Introduction A basic result from classical univariate extreme value theory is expressed by the Fisher-Tippett theorem. It states that the limit distri

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1 The dependence function for bivariate extreme value distributions { a systematic approach Claudia Kluppelberg Angelika May October 2, 998 Abstract In this paper e classify the existing bivariate models by means of their dependence function and dependence measure. We sho that ithin certain classes the mixed and the logistic are the only possible models. We give a complete discussion of the class of polynomial dependence functions and sho that polynomials of higher order do not model high dependence. We also sho ho the concentration of the dependence measure translates to dependence of the bivariate distribution. Finally, e introduce some crude dependence measures and discuss their analytic and probabilistic properties. AMS 99 Subject Classications: primary: 6G7 secondary: 6G55 Keyords: Bivariate extreme value distributions, dependence function, dependence measure, Pickands' representation theorem. Center of Mathematical Sciences, Munich University of Technology, D-829 Munich, Germany, fcklu,mayg@mathematik.tu-muenchen.de,

2 Introduction A basic result from classical univariate extreme value theory is expressed by the Fisher-Tippett theorem. It states that the limit distribution of normalized maxima from a sequence of independent and identically distributed (iid) random variables must belong to a 3-parameter family. In a multivariate setting one cannot expect a similar result ith a nite dimensional parametrization for the possible limit distributions, the obvious reason being the numerous possible models for dependence in the extremes. The analysis of multivariate extremes is usually based on componentise ordering: for a sequence of iid vectors X i = (X i; ; : : : ; X i;d ) for i = ; : : : ; n, the vector of componentise maxima is given by M n = (M n; ; : : : ; M n;d ), here M n;j = maxfx ;j ; : : : ; X n;j g for j = ; : : : ; d. We assume that the X i are in the maximum domain of attraction of some multivariate extreme value distribution. Then obviously the marginal distributions must be in the maximum domain of attraction of some univariate extreme value distributions. Since each univariate extreme value distribution can be transformed into a standard Frechet distribution, it is no restriction to consider only this standard Frechet distribution as limit distribution for all margins. It has the form (x) = e?=x for x >, and e rite X 2 MDA( ) if for iid X; X ; : : : ; X n the maximum M n = maxfx ; : : : ; X n g properly normalized converges in distribution to. For details on univariate extreme value theory see Resnick (987) or Embrechts, Kluppelberg and Mikosch (997). An impressive list of multivariate extreme value models has been introduced, motivated by analytic tractability and/or by practical requirements. Parametric models have been knon since Gumbel (96) and Galambos (987), a list of examples can be found for instance in Resnick (987), Coles and Tan (99), Smith (994) or Joe (997). In a multivariate setting, a systematic approach of models seems at rst not feasible. Nevertheless there exist various equivalent characterizations among those the one based on Pickands' A-function hich e shall deal ith in the sequel. For an early alternative representation check e.g. de Haan and Resnick (977). One possible ay to put the representation theorem is the folloing: Any d-dimensional random vector X ~ ith unit Frechet margins follos an extreme value distibution if and only if its joint distribution function G: R d! [; ] can be expressed as G(x) = G(x ; : : : ; x d ) = P ( ~ X x ; : : : ; ~ Xd x d ) = exp "? R S d max jd n qj x j o H(d(q ; : : : ; q d )) # (.) 2

