? +N?Xe +?p?|? +?|? +n?.|? IK ?3K ?O Copula 8? z ? 122 / ?36 - , ? | 4 1+6?
9、2U? |?1N? 2+?3/y/? 2?|?1N? 3du.|?lz| L?|T|? ?S,? | ?t?copula?X ZIP?. Pr(Y = yp,) = p+(1 p)e, y = 0 (1 p) y y!e , y = 1,2,. b?Y?GgClog() = ,logit(p) = ?. IK ?3K ?O Copula 8? z ? 129 / ?36 - , ? | 4 l du?GglC?|copula 3(J d d C X ) k b ?Hj(yij;j)ZIP?.?1ji?S yijL1ji?1i?Ggg b?lskew-t?dCxijv Hj(y ij;j) FS
10、T,j(xij;j,) S?O ?xiE(Yi)Pr(Yi= 0) yi#(yi= 0) ?1.05445.53070.13780.8040.12220.796 S1.06942.22550.15350.7430.13870.735 ?2.33993.95910.18380.7330.17130.735 Scopula 2Nx?Gg?S u ?t?copula 3c?ximJ=? copula IK ?3K ?O Copula 888? z ? 132 / ?36 - , ? | 4 4888? k?C?8 |Copula5?Bayesian u ?8?.?O?K u ?8?.?CJ IK ?
11、3K ?O Copula 8? zzz ? 133 / ?36 - , ? | 4 5zzz 1. Azzalini, A., Genton, M.G., 2008. Robust likelihood methods based on the skew-t and related distributions. Inter. Stat. Revi. 76, 106-129. 2. Azzalini, A., 1985. A class of distributions which includes the normal ones. Scand. J. Stat. 12, 171-178. 3.
12、 Azzalini, A., Capitaino, A.,2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t -distribution. J. R. Stat. Soc. B. 65, 367-389. 4. G omez, H.W., Venegas, O., Bolfarine, H.,2007. Skew-symmetric distribu- tions generated by the distribution function of the
13、 normal distribution. Envi- ronmetrics 18, 395-407. 5. Azzalini, A., Dalla-Valle, A., 1996. The multivariate skew-normal distri- bution. Biometrika, 83, 715-726. 6. Azzalini, A., Capitanio, A., 1999. Statistical applications of the multivari- ate skew normal distributions. J. R. Stat. Soc. B. 61, 57
14、9-602. 7. Ma, Y., Genton, M.G., 2004. A fl exible class of skew-symmetric distribu- tions. Scand. J. Stat. 31, 459-468. 8. Zeller, C.B., Lachos, V.H., Vilca-Labra, F.E., 2011. Local infl uence analysis for regression models with scale mixture of skew-normal distributions. J. Appl. Stat. 38, 343-368.
15、 IK ?3K ?O Copula 8? zzz ? 134 / ?36 - , ? | 4 zzz(YYY) 9. Cancho, V., Dey, D.K., Lachos, V.H., Andrade, M., 2011. Bayesian non- linear regression models with scale mixutures of skew-normal distributions: estimation and case infl uence diagnostics. Comput. Stat. Data Anal. 55, 588- 602. 10. Xie, F., Wei, B., Lin, J., 2009. Homogeneity diagnostics for skew-normal nonlinear regression models. Stati. Proba. Lett. 79, 821-827. 11. Vanegas, L., Cysnei
