New Classes of Multivariate Survival Functions

Transcription

1 Xiao Qin 2 Richard L. Smith 2 Ruoen Ren School of Economics and Management Beihang University Beijing, China 2 Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC, USA SAMSI workshop, May 2, 2008

2 Outline of Topics

3 Multivariate Extreme Value model Define i.i.d p-dimensional random vectors: X k = (X,k,, X p,k ), k =,, n with marginal distribution Pr(X i,k x i ) = F Xi (x i ), i =,, p. The ith component maximum: M i,n = max(x i,,, X i,n ) Choose normalising constants a n = (a,n,, a p,n ) and b n = (b,n,, b p,n ) such that, { } lim Pr Mi,n b i,n x i = G(x,, x d ) n a i,n Denoting by vectors, we get: { } lim Pr Mn b n = G(x) n a n

4 Multivariate Extreme Value model It is easily checked that all the marginal distributions of G are Generalised Extreme Value distribution (GEV), which belongs to one of the following three families (Fisher-Tippett theorem). F Xi (x i ) = G(,,, x i,,, ) Gumbel : exp { exp( x i )} = Fréchet : exp { x α } i Weibull : exp { ( x i ) α } A common practice is to choose the Fréchet marginals, and usually the unit Fréchet case when shape parameter = /α =. This is WOLG, as any marginal distributions can be changed into a unit Fréchet by a probability integral transformation. Details can be found in Smith (2003).

5 Multivariate Extreme Value model Pickands representation: For x with unit Fréchet marginals, x follows a multivariate extreme value distribution iff its joint distribution function can be represented as: G(x) = exp { V (x)} () Here, V (x) is the exponent measure as in de Haan and Resnick (977) and can be expressed in terms of an angular measure H (density h) as: V (x) = max (w i ) dh(w) (2) i p x i

6 Multivariate Extreme Value model Where, H is a positive finite measure on the (p )-simplex, p = (w,..., w p ) : w j =, w j 0, j =,, p j= satisfying the following constraint. MEV constraint: Suppose x i = u, x j =, j i, i =,, p, then, V (,,, u,,, ) = u. Thus, = w i dh(w) = V (,,,,,, ) (3) wherever the is.

7 Multivariate Extreme Value model General case: V (x) = { } max (w i ) dh (w) (4) i p x i MEV constraint (general): = wi dh (w) = V (,,,,,, ) (5) wherever the is. In this case, G (x) has Fréchet marginals with shape parameter and scale parameter.

8 Limitation of the Multivariate Extreme Value model As stated in Ledford and Tawn (996), all Bivariate Extreme Value models with Fréchet marginals have: { r lim Pr(X asymptotic dependence > r, X 2 > r) = r r 2 exact independence However, strictly speaking, there should also be negative association and positive association under asymptotic independence.

9 Ramos-Ledford-Tawn model Ledford-Tawn model: Ledford and Tawn (997) For (Z k, Z 2k ), k =,, n with unit Fréchet marginals but unknown dependence structure, the joint survival functions F Z Z 2 has the following asymptotic form: F Z Z 2 (z, z 2 ) = L(z, z 2 )z c z c 2 2 where, c + c 2 = ( is called the Coefficient of tail dependence), L(z, z 2 ) is a bivariate slowly varying function (BSV) L(tz with limiting function g, i.e., lim,tz 2 ) t L(t,t) = g(z, z 2 ) and g(cz, cz 2 ) = g(z, z 2 ), c > 0, z, z 2 > 0.

10 Ramos-Ledford-Tawn model Ramos and Ledford (2007) reformulated the Ledford-Tawn model: F Z Z 2 (z, z 2 ) = L(z, z 2 )(z z 2 ) 2 where, L(z, z 2 ) = L(z, z 2 )( z ) κ/2, κ = c 2 c z 2 Consider the limiting joint survival distribution of the conditional variables (X, X 2 ) = (Z /u, Z 2 /u) (Z > u, Z 2 > u), thus, X, X 2. Then, F X X 2 (x, x 2 ) = lim u Pr(Z >ux,z 2 >ux 2 ) Pr(Z >u,z 2 >u) L(ux,ux = lim 2 ) u L(u,u) (x x 2 ) /2 = g(x,x 2 ) = g ( x ) x +x 2 (x x 2 ) /2 (x x 2 ) /2

11 Ramos-Ledford-Tawn model Ramos-Ledford spetral model: By taking a polar-coordinate transformation, the following relationship of the limiting survivor in terms of an angular measure H is obtained: F X X 2 (x x 2 ) = 0 { min( w, w } ) x x 2 dh (w) Ramos-Ledford Constraint: Suppose x = x = u, then, F X X 2 (u, u) = u. Thus, the following normalising condition should hold: { = min( w, w } ) dh x x (w) = F X X 2 (, ) 2 0

