The Generalized Pareto process; with application

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1 The Generalized Pareto process; with application Ana Ferreira ISA, Universidade Técnica de Lisboa and CEAUL Tapada da Ajuda Lisboa, Portugal Laurens de Haan Erasmus University Rotterdam and CEAUL P.O. Box 1738, 3000 DR Rotterdam, The Netherlands February 9, 2012 Abstract In extreme value statistics the peaks-over-threshold method is widely used. The method is based on the Generalized Pareto distribution [1], [8] in univariate theory and e.g. [3], [14] in multivariate theory characterizing probabilities of exceedances over high thresholds. We present a generalization of this concept in the space of continuous functions. We call this the Generalized Pareto process. Different from earlier papers our definition is not based on a distribution function but on functional properties. As an application we use the theory to produce wind fields connected to disastrous storms on the basis of observed extreme but not disastrous storms. Keywords: domain of attraction, extreme value theory, generalized Pareto process, max-stable processes, regular variation 1 Introduction We say that a stochastic process X in CS the space of continuous functions whith S a compact subset of R d is in the domain of attraction of a max-stable process if there are continuous functions a s n positive and b s n on S such that the processes { max i n } X i s b s n, a s n with X,X 1,...,X n independent and identically distributed, converge in distribution to a max-stable process Y in CS. Necessary and sufficient conditions 1

2 for this to happen are: uniform convergence of the marginal distributions and a convergence of measures in fact a form of regular variation: where η t s := 1+γs Xs bst a st lim tp η t A = νa 1.1 t 1/γs for all s S, ν is a homogeneous of order -1 measure on C + S := {f CS : f 0} and A any Borel set of C + S with the properties: ν A = 0 and inf{sup fs : f A} > 0 de Haan and Lin 2001, cf. de Haan and Ferreira Section 9.5. The functions a s n and b s n are chosen in such a way that the marginal distributions are in standard form: exp 1+γsx 1/γs. Here γ is a continuous function. In particular one may take b s t := inf{x : PXs x 1 1/t}. This is how we choose b s t from now on. From 1.1 it follows that P 1/γ 1+γ X b t a t A P sup Xs b st a st > 0 converges as t and so does P 1+γ X b t 1/γ A Xs b s t sup > 0. a t a s t The limit constitutes a probability distribution on C + S. This reasoning is quite similar to how one gets the generalized Pareto distributions in R Balkema and de Haan, 1974 and in R d Rootzén and Tajvidi, 2006; Falk, Hüsler and Reiss, It leads to what we call generalized Pareto processes. As in the finite dimensional context it is convenient to study first generalized Pareto processes in a standardized form. This is done in Section 2. The general process is discussed in Section 3 and the domains of attraction in Section 4. In finite dimensional space the peaks-over-threshold method for estimating distribution tails is well known see e.g. Coles 2001 book, Chapters 4 and 8. In the same vein, in Section 5, we show that by using the stability property of generalized Pareto processes one can create extreme storm fields starting form independent and identically observations of storm fields. A note on notation. Operations like w 1 + w 2 or w 1 w 2 with w 1,w 2 CS mean respectively {w 1 s + w 2 s} and {w 1 s w 2 s}. Then with abuse of notation, operations like w + x or w x with w CS and x R mean respectively {ws + x} and {ws x}. Similarly for 1/γ products and powers. Then e.g. we shall simply write 1+γ X bt at for { } 1/γs 1+γs Xs bst a st, when X = {Xs}, at = {a s t}, bt = {b s t} and γ = {γs}. 2

3 2 The simple Pareto process Let C + S be the space of non-negative real continuous functions in S, with S some compact subset of R d. We denote the Borel subsets of a metric space by B. Theorem 2.1. Let W be a stochastic process in C + S and ω 0 a positive constant. The following three statements are equivalent: 1. POT - peaks-over-threshold - stability a EWs/sup Ws > 0 for all s S, b P sup Ws/ω 0 > x = x 1, for x > 1 standard Pareto distribution, c P ω 0 W sup Ws B sup Ws > r = P for all r > ω 0 and B B C+ ω0 S with 2. Random functions a P sup Ws ω 0 = 1, ω 0 W sup Ws B, 2.1 C ω + 0 S := {f C + S : supfs = ω 0 }. 2.2 b EWs/sup Ws > 0 for all s S, c PW ra = r 1 PW A, 2.3 for all r > 1 and A B C ω + 0 S, where ra means the set {rf,f A}, and C ω + 0 S := {f C + S : supfs ω 0 } Constructive approach Ws = YVs, for all s S, for some Y and V = {Vs} verifying: a V C + S is a stochastic process verifying sup Vs = ω 0 a.s., and EVs > 0 for all s S, b Y is a standard Pareto random variable, F Y y = 1 1/y, y > 1, c Y and V are independent. Definition 2.1. The process W characterized in Theorem 2.1, with threshold parameter ω 0, is called simple Pareto process. The probability measure in 2.1 i.e., QB = P is called the spectral measure. ω 0 W sup Ws B, for B B C+ ω0 S, 2.5 3

