Estimating Bivariate Tail: a copula based approach Elena Di Bernardino, Université Lyon 1 - ISFA, Institut de Science Financiere et d'assurances - AST&Risk (ANR Project) Joint work with Véronique Maume-Deschamps and Clémentine Prieur 11ème Forum des Jeunes Mathématicien-ne-s - Université Toulouse 3 21-23 Novembre 2011 1 / 29
Plan Introduction 1 Introduction 2 3 Estimating the tail dependence structure Estimating the tail 4 2 / 29
Univariate POT and Univariate Pickands-Balkema-de Haan Theorem The univariate POT (Peaks-Over-Threshold) method is common for estimating extreme quantiles or tail distributions (see e.g. McNeil 1997, 1999 and references therein). A key idea of this method is that a distribution is in the domain of attraction of an extreme value distribution if and only if the distribution of excesses over high thresholds is asymptotically generalized Pareto (GPD) (e.g. Balkema and de Haan, 1974; Pickands, 1975): V ξ,σ (x) := { 1 ( ) 1 ξx 1 ξ σ, if ξ 0, σ > 0, 1 e x σ, if ξ = 0, σ > 0, and x 0 for ξ 0 or 0 x < σ ξ for ξ > 0. 3 / 29
Univariate POT and Univariate Pickands-Balkema-de Haan Theorem From the Univariate Pickands-Balkema-de Haan Theorem: β( ) such that lim sup Fu (x) V u x k,β(u)(x) = 0, F 0 x<x F u where F u (x) = P[X u x X > u] and x F := sup{x R F (x) < 1} (end point of function F ). POT estimator for univariate tail distribution: ( ) 1 1 F N (y u) ξ 1 ξ (y) = n σ, if ξ 0, ) (e (y u) σ, if ξ = 0, N n with u < y < (if ξ 0) or u < y < σ ξ number of excesses above the threshold. (if ξ > 0) and N the random 4 / 29
Framework Goal : estimating the tail of a bivariate distribution function. Idea : a general extension of the Peaks-Over-Threshold method. Tools : a two-dimensional version of the Pickands-Balkema-de Haan Theorem, Juri & Wüthrich's and Charpentier & Juri's approach of the tail dependence. dependence modeled by copulas. Asymptotic dependence as well as asymptotic independence are considered. 5 / 29
Conditionally versus unconditionally on N Summary of results Construction of a two-dimensional tail estimator, study of its asymptotic properties. A parameter that describes the nature of the tail dependence is introduced and estimated. X 1,..., X n i.i.d. F. We x some high threshold u, let N denote the number of excesses above u. In the following, two approaches will be considered: Conditionally on N. If n is the sample size and u n the associated threshold, the number of excesses is m n, with lim n m n = and lim n m n /n = 0. Unconditionally on N, i.e., N n as a binomial random variable N n Bi (n, 1 F (u n )) with lim n 1 F (u n ) = 0 and lim n n(1 F (u n )) =. 6 / 29
Upper-tail dependence copula C(u, v) is a 2-dimensional copula and C (u, v) is the associated survival copula. Let X and Y be uniformly distributed on [0, 1]. Let u be a threshold in [0, 1) such that C (1 u, 1 u) > 0. The Upper-tail dependence copula at level u [0, 1) is C up u (x, y) := P[ X F 1 X, u(x), Y F 1 Y, u(y) X > u, Y > u ], (x, y) [0, 1] 2, where F X, u, F Y, u conditioned on {X > u, Y > u}: are the distribution of X and Y F X, u(x) := P[ X x X > u, Y > u ] = 1 C (1 x u, 1 u) C. (1 u, 1 u) 7 / 29