12 Ramos-Ledford-Tawn model Multivariate version: Define i.i.d p-dimensional random vectors Z k = (Z,k,, Z p,k ), k =,, n, with unit Fréchet marginals. By assuming multivariate regular variation, we have F Z Z p (z,, z p ) = L(z,, z p )z c z cp p (6) where, z,, z p > 0, c + + c p =, L is a multivariate regularly varying function (MRV) satisfying: L(tz,, tz p ) lim = g(z,, z p ) t L(t,, t) and g(cz,, cz p ) = g(z,, z p ), c > 0, z,, z p > 0.

13 Ramos-Ledford-Tawn model Rewrite the Eq. (6), we get, F Z Z p (z,, z p ) = L(z,, z p )(z z p ) p (7) ( ) κ2/p ( ) κ23/p ( ) κp/p where, L = L z z2 z 2 z 3 z z p, κ 2 = c 2 c, κ 23 = c 3 c 2,, κ p = c p c. Define conditional variables: (X,, X p ) = (Z /u,, Z p /u) (Z > u,, Z p > u), thus, X,, X p. Then, F X X p (x,, x p ) = lim u Pr(Z >ux,,z p>ux p) = lim u L(ux,,ux p) = g(x,,x p) (x x p) /p = L(u,,u) ( g Pr(Z >u,,z p>u) (x x p) /p ) x x,, p p i= x i p i= x i (x x p) /p (8)

14 Ramos-Ledford-Tawn model By taking a polar-coordinate transformation again, we have: F X X p (x,, x p ) = Ramos-Ledford Constraint (general): Suppose { } min (w i ) dh i p x (w) (9) i x i = u, i =,, p, then, F X X p (u,, u) = u. Thus, the following normalising condition should hold: = { } min w i dh (w) = F X X p (,, ) (0) i p

15 From h 0 to h Theorem : Given any positive function h 0 with finite moments on, then, where, h (w) = (m w) (p+ ) p i= m i h 0( m w m w,, m pw p m w ) () m i = ui h 0 (u) du, i =,, p (2) is the density of a valid measure function H, satisfying the MEV constraint (general), i.e., Eq. (5). The special case when = reduces to the result in Coles and Tawn (99).

16 From h to h Theorem 2: Given any function h satisfying the MEV constraint (general), i.e., Eq. (5), then, where, h (w) = h (w) δ (3) δ = p + ( ) 3 V (,,,,,,,,,, ) + + ( ) p+ V (,, ) (4) is a valid density function of a measure function H, satisfying the Ramos-Ledford Constraint (general), i.e., Eq. (0). The special case when d = 2 and = is given in Ramos and Ledford (2007), where, h (w) = h(w)/(2 V (, )).

17 From h to h Sketch of the proof: It is straightforward to check that, = { min i p w i { min i p w i } h (w) dw } h (w) { } dw = min w i h (w) dw i p

18 From h to h According to the inclusion-exclusion principle, σ = { min = ( ) 2 p i= i p w i w } h (w) dw i +( ) 3 p i= p j=i+ h (w) dw { + + ( ) + p max i p w i max i j, i p, i+ j p } (w i, w j ) h (w) dw h (w) dw

19 General survival functions Following the definition of F X X p (x,, x p ) in terms of angular measures, we get, F X X p (x,, x p ) { = min (w i ) i p x i { = δ S min p = δ i p (w i { min i p (w i } h (w) dw } ) h (w) dw x i } (m w) (p+ ) ) x i p i= m i h 0( m w m w,, m p w p m w ) dw (5)

20 General survival functions As F Z Z p (z,, z p ) = λ F X X p ( z u,, zp u ), where λ = Pr(Z > u,, Z p > u). F Z Z p (z,, z p ) { = λu δ min (w i ) i p z i p i= m i h 0( m w m w,, m p w p m w ) dw } (m w) (p+ ) where, as defined in Theorem and 2, m i = ui h 0 (u) du, i =,, p δ = p + ( ) 3 V (,,,,,,,,,, ) + + ( ) p+ V (,, ) (6)

21 Validility check We check the validility of F Z Z 2 x x 2, y y 2, as a survival function, i.e., for F Z Z 2 (x 2, y 2 {[ ) F Z Z 2 (x, y 2 ) F Z Z 2 (x 2, y ) + F Z Z 2 (x, y ) = λu δ x x x +y w h x (w) dw x x 2 ] x 2 +y 2 w h x (w) dw 2 [ x +y 2 x 2 +y y y 2 x 2 x 2 +y ( w) x x +y x 2 x 2 +y 2 ( w) x x +y 2 h (w) dw h (w) dw ]}

