46 2. Coula Coula Coula [7], Coula. Coula C(u, v) = φ [ ] {φ(u) + φ(v)}, u, v [, ]. (2.) φ( ) (generator), : [, ], ; φ() = ;, φ ( ). φ [ ] ( ) φ( ) []

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1 2 Chinese Journal of Alied Probability and Statistics Vol.26 No.5 Oct. 2 Coula,2 (,, 372; 2,, 342) Coula Coula,, Coula,. Coula, Coula. : Coula, Coula,,. : F83.7..,., Coula,,. Coula Sklar [],,, Coula., Coula. Coula,, [2 5].,,. Coula. Coula, Coula, Coula,, Frank Gumbel Clayton Coula ( []). Coula,., Coula,. [4, 6] GARCH-t, Coula Coula, Coula,. Coula., Coula,. (757377)

2 46 2. Coula Coula Coula [7], Coula. Coula C(u, v) = φ [ ] {φ(u) + φ(v)}, u, v [, ]. (2.) φ( ) (generator), : [, ], ; φ() = ;, φ ( ). φ [ ] ( ) φ( ) [], Coula. φ( ) Coula C(u, v), [8] C(U, V ) K() = P{C(U, V ) } = φ() φ, [, ]. (), Coula, Kendall [8] φ() τ = 4E{C(U, V )} = + 4 φ () d. ρ,, ρ. Coula, Coula τ,, (concordant) []., ρ, τ., Coula, τ, Coula ( ). Gumbel Frank Clayton Coula Coula ( ( θ } /θ ) Coula), Kendall (θ )/θ 4 { θ t/(e t )dt θ/(θ + 2), θ Coula. Coula C θ (u, v) φ() θ Gumbel ex{ [( log u) θ + ( log v) θ ] /θ } ( log ) θ [, ] Frank { θ log + (e θu )(e θv ) } e θ log e θ e θ (, + )/{} Clayton max((u θ + v θ ) /θ, ) ( θ )/θ [, )/{} Coula, t,, t n n, < t < < t n <, t =, t n+ >, Coula φ(), S() = log(φ()), Q() = n {θ,i + θ i S()}I(t i < t i+ ), (2.2) i=

3 : Coula 46 I( )., ex( Q) Coula : i =,, n, θ,i = i k= S(t k )(θ k θ k ); θ n θ. θ,, ex( Q) Coula, θ, =. : ex( Q),, ex( S). Q() Q (x) = n i= S ( x θ ),i I{Q(t i ) x < Q(t i+ )}, θ i S, Q, (2.), ex( Q()) Coula C(u, v) = n i= Coula ( Coula), [ x = log ex S ( x θ ),i I{Q(t i ) x < Q(t i+ )}, (2.3) θ i { n i= } (θ,i + θ i S(u))I(t i u < t i+ ) { + ex n }] (θ,i + θ i S(v))I(t i v < t i+ ). i= φ (2.3) Coula. n, S() m, Coula m + n +. Coula S(). S(), Coula ;, (θ,, θ n ) Coula. φ(), S(), Coula C(u, u) S (S(u) log 2/θ ) λ L = lim = lim, u + u u + u (2.4) 2u + C(u, u) 2u + S (S(u) log 2/θ n ) λ U = lim = lim. u u u u (2.5) Coula S() θ ( θ n ),, Coula, Coula,., φ() = log, Coula C(u, v) = uv,, ex{ Q()} = n [e θ,i ( log ) θ i ] I(t i <t i+ ), (2.6) i= Coula. θ i = θ, i =,, n, ex( Q()) = ( log ) θ, Gumbel Coula.

4 (X,k, X 2,k ), k =,, m, V k = m m I(X,j X,k, X 2,j X 2,k ) k =,, m. (3.) j= K m () C(U, V ), [, ] K m () = m m I(V k ). k= Coula (2.3), C(U, V ), (2.6), K() = + S () n i= K() = log() n i= θ i I(t i < t i+ ), θ i I(t i < t i+ ). β = (β = /θ,, β n = /θ n ), Y = {K m (V k ) V k, j =,, m} X = {/S (V k )I(t i V k < t i+ ), i =,, n, k =,, m}. A, β β n A T β. : Y = Xβ + ɛ, β β = arg min β (Y Xβ) T (Y Xβ), A T β. (3.2),, β = arg min β {(Y Xβ) T (Y Xβ) + λβ T T r r β}, A T β. (3.3) β = (β i β i, i =,, n) t, r β = ( r β) r. λ. r =. r = 2,, λ +, β i., 2. (3.3)., (bootstra). R, [9] :

5 : Coula 463. {(X,k,X 2,k ), k =,, m} (X,,X 2, ),, (X,m,X 2,m ); 2. (3.) V,, V m; 3. λ, V k, k =,, m (3.3), β r. R, β r, r =,, R β. K() K m (), Coula. Coula K() K m (). K() K m (), Coula., n (K( i ) K m ( i )) 2, < i <,, Coula i=. χ 2 Coula, Coula. {u t } {v t }, t =,, m, [, ]. k < a < < a k <, a =, a k =, [, ] [, ] k k, i j A(i, j), i, j =,, k. (u t, v t ), a i u t < a i a j v t < a j, (u t, v t ) A(i, j), a i, i =,, k,,. E ij A(i, j), T ij Coula A(i, j), Coula N = k k i= j= (E ij T ij ) 2 T ij, (3.4) N (k ) 2 χ 2.,,, q, (k ) 2 (q ). 4. / / / / / / /, EUR CAD AUD JPY CHF DKK GBP. {P t }, {R t } R t = log P t log P t , 479. (CHF, EUR) (DKK, EUR) (AUD, GBP) (CAD, JPY) Kendall τ , CHF EUR, DKK EUR, CAD JPY. [, ],, (CHF, EUR),, ; (DKK, EUR),, (, ) (, ), ; (AUD, GBP), (, ) (, ),