3 here H induces a positive nite measure H on S d def = f(q ; : : : ; q d ): q + + q d = ; q j for j = ; : : : ; dg, the (d? )-dimensional unit simplex in R d. In order to get unit Frechet margins e need R S d q j H(dq) = for j = ; : : : ; d. In this paper e restrict ourselves to bivariate extreme value models. Then the above representation theorem takes a simpler form as the simplex S 2 = f(q;? q): q 2 [; ]g is the upper right boundary of the triangle (; ); (; ); (; ). The bivariate extreme value distribution can thus be described by a function of just one real variable, the dependence function. The aim of this paper is to characterize parametric models via this dependence function A. We are ell aare of the fact that in dimensions higher than to the dependence function has no concise equivalent. It is hoever clear that the bivariate extreme value models occur as bivariate marginals for higher dimension models. Hence any extension to higher dimensions must be an extension of a bivariate model. The paper is organized as follos: In Section 2 e introduce the dependence function and sho that the class of reasonable parametric models is fairly limited. The systematic approach leads to a ne 2-parametric model based on a polynomial of order 4 in Proposition 2.4(iii). In Section 3 e discuss consequences of Pickands' representation (.), in particular the measure H. Concentration of H in nitely many points translates to dependence structures and vice versa. The analytic results of this section are illustrated by an example. We conclude this section ith the introduction of some crude measures of dependence hich can be interpreted analytically and are statistically meaningful. Example 3.7(iii) presents a ne family of 2-parametric models. Section 4 discusses the concentration of the measure H on subintervals of [; ]. We sho that this relates to stochastic ordering of the marginal random variables. 2 Dependence functions Let (X j ; Y j ); j = ; 2; : : :, be iid non-negative bivariate random vectors ith common distribution function F. We rite F and F 2 for the corresponding marginal distribution functions, so that F (x) = F (x; ) and F 2 (y) = F (; y) for x; y 2 R. Without loss of generality e henceforth assume that X j and Y j are in MDA( ) and that lim F j n (nx) = (x) for x 2 R, j = ; 2. (See Resnick (987), section 5.4.) We denote n! an extremal random vector by ( X; ~ Y ~ ) and assume that it follos a bivariate extreme value distribution G ith unit Frechet margins. 3

4 We consider the point process P n = f n (X ; Y ); : : : ; n (X n; Y n )g in R 2. Under the above conditions on G, the process P n converges eakly to a non-homogeneous Poisson process on (R 2 def ) = [; [[; [nf(; )g ith intensity measure : B? (R 2 )! R > here B? (R 2 ) denotes the set of Borel sets B that are bounded aay from (; ). The intensity measure satises the homogeneity property (8m > )? 8B 2 B? (R 2 ) (B=m) = m(b): (2.2) The measure is linked to the exponent in (.) as follos. For x; y > dene the so-called exponent measure. Then Z q (x; y) = max x ;? q y? (x; y) def = (R 2 ) n ( ]; x] ]; y] ) ; H(dq) def = x + x A ; x; y > : (2.3) y x + y Note that inherits the homogeneity property from, i.e. is homogeneous of order?. Since G(x; y) = exp (?(x; y)) for x; y >, the folloing is obvious by (.): Denition 2. In the above representation, the function A: [; ]! [; ]; 7! A() = Z maxf(? q); (? )qgh(dq) (2.4) is called the dependence function. We deduce directly that A() = = A(). In addition, due to (2.4) e have that (8 2 [; ]) maxf;? g A() : The function A: [; ]! [; ] is convex since A is the integral over a convex function. Furthermore, the class of dependence functions is a convex set, i.e. if A ; : : : ; A m are dependence functions, then A def = A + + m A m is a dependence function for every choice of the constants f ; : : : ; m g R ith + + m =. Hence ne dependence functions can be constructed by convex combinations of existing ones. Examples of a construction of that type are given in Tajvidi (996). Since the marginal distributions are assumed as unit Frechet, e have Z qh(dq) = = Z (? q)h(dq): 4