Limit of the upper-tail dependence copula Proposition Assume that C (1 u, 1 v)/ u < 0 and C (1 u, 1 v)/ v < 0 for all u, v [0, 1). Furthermore, assume that there is a positive function G such that lim u 1 C (x (1 u), y (1 u)) C (1 u, 1 u) Then for all (x, y) [0, 1] 2 lim u 1 C up u (x, y) = x + y 1 + G(g 1 X = G(x, y), for all x, y > 0. 1 (1 x), g (1 y)) := C G (x, y), where g X (x) := G(x, 1), g Y (y) := G(1, y). Moreover there is a constant θ > 0 such that, for x > 0 { x θ g Y ( y x G(x, y) = ) for y [0, 1], x y θ g X ( x ) for y (1, ). y x 8 / 29 Y
Standing assumptions X and Y are two continuous real valued random variables, with marginal distributions, F X, F Y, and copula C. F X MDA(H ξ1 ), F Y MDA(H ξ2 ) Let X 1, X 2,..., X n be a sequence i.i.d. F. If (a n ) n>0 (sequence of positive values), and (b n ) n>0 (sequence of real values) such that [ ] lim P max{x1, X 2,..., X n } b n x = H k (x), x R, n a n then H k (x) is a Generalized Extreme Value Distribution: { e H k (x) = (1 k x) 1 k, if k 0, e e x, if k = 0, with 1 k x > 0, k R (k < 0 Fréchet family, k = 0 Gumbel, k > 0 Weibull).We say F MDA(H k ). C as in the above Proposition. V ξ1,a1( ) (resp. V ξ2,a2( )) is the univariate GPD distribution with parameters ξ 1 (resp. ξ 2 ) and a 1 ( ) (resp. a 2 ( )) of X (resp. Y ). 9 / 29
A two dimensional Pickands- Balkema-de Haan Theorem Theorem Under the standing assumptions, sup P[ X u x, Y F 1 (F Y X (u)) y X > u, Y > F 1 (F Y X (u)) ] A C G ( 1 g X (1 V ξ1,a1(u)(x)), 1 g Y (1 V ξ2,a2(f 1 Y (F X (u))) (y))) A := {(x, y) : 0 < x x FX u, 0 < y x FY F 1 (F Y X (u))}, with x FX := sup{x R F X (x) < 1}, x FY := sup{y R F Y (y) < 1}. u x FX 0, Proof: We generalize the proof by Jury and Wüthrich (2004) in the case of a symmetric copula (and same marginal distributions). 10 / 29
Stable tail dependence function Estimating the tail dependence structure Estimating the tail Assume that the bivariate distribution function F has stable tail dependence function l: lim t 0 or equivalently lim t 0 1 t P[1 F X (X ) t x or 1 F Y (Y ) t y] := l(x, y) (1) 1 t P[1 F X (X ) t x, 1 F Y (Y ) t y] := R(x, y) = x + y l(x, y), see e.g. Huang (1992). If F X, F Y are in the maximum domain of attraction of two extreme value distributions H X, H Y and if (1) holds then F is in the domain of attraction of an extreme value distribution H with marginals H X, H Y and with copula determined by l. 11 / 29
Estimating the tail dependence structure Estimating the tail Relation with the upper tail dependence coecient Recall the upper tail dependence coecient: λ := lim t 0 P[F 1 X 1 (X ) > 1 t F (Y ) > 1 t]. Y Asymptotic dependence: λ > 0. Asymptotic independence: λ = 0. λ = R(1, 1). Asymptotic independence i.e., when the data exhibit positive or negative association that only gradually disappears at more and more extreme levels. 12 / 29
Estimators Estimating the tail dependence structure Estimating the tail We introduce the non parametric estimators for R (see Einmahl et al., 2006): R(x, y) = 1 n 1 {R(X k i )>n k n x+1, R(Y i )>n k n y+1}, n i=1 where k n, k n /n 0 and R(X i ) is the rank of X i among (X 1,..., X n ), R(Y i ) is the rank of Y i among (Y 1,..., Y n ), for i = 1,..., n. The functions g X, g Y, G are estimated by ĝ X (x) = R(x, 1) R(1, 1), ĝ Y (x) = R(1, y) R(1, 1), and Ĝ(x, y) = R(x, y) R(1, 1), 13 / 29