6 464 (, ) (, ) ; (CAD, JPY), (AUD, GBP). τ,., Coula. EUR GBP CHF AUD EUR JPY DKK CAD n = t < < t, t =, t =.. S() = log( log()) Q(), ex( Q()) Coula., r = 2, λ =, β. R = 5, β. τ Coula, 2. 2 β (CHF, EUR) (DKK, EUR) (AUD, GBP) (CAD, JPY) θ θ θ θ Gumbel Frank Clayton

7 : Coula 465 (CHF EUR) (DKK EUR) emirical semiarametric model 95% confidence bands (AUD GBP) emirical semiarametric model 95% confidence bands emirical gumbel model 2 frank model clayton model (CHF EUR) (AUD GBP) k() k()..5 emirical semiarametric model 95% confidence bands (CAD JPY) emirical semiarametric model 95% confidence bands Coula K() emirical gumbel model frank model clayton model k() k().5..5 (DKK EUR) emirical gumbel model frank model clayton model (CAD JPY) emirical gumbel model frank model clayton model Coula K()

8 Coula K() K m (). 2 β 95% Coula K(). Coula 4,., 4, K() K m (), K() K m (). (CAD, JPY), K() K m (). (CAD, JPY), K() K m (), 2,, (CAD, JPY)., 3 Coula β, 4, 7.48%.%, (DKK, EUR), K() 95% K m (), (DKK, EUR), 95% K m ()., 4, K() K m (),,. Clayton K() K m (), K m (), (CHF, EUR) (DKK, EUR) ; Gumbel K(), K m (), K m (), (CHF, EUR) (DKK, EUR) ; Frank K(), (CAD, JPY), K m (),., 2, Coula Coula., 4, K() K m (), Coula Coula,. 3 β (CHF, EUR) (DKK, EUR) (AUD, GBP) (CAD, JPY).92%.%.7% 2.2% 4.39%.8% 3.65% 7.48%, k a i, i =,, k, k k A., (u t, v t ), t =,, 478 A(i, j) E ij, Coula A(i, j), T ij, i, j =,, k, (3.4) N. 4 N, Coula 4 : Coula 4, Coula.,.5, Coula 3, Coula Coula ;., Coula (CAD, JPY).

9 : Coula df χ 2 N (CHF, EUR) (DKK, EUR) (AUD, GBP) (CAD, JPY) df N df N df N df N Frank Gumbel Clayton : , Coula Gumbel Frank Clayton Coula, Coula, Coula. 4,, Coula,. Coula, Coula, S() Coula,. [] Nelsen, B., An Introduction to Coulas, New York: Sringer, 999, [2],, VaR,, 22(9)(24), [3],, Coula,, 24(4)(24), [4],,,,, 9(4)(24), [5],,, Coula, ( ), 3(5)(23), 97. [6] Hu, L., Essays in Econometrics with Alications in Macroeconomic and Financail Modelling, New Haven: Yale University, 22. [7] Vandenhende, F. and Lambert, P., Local deendence estimation using semiarametric Archimedean coulas, The Canadian Journal of Statistics, 33(3)(25), [8] Genest, C. and MacKay, R.J., Coules archimédiennes et familes de lois bidimensionnelles dont lesmarges sont données, The Canadian Journal of Statistics,4(2)(986), [9] Davison, A.D. and Hinkley, D.V., Bootstra Methods and Their Alication, Cambridge, England: Cambridge University Press, 999, [] Waldmann, K.H., On the exact calculation of the aggregate claims distribution in the individual life model, ASTIN Bulletin, 24(994), [] Genest, C. and Rivest, L.P., Statistical inference rocedures for bivariate Archimedean coulas, Journal of the American Statistical Association, 88(993),

10 468 [2] Embrechts, P., Lindskog, F. and McNeil, A., Modelling deendence with coulas and alication to risk management, Handbook of Heavy Tailed Distributions in Finance, ed. S. Rachev, Elsevier, 23, Chater 8: Emirical Research on the Semiarametric Archimedean Coula Shi Daoji Guo Hui,2 Luo Juneng ( Institute of Science, TianJin University, TianJin, 372 ) ( 2 The Real Estate Market Administrative Deartment of Tianjin, Tianjin, 342 ) Semiarametric Archimedean coulas, which have a fexible deendence structure because of the secial way constructed by using the existing archimedean generator, can describe the deendence structure between the financial data auto-adatively. The emirical results on the exchange rate market suggest that the semiarametric Archimedean coula is more flexible than the other three coulas, and is suggestive when selecting coulas. Keywords: Selection of coulas, semiarametric Archimedean coula, exchange rate, tail deendence. AMS Subject Classification: 62P2.

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