5 From that e calculate easily that H=2 is a distribution function ith mean =2. We start ith to important examples that ill give the upper and loer boundary for A in the class of dependence functions, equipped ith pointise ordering. The folloing results immediately from (2.4). Proposition 2.2 (i) If X ~ and Y ~ are independent, then A: [; ]! [; ]; 7!, corresponding to G(x; y) = P ( X ~ x; Y ~ y) = exp (?(x; y)) = exp(?=x) exp(?=y) for x; y >. (ii) If X ~ and Y ~ are totally dependent, i.e. P ( X ~ = Y ~ ) =, then A: [; ]! [; ], 7! maxf;? g. Further properties of the dependence function A can be deduced if e restrict ourselves to dierentiable functions. If A is tice dierentiable, this yields the folloing constraints: A() = = A()? A () A () A () for all 2 [; ], since A is convex. (2.5) These properties imply in particular that the graph of A lies in the triangle (; ); (=2; =2); (; ). We ill refer to models that corresponds to an A-function ith properties (2.5) as dierentiable models. The examples best knon among the dierentiable parametric models are the mixed model and the logistic model. Example 2.3 (i) The mixed model has dependence function A: [; ]! [; ]; 7! 2? + ; here. This corresponds to a joint distribution function G(x; y) def = P ( X ~ x; Y ~ y) = exp? x + y + x + y for x; y > : (ii) The asymmetric mixed model has dependence function A: [; ]! [; ]; 7! 3 + 2? ( + ) + ; here ; + 2 ; + 3. The corresponding exponent measure is (x; y) = x + y? (2 + )x + ( + )y (x + y) 2 for x; y > : 5

6 (iii) The logistic model has dependence function A: [; ]! [; ]; 7! f(? ) r + r g r ; here r. This yields as joint distribution function G(x; y) = exp (? x + ) r r y r for x; y > : (iv) The asymmetric logistic model has dependence function A: [; ]! [; ]; 7! f[(? )] r + [] r g r +? (? )? ; here,, r. The graph of A is shon in Figure. The corresponding exponent measure is (x; y) = x + y? x? y + ([x]r + [y] r ) r xy for x; y > : A() A() A() Figure : Asymmetric logistic model. A-function for r = 2, () = :3; = :4; (2) = :3; = :6; (3) = :8; = :2. The minimum varies from () (:46; :899), (2) (:4; :88) to (3) (:69; :898). The models (i), (ii) and (iii) play a specic role, as the constraints in (2.5) force A to be exactly the function in (i), (ii) and (iii) and do not allo for similar functions of the same type, but another choice of the parameters. This is illustrated by the folloing Proposition. For a further interpretation of these models in connection ith total dependence and independence, see section 4. Proposition 2.4 (i) If A: [; ]! [; ]; 7! a 2 + b + c is a quadratic polynomial, then a =?b def = ; c = ;. So the mixed model is the only polynomial of order 2 that satises (2.5). 6

7 (ii) If A: [; ]! [; ]; 7! a 3 + b 2 + c + d is a polynomial of order 3, then a def = ; b def = ; c =?( + ); d =, here, + 2, + 3. So the asymmetric mixed model is the only polynomial of order 3 that satises (2.5). (iii) If A: [; ]! [; ]; 7! a 4 + b 3 + c 2 + d + is a polynomial of order 4 and A (=2) =, then a def =, b =?2, c def = ; d =?, here,?. This denes a ne 2-parametric model ith dependence function A() = 4? (? ) +. It is the only possible symmetric (ith respect to ) polynomial of 2 order 4 that satises (2.5). (For the number of parameters in this model see also the remark in Example 3.7 (iii).) (iv) If A: [; ]! [; ]; 7! f(? ) p + q g r gives the only function of that type that satises (2.5). Proof. (i), (ii) and (iii) follo from simple analysis using (2.5). (iv) We calculate A () = r A () = r e get r = p = q, and the logistic model? q q?? p(? ) p? f(? ) p + q g r? and? r? f(? ) p + q g r?2? q q?? p(? ) p? 2? + r f(? )p + q g r? q(q? ) q?2 + p(p? )(? ) p?2.? r? (?p) 2 + r No,? A () ) p r, but A () = r Analogously, A () and A () yield r = q. (p(p? )), p r ) r = p. In general, a polynomial A: [; ]! [; ] of order n is a dependence function if and only if P A() = a n n + a n? n? + + a 2 2? ( n P a j ) + for 2 [; ], ith a 2, n a j, j=2 j=2 np P (j? )a j and n j(j? )a j. We get an (n? 2)-parametric model ith corresponding j=2 exponent measure j=2 (x; y) = x + y? np k=2 n?k P a k j=? n?k j x j+k? y n?k?j? : (x + y) n? 3 A measure for dependence Given a pair of extremal random variables ( X; ~ Y ~ ), e may pose the question ho extreme movements of X ~ and Y ~ are related. This can be described by the measure or, equivalently, the distribution function H=2. The measure dening function H has compact support [; ] and induces a measure H hich has total mass 2 on [; ]. We ill use H to measure the 7