22 A theoretical example Take the Dirichlet distribution for example, { p p h 0 (w) = Γ(α i )} Γ(α ) i= According to the definition of m i, we have, i= w α i i m i = Γ(α )Γ(α i + ) Γ(α i )Γ(α + i =,, p ),

23 A theoretical example By Theorem, we get, denoting h (w) = p i= p i= = p i= γ = (γ,, γ p ) = Γ (α i + ) Γ(α + ) Γ + (α i ) { p Γ (α i + } p+ ) i= Γ w (α i ) i Γ (α i + α ) Γ w (α i ) i i p Γ (α i + ) i= Γ w (α i ) i γ Γ(α + i ) p Γ(α i ) (γ w) p+ i= ( Γ (α + ) Γ (α ) { } αi γi w i γ w (7),, Γ (α p + ) ) Γ (8) (α p )

24 A theoretical example Following Theorem 2, we get, p h(w) = δα γ i Γ(α + ),,α p, Γ(α i ) (γ w) p+ where, i= δ α,,α p, = p + ( ) 3 p p i= ( ) p+ γ Γ(α + i ) Γ(α i ) (γ w) p+ { p i= max i p w j i= p j=i+ } { γi w i γ w p i= p i= { } γi w αi i γ w max } ] αi dw + + i j, i p, i+ j p γ i Γ(α + ) Γ(α i ) (γ w) p+ p (9) (w i, w j ) i= { } γi w αi i dw γ w

25 A theoretical example For d = 2, then, h (w) = γ γ 2 { γ w γ w+γ 2 ( w) Γ(α +α 2 + ) Γ(α )Γ(α 2 ) {γ w+γ 2 ( w)} 2+ } α2 } α { γ2 ( w) γ w+γ 2 ( w) (20) V (x, x 2 ) = x { I γ x (α + } γ x, α 2) + x 2 I γ x γ x (α, α 2 + ) (2) where, I v (a, b) = Γ(a+b) v Γ(a)Γ(b) 0 ta ( t) b dt is the regularized incomplete beta function, and a, b > 0.

26 A theoretical example Thus, where, h(w) = δ { γ w γ w+γ 2 ( w) γ γ 2 Γ(α +α 2 + ) α,α 2, Γ(α )Γ(α 2 ) {γ w+γ 2 ( w)} 2+ } α2 } α { γ2 ( w) γ w+γ 2 ( w) (22) δ α,α 2, = + I γ (α + γ, α 2) I γ (α, α 2 + γ ) (23) Then, F X X 2 (x, x 2 ) = δ α,α 2, [ x I γ x γ x (α +, α 2) + x 2 { } ] I γ x (α, α 2 + γ x ) (24)

27 A theoretical example As F Z Z 2 (z, z 2 ) = λ F X X 2 ( z Then, F Z Z 2 (z, z 2 ) = λu δ α,α 2, [ z I γ z γ z u, z 2 u ), where λ = Pr(Z > u, Z 2 > u). (α +, α 2) + z 2 { I γ z γ z (α, α 2 + ) } ] (25) We further get the density of the distribution function as follows: f Z Z 2 (z, z 2 ; Θ) = λu γ δα γ 2 Γ(α + α 2 + ) (γ z ) α (γ 2 z 2 ) α 2,α 2, Γ(α )Γ(α 2 ) (γ z + γ 2 z 2 ) α +α 2 + (26) where, parameter set Θ = (α, α 2,, λ).

28 Survival distribution plots (α, α 2 ) = (2, 9), λ = 0.3, u = 0.5

29 Inference Censored maximum likelihood estimation - Smith et al. (997) Numerical examples Simulation Practical examples

30 Coles, S. G. and J. A. Tawn (99). Modelling extreme multivariate events. Journal of the Royal Statistical Society, Series B: Methodological 53, de Haan, L. and S. I. Resnick (977). Limit theory for multivariate sample extremes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 40, Ledford, A. W. and J. A. Tawn (997). Modelling dependence within joint tail regions. Journal of the Royal Statistical Society, Series B: Methodological 59, Ramos, A. and A. W. Ledford (2007). A new class of models for bivariate joint tails. Smith, R. L. (2003). Statistics of extremes, with applications in environment, insurance, and finance. Extreme Values in Finance, Telecommunications and the Environment, 78. Smith, R. L., J. A. Tawn, and S. G. Coles (997). Markov chain models for threshold exceedances. Biometrika 84,