Estimators Estimating the tail dependence structure Estimating the tail The functions g X, g Y, G are estimated by ĝ X (x) = R(x, 1) R(1, 1), ĝ Y (x) = R(1, y) R(1, 1), and Ĝ(x, y) = R(x, y) R(1, 1), The coecient θ is estimated by θ y x = log(ĝ(x,y)) log(ĝ Y ( y x )) log(x) if y x log(ĝ(x,y)) log(ĝ X ( x y )) log(y) if y x [0, 1], (1, ). Symmetric copula case, g X (x) = g Y (x) = g(x) = x θ g(1/x) for x > 0, and θ x = log(ĝ(x)) log(ĝ( 1 )) x. log(x) 14 / 29
Estimating the tail dependence structure Estimating the tail Convergence results (asymptotically dependent case) Theorem 2.2 in Einmahl et al. (2006) leads to the following consistency result. Proposition Under our standing assumptions, for v n such that v n / k n 0, for n, and λ > 0, v n sup 0<x 1 v n sup 0<x,y 1 Ĝ(x, y) G(x, y) P 0, n ĝx (x) g X (x) P 0, v n sup n 0<y 1 with k n, k n /n 0 and k n = o ( n 2α 1+2α ). ĝy (y) g Y (y) P n 0. 15 / 29
Estimating the tail dependence structure Estimating the tail Convergence results (asymptotically independent case) The case λ = R(1, 1) = 0 requires second order conditions. As in Draisma et al. (2004), we assume that: lim t 0 C (tx, ty) C (t, t) G(x, y) q 1 (t) := Q(x, y), for all x, y 0, x + y > 0, with q 1 is some positive function and Q is neither a constant nor a multiple of G. The above convergence is uniform on {x 2 + y 2 = 1}. Denote q(t) := P[1 F X (X ) < t, 1 F Y (Y ) < t]. 16 / 29
Estimating the tail dependence structure Estimating the tail Convergence results (asymptotically independent case) Proposition Under our standing assumptions and second order conditions, for a sequence k n such that a n := n q(k n /n) ψ n sup 0<x 1 ψ n sup 0<x,y 1 Ĝ(x, y) G(x, y) P 0, n ĝx (x) g X (x) P 0, ψ n sup n with ψ n = o( a n ). 0<y 1 ĝy (y) g Y (y) P n 0, 17 / 29
A new tail estimator Estimating the tail dependence structure Estimating the tail Main ingredients of our estimator. A high threshold u; 1 F Y Dene û Y = function and F 1 Y ( F X (u)), with F X (u) the empirical distribution the empirical quantile function of Y. k X, σ X (resp. k Y, σ Y ) the MLE based on the excesses of X (resp. Y). F X (x) (resp. FY (y)) the univariate tail estimator (see McNeil (1999)): F X (x) = (1 FX (u))v k, σ (x u) + F X (u), for x > u. F 1 (u, y) = exp{ l n ( log( F X (u)), log( F Y (y)))}, and F 2 (x, û Y ) = exp{ l n ( log( F X (x)), log( FY (û Y )))}. 18 / 29
A new tail estimator Estimating the tail dependence structure Estimating the tail We estimate F (x, y), for x > u, y > u Y = F 1 (F Y X (u)) and u large, by: F (x, y) = ( 1 n n 1 {Xi >u, Y i >u Y } ) (1 ĝx (1 (x u)) V ξx, σx i=1 ĝ Y (1 V ξy, σ Y (y u Y )) + Ĝ( 1 V ξx, σ X (x u), 1 V ξy, σ Y (y u Y ) )) + F 1 (u, y) + F 2 (x, u Y ) 1 n n 1 {Xi u, Y i u Y }, i=1 where ξ X, σ X Y ). (resp. ξ Y, σ Y ) are MLE based on the excesses of X (resp. 19 / 29
A new tail estimator Estimating the tail dependence structure Estimating the tail But u Y = F 1 (F Y X (u)) is unknown. Then, in order to compute in practice F 1 (x, y), we propose to estimate u Y by û Y = F ( F Y X (u)), with F X (u) = 1 n 1 1 n i=1 {X i u} and F the empirical quantile function of Y. Y We estimate F (x, y), for x > u and y > û Y, by: F (x, y) = ( 1 n n 1 {Xi >u, Y i >û Y } ) (1 ĝx (1 (x u)) V ξx, σx i=1 ĝ Y (1 V ξy, σ Y (y û Y )) + Ĝ( 1 V ξx, σ X (x u), 1 V ξy, σ Y (y û Y ) )) + F 1 (u, y) + F 2 (x, û Y ) 1 n n 1 {Xi u, Y i û Y }, i=1 20 / 29