8 dependence of extremal events. If most of the mass is concentrated around, the extremes are 2 highly dependent, i.e. an extremal event in ~ X is likely to be accompanied by an extremal event in ~ Y, and vice versa. Conversely, most of the mass concentrated around and translates the situation hen the extremes in ~ X and ~ Y are (almost) independent. We start ith an example (see Joe [7], p. 78) hich e shall discuss in more detail later.? Example 3. Let Z ; : : : ; Z N be iid unit Frechet random variables, i.e. P (Z x) = exp? x P for x >. Choose a k;x ; a k;y 2 R and k > for k = ; : : : ; N such that N k a k;x = and NP k a k;y =. We dene random variables X def = max ka k;x Z k and Y def = max ka k;y Z k. kn kn The common distribution function G for the vector (X; Y ) can be ritten as follos P (X x; Y y) = P max kn k a k;x Z k x; max kn = P Z k x k a k;x ; Z k Q n = N P Z k min P n = exp? N = min P = exp? N y k a k;y ; x k a k;x ; x k a k;x ; y k a k;y ) maxn k a k;x ; ka k;y x y k a k;y Z k y k = ; : : : N o o y k a k;y o : We check that the marginal distributions are unit Frechet: P max ka k;x Z k x kn Q = N Q P Z k x k a k;x = N? NP exp? a k;x x = exp? x k a k;x = e? x : The corresponding dependence function can be calculated via the exponent measure (x; y) = x + xy NX max k! a k;x ; ka k;y y x + y x y = x + y NX max? x x + y k a k;x ; P This gives A() = N max f(? ) k a k;x ; k a k;y g for 2 [; ].! x x + y ka k;y : From (3.6) e kno that is a discrete measure on [; ] 2 ith atoms (a k;x ; a k;y ) of mass k. Hoever, is not uniquely dened. We x by forcing atoms (a k;x ;? a k;x ) of mass k, i.e. is concentrated on the simplex S 2 = f(q;? q): q 2 Rg. Then has total mass 2 because of = P N k(? a k;x ) = P N k?. Furthermore, can be vieed as a measure on [; ]. From representation (2.3) e kno that the dependence function A and the measure H (the (3.6) 8

9 distribution function H, respectively) correspond uniquely to one another by H () (x; here H() = A() + A() = 2, and A() = A()(? ) + y=?;x= for ; Z H(q)dq for < ; R here A() = (H()? H(q))dq. (This is Theorem 3., p. 87, in Pickands (98).) So equivalence is clear in the folloing Proposition, and e may sho the statements either for A or H. Proposition 3.2 (i) The independent case. The case A() = for all is equivalent to the situation that the corresponding measure H puts mass at both and. (Note that this comes out of Example 3. for N = 2 and atoms (; ) and (; ) giving X = Z and Y = Z 2.) (ii) The totally dependent case. The case A() = maxf;? g for 2 [; ] is equivalent to the situation that the measure H puts mass 2 at. (This refers to the case N = in 2 the preceding Example 3. after multiplication ith =2.) (iii) The measure H (or A, equivalently) is symmetric about 2 and ~ Y are exchangeable. if and only if the variables ~ X R Proof. (i) If H fg = H fg = and H (q) = for < q <, then A() = q); (? )qgh(dq) = + (? ) = for all 2 [; ]. maxf(? (ii) If H f=2g = 2 and H fqg = for all q ; q 6= =2 it follos directly that R A() = maxf(? q); (? )qgh(dq) = maxf; (? )g for all 2 [; ]. (iii) If A() = A(? ) e have x A x + y No, (.) gives = A? x y = A : x + y x + y F (x; y) = exp (?(x; y)) = exp (?(y; x)) = F (y; x) : 9

10 Proposition 3.3 The measure H is a point measure ith all mass concentrated in one point a 2]; [ if and only if a =, corresponding to totally dependent random variables. 2 Proof. Let X H=2. Since 2 = E(X) = 2 a P (X = a), this gives P (X = a) = a ; < a <. For the dependence function, e calculate from (2.4) From = A() = a?, e get a = 2. A() = maxfa(? ); (? a)g a = 8 < :? ; a? a? ; a < : Proposition 3.4 If H is a point measure on [; ] that is concentrated in 2 points a; b 2 [; ] ith a < b, say, then (i) if a < 2 < b, the positive measure H puts mass 2b? b?a in a and?2a b?a to N = 2 in Example 3. and atoms (a;? a) and (b;? b).); (ii) if a; b 2 [; ] or [ ; ], there is no such positive measure; 2 2 (iii) the case a = and b = is the only one that admits independence. Proof. in b (This corresponds (ii) and (iii) follo directly from (i). For (iii) note that H (fg) = = H (fg) if and only if a = and b = from (i). (i) Let X H=2. Since the total mass of H on [; ] is 2, e have = P (X = a)+p (X = b), and = a P (X = a) + b P (X = b) because of the mean value. This proofs the assertion. The dependence function no gives rise to a quantity that measures the dependence of ~ X and ~ Y. We have seen that independence can be reached if and only if A =, the constant function, hereas for complete dependence it is necessary and sucient that A equals the loer boundary in the function space, 7! maxf? ; g; 2 [; ]. This motivates the folloing crude measure of dependence.? Denition 3.5 Let # = #(A) def =? A 2, the coecient of dependence. If # =, this corresponds to independent random variables, hereas # = 2 completely dependent case. In Example 3., e calculate NX A = maxn k a k;x ; ka k;y o = NX max f k a k;x ; k a k;y g : belongs to the

11 A() A() A() Figure 2: Logistic model. A-function for parameters () r = :, (2) r = :8 and (3) r =. The graphs illustrate that the minimum tends to =2 as r increases. Remark 3.6 Note also that # is linked to Sibuya's (96) denition of the dependence function (F (x); F 2 (y)) def P (X < x; Y < y) = in the folloing ay: For the exponent measure e have P (X < x)p (Y < y) (x; y) =? (R 2 ) n ( ]; x] ]; y] ) (3.7) = (]x; [ R) + (R ]y; [)? (]x; [ ]y; [) (3.8) = x + x y?? A x + y x + : (3.9) y Consequently, (]x; [ ]x; [) = # 2 x and def =? # = # (x; x) (]x; [ ]x; [) (3.) is another dependence parameter. Note that is independent of x by homogeneity. For # =, i.e. 2 the completely dependent case, = ; for # =, corresponding to the independent case, =. The parameter measures the concentration of in the cone ]x; [ ]x; [. From a statistical point of vie is more attractive than #, it can be estimated by the empirical counterpart of the lhs of (3.). A recent discussion of this quantity can be found in Smith, Coles and Tan (997). Example 3.7 (i) Let A r : 7! f(? ) r + r g r ; r, belong to the logistic model. Then #(r) def =? A r? 2 =? 2 2 r. This gives #() =? = for r =, the independent case, and lim r! #(r) =? 2 = 2 in case of complete dependence. (See Figure 2.) (ii) For the mixed model ith A : 7! 2? + for e obtain #() =??? + =. So complete dependence is not contained ithin this model