Estimating the tail dependence structure Estimating the tail Convergence result for our bivariate POT estimator (asymptotically dependent case) λ > 0, standing assumptions, rst and second order conditions on the marginal laws and Smith's hypothesis on the thresholds above. Theorem k n (F (x n, y n ) F (x n, y n )) P 0, n with x n = f 1 (n)f 1 (n), y n = f 2 (n)f 2 (n). Moreover if f 2 (n) satises the thershlod conditions in probability then k n (F (x n, ŷ n ) F (x n, ŷ n )) P 0, n with ŷ n = f 2 (n)f 2 (n). We have k n, k n /n 0, k n = o(n 2α 1+2α ), α > 0. 21 / 29
Estimating the tail dependence structure Estimating the tail Convergence result for our bivariate POT estimator (asymptotically independent case) λ = 0, standing assumptions, rst and second order conditions on the marginal laws, second order condition on the joint distribution and Smith's hypothesis on the thresholds above. Theorem a n (F (x n, y n ) F (x n, y n )) P 0, n where x n = f 1 (n)f 1 (n), y n = f 2 (n)f 2 (n). Moreover if f 2 (n) satises the threshold conditions in probability then a n (F (x n, ŷ n ) F (x n, ŷ n )) P 0, n with ŷ n = f 2 (n)f 2 (n) and a n = n q(k n /n). k n /n 0, a n q 1 (q (a n /n)) 0 and k n = o(n 2α 1+2α ), for some α > 0. 22 / 29
s of the dependence structure : simulations (sensitivity of θ x to the sequence k n ) Clayton copula. Mean curve on 100 samples of size n = 1000 (full line) and the empirical standard deviation (dashed lines). Mean squared error for θ x is calculated on 100 samples of size n = 1000. Asymptotic independent case. Figure: Clayton copula with parameter 0.05: (left) estimator for θ, (k, θ x) with x = 0.7; (right) mean squared error for θ x with x = 0.7. 23 / 29
Real data Figure: Logarithmic scale (left) ALAE versus Loss; (right) Wave heights versus Water level. 24 / 29
Real data Stability of our estimation compared to the one of F, as well as the 1 estimation of parameter θ of these real cases. F 1 (y 1, y 2 ) = exp{ l( log( F Y1 (y 1)), log( F Y2 (y 2)))}, is known to produce a signicant bias for asymptotically independent data. 25 / 29
Loss / ALEA data: asymptotic dependent case Loss-ALAE data: Each claim consists of an indemnity payment (the loss, X ) and an allocated loss adjustment expense (ALAE, Y ). We estimate F (2.10 5, 10 5 ). Figure: (left) θ 0.04; (right) F (2.10 5, 10 5 ) (full line), F 1 (2.10 5, 10 5 ) (dashed line), with the empirical probability indicated with a horizontal line. 26 / 29
Wave height vs Water level: asymptotic independent case Wave height versus Water level data: recorded during 828 storm events spread over 13 years in front of the Dutch coast near the town of Petten. Figure: (left) θ 0.1 0.11 = θ 0.91; (right) F (5.93, 1.87) (full line), F 1 (5.93, 1.87) (dashed line), with the empirical probability indicated with a horizontal line. 27 / 29
Ideas for future developments Get the optimal rate (choice of k n ), a central limit theorem for the dependence parameter θ, the dependence function g X, g Y,G and the bivariate POT tail estimator F? Optimal choice of k n and deep comparison with estimators in literature (Ledford and Tawn; 1997,1998). Use the bivariate tail estimator F (x, y) to obtain estimation of bivariate upper-quantile curves, for high levels α (see article Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory, written with Thomas Laloë, Véronique Maume-Deschamps and Clémentine Prieur, accepted for publication in ESAIM: Probability and Statistics journal. 28 / 29
Thanks for your attention 29 / 29