12 A() A() A() Figure 3: A-function for the polynomial model ith = :4 and n = 2; 4; 8. (iii) Let n = 2m 2 N be an even integer and set A n () def = (? ) n +. From condition (2.5) e get the folloing equations A n () = = n + A n () = = (? ) n + : The only non-trivial solution is = 2 giving A n () = n (2? ) n +? n : The folloing inequality A () = 2n n gives an upper bound for the coecient of dependence: #(A n ) =? A n ( 2 ) = n 2n < 2 for n 2. This shos that complete dependence cannot be reached for any of the polynomial dependence functions of the above form. Moreover, the coecient of dependence ill decrease ith increasing order n, hence the larger n e get even less chance to model high dependence. See Figure 3 for polynomials ith constant, but of dierent order. Note also that Example 2.4 (iii) is included in the set of functions A n if e set n = 4, = 6 4, and = So all possible polynomial models of order 4 that allo for exchangeability of the variables ~ X and ~ Y are in fact one-parametric. 4 Order relations A special case occurs if the measure H is concentrated on a proper subinterval of [; ]. 2

13 Theorem 4. (i) If H is concentrated on [; p y ], 2 < p y <, then?py p y ~ X ~ Y. (ii) If H is concentrated on [p x ; ]; < p x < px, then ~ 2?p x Y X. ~ Proof. (i) We set p y = +b for < b < and assume H([ +b ; ]) =. For any x; y 2 R > ith p y x x+y e calculate G(x; y) = exp? y = exp? y = exp? y Z ];p y] (? q)h(dq) H(p y )? : Z! ];p y] qh(dq) Equivalently, P (Y y) = P (X x; Y y) = P (X x j Y y)p (Y y) for all y > and x y b. So for all y > e have P (X x j Y y) = for all x y b, i.e. for any given y > is for given Y y the random variable X y b the proof.!!?py ith probability. Since b = p y, this completes We apply this result to a measure that is symmetric about 2 variables ~ X and ~ Y are exchangeable by Proposition 3.2 (iii). Corollary 4.2 indicating that the random (i) If H is concentrated on [p x ;?p x ] here p x def = =(+a) ith a >, then Y ~ ~ a X a Y ~. So X ~ is concentrated in a cone that is symmetric around the diagonal in the rst quadrant. (ii) If H is concentrated on [p x ; p y ] ith < p x < =2 and =2 < p y <, then p x?p x ~ Y ~ X. px?p x?py p y ~ X Proof. We have p y def =? p x, and by Theorem 4. e get p x?p x ~ Y ~ X p y?p y ~ Y =?p x p x ~ Y. Example 4.3 Take < p x < =2 and =2 < p y < arbitrary. (i) If the measure H is concentrated in p x and p y ith H (p x ) = = H (p y ), the corresponding dependence function is A() = 8 >< >:? ; < p x?p x?p y p x?p y + py(2px?) p x?p y ; p x < p y ; p y : 3

14 A() A() A() Figure 4: Pieceise linear A-function ith () p x = :; p y = :7, (2) p x = :2; p y = :6, (3) p x = :; p y = :8. (ii) If the measure H is concentrated in any N equidistant points q ; : : : ; q N on [p x ; p y ] then the corresponding dependence function is equal to? on [; p x ], equals on [p y ; ] and is pieceise linear in [p x ; p y ]. If e furthermore have equidistant points q = p x ; q 2 ; : : : ; q N = p y and put the same mass 2=N at each q j this is only possible for a symmetric interval, i.e. p y =? p x. Proposition 4.4 (i) If H is concentrated on [; p y ] for < p y < =2 and H=2 is uniformly distributed on [; p y ], the corresponding dependence function is given by A() = 8 >< >: 2 p y p y? p y + ; < p y ; p y (ii) Analogously, if H is concentrated on [p x ; ] (uniformly), e have Proof. A() = 8 >< >:? ; < p x 2 p x p x? p x + ; p x R (i) Recall A() = maxfq(? ); (? q)gh(dq). If H(dq) = on ]p y ; ], e get A() = for < p y. Let A() def = 2 + +, i.e. A () = 2 +. Since A is dierentiable, e have? = A () = and = A (p y ) = 2p y?. So = p y?. Since A(p y ) = p y, e claim p y = lim!p y? p y? 2? + =. So, A() = p? y 2? + p y = p y p y 2? p y +! 4

15 Conclusions In this paper e have tried to give a systematic overvie of bivariate extreme value theory by means of the dependence function as a parametric model for extremal behaviour. At rst sight the restrictions on the function A given by (2.5) seem to be rather eak, alloing for a considerable variety of models. Hoever, once e consider certain reasonable classes of functions, there are only fe models left. A dependence funtion hose graph coincides partly ith the sides of the triangle corresponds to some order relation concerning the rvs e X and e Y. A polynomial dependence function can only model eak dependence, it never reaches anyay near total dependence. The only \nice" model hich is able to model all the ay from independence to complete dependence is the logistic model. Hence our conclusion is that for data exhibiting strong dependence (ithout being stochastically ordered) the logistic function oers is a very reasonable rst model. Acknoledgement We ould like to thank Holger Rootzen for the invitation to the inspiring conference \Extremes { Risk and Safety" in Gothenburg hich gave us the possibility to present this ork. We also thank the attendants of this conference for various discussions and useful remarks on the topic. References [] Coles, S.G. and Tan, J.A. (99) Modelling extreme multivariate events. J. R. Statist. Soc. B, 53, No.2, pp [2] Davison, A.C. and Smith, R.L. (99) Models for exceedances over high thresholds (ith discussion). J. R. Statist. Soc. B 52, pp [3] Embrechts, P., Kluppelberg, C. and Mikosch, T. (997) Modelling Extremal Events for Insurance and Finance. Springer, Heidelberg. 5

16 [4] Galambos, J. (987) The Asymptotic Theory of Extreme Order Statistics. 2nd Edition. Krieger Publishing Co., Malabar, Florida. [5] Gumbel, E.J.(96) Bivariate exponential distributions. J. Am. Statist. Assoc., 55, pp [6] de Haan, L., Resnick, S. I. (977) Limit Theory for Multivariate Sample Extremes. Z. Wahrscheinlichkeitstheorie ver. Gebiete, 4, pp [7] Joe, H. (997) Multivariate Models and Dependence Concepts. Chapman and Hall, London. [8] Joe, H., Smith, R.L. and Weissmann, I. (992) Bivariate threshold methods for extremes. J. R. Statist. Soc. B, 54, pp [9] Ledford, A., Tan, J. A. (997) Modelling Dependence ithin Joint Tail Regions. J. R. Statist. Soc. B, 59, No. 2, pp [] Pickands, J. (98) Multivariate extreme value distributions. Bull. Int. Statist. Inst., pp [] Resnick, S.I. (987) Extreme Values, Regular Variation, and Point Processes. Springer, Berlin. [2] Smith, R.L. (994) Multivariate threshold methods. In Extreme Value Theory and Applications, eds. J. Galambos, J. Lechner, and E. Simiu. Kluer Academic Publishers, Dordrecht, pp

17 [3] Smith, R.L., Tan, J.A. and Coles, S.G. Markov chain models for threshold exceedances. Biometrika, 84, pp [4] Smith, R.L., J.A. Tan and H.K. Yuen (99) Statistics of multivariate extremes. Int. Stat. Revie 58,, pp [5] Sibuya, M. (96) Bivariate extreme Statistics. Ann. Inst. Stat. Math.,, pp [6] Tajvidi, N. (996) Characterisation and Some Statistical Aspects of Univariate and Multivariate Generalised Pareto Distributions. Ph.D. thesis, Department of Mathematics, Goteborg University. [7] Tan, J.A. (988) Bivariate extreme value theory: models and estimation. Biometrika 75, 3, pp