4 Proof of Theorem 2.1. We start by proving that 1. implies 3. By compactness and continuity, sup Ws < a.s. Take: Y = sup Ws ω 0 W and V = ω 0 sup Ws. Then a, b and c are straightforward. Next we prove that 3. implies 2. Let { } A r,b = f C + ω 0 f S : supfs/ω 0 > r, sup fs B = r A 1,B, for all r > 1 and B B C+ ω0 S. Then, P W A r,b = P supws/ω 0 > r, ω 0 W sup Ws B = PY > r, V B = PY > rpv B = 1 sup r P ω 0 W Ws/ω 0 > 1, sup Ws B = 1 r P W A 1,B using in particular the independence of Y and V and P sup Ws/ω 0 > 1 = 1. Since PtA = t 1 PA holds for any of the above sets, it holds for all Borel sets in the statement. Finally, check that 2. implies 1. For any r 1, sup Ws P = P ω 0 > r sup Ws ω 0 W > r, ω 0 sup Ws C ω + 0 S = 1 r P C+ ω0 S = 1 r and also for any B B C+ ω0 S, P supws/ω 0 > r, = 1 sup r P Ws/ω 0 > 1, ω 0 W sup Ws B ω 0 W sup Ws B = 1r P ω 0 W sup Ws B since sup Ws > ω 0 holds a.s. That is, it follows that sup Ws/ω 0 is univariate Pareto distributed and, sup Ws and W/sup Ws are independent. The following properties are direct consequences: Corollary 2.1. For any simple Pareto process W, the random variable ω 1 0 sup Ws has standard Pareto distribution. 4

5 Corollary 2.2. W C + S is a simple Pareto process if and only if any of the two equivalent statements hold: 1. a EWs/sup Ws > 0 for all s S, b P sup Ws/ω 0 > x = x 1, for x > 1, c P W ra sup Ws > rω 0 = P W A 2.6 for all r > 1 and A B C + ω 0 S. 2. a EWs/sup Ws > 0 for all s S, b P Ws sup > r, ω 0 for all r > 1 and B B C+ ω0 S. ω 0 W sup Ws B = QB, 2.7 r From 2.6 we see that the probability distribution of W serves in fact as the exponent measure in max-stable processes cf. de Haan and Ferreira 2006, Section 9.3. Characterization 2. suggests ways for testing and modelling Pareto processes. Let w,w C + S. Conditions W >w define the sets {f C + S : fs >ws for all s S} and W w defines the set {f C + S : fs > ws for at least one s S}; in the later if additionally inf ws > ω 0 then {sup f C + S : fs > ws for at least one s S} > ω 0. Note also that W > w is not the complement of W w. Proposition 2.1. Distribution functions Let w,w C + S, with W simple Pareto process. Then, Vs Vs PW w = E sup E sup 2.8 ws ω 0 ws with the difference interpreted as zero if inf ws = 0. In particular, PW ω 0 = 1. Corollary 2.3. For all w C + S such that sup s R ws > ω 0, PW > w = E inf Corollary 2.4. For x R, Vs = ω 0 E ws inf Ws/ws. 2.9 sup Ws P Ws > x for all s S W > ω 0 = ω 0 x, x > ω 0, 2.10 P Ws > x Ws > ω 0 = ω 0 x, x > ω 0, for all s S

6 Relation 2.8 is an analog of Definition 2.1 in Rootzén and Tajvidi 2006, and also 2.11 is in agreement with their results on lower-dimensional distributions. Proof of Proposition 2.1. P sup Ws ω 0 = 1 implies that, for all 0 ε < ω 0 there exists s S such that Ws > ε with probability one, which implies, for w C + S such that inf ws = 0, PW w = P {f C + S : fs > ws for at least one s} = 1. Hence from now on take w such that inf ws > 0. If even inf ws ω 0, ws Vs PW w = PYV w = P Y inf = 1 E sup 2.12 Vs ws hence, Vs PW w = E sup = ω 0 E ws sup Ws/ws sup Ws On the other hand, the homogeneity property 2.3 allows us to extend the probability measure of W that lives on the space C + ω 0 S to all of C + S. We call the resulting measure ν. It plays the same role as the exponent measure ν in the theory of max-stable processes cf. de Haan and Ferreira 2006, Section 9.3. Then, for B C + S, PW B = ν{f C + S : f B,supfs > ω 0 } In order to determine PW w for any w C + S we use 2.13, 2.14 and Theorem 2.1: ν { f C + S : f w } { } ω 0 = inf ws ν f C + wω 0 S : f inf ws ω 0 = inf ws P wω 0 Vs W = E sup, inf ws ws hence, PW w = ν { f C + S : f w,f ω 0 } = ν { f C + S : f w } +ν { f C + } { S : f ω 0 ν f C + } S : f w ω 0 Vs Vs = E sup +1 E sup. ws ws ω 0 Proof of Corollary 2.3. Similar to

7 Proof of Corollary 2.4. Relation 2.10 is direct from Corollary 2.3. For 2.11: for x R, P Ws > x = P YVs > x = P Y > x = E Vs x 1 = Vs {vs/x 1} which simplifies to x 1 EVs for x > ω 0. The rest follows. Finally, we give two simple examples. vs x df Vs +P Vs > x Example One of the simplest process is when W Y, that is a Pareto random variable governing through the whole space. Then Q concentrates on the constant function w 1. Some examples of probabilities in space are then, for all w with inf s R ws > 1, PW w = P Y inf ws 1 = 1 s R inf s R ws and, for w with sup s R ws > 1, 1 PW > w = P Y > supws = s R sup s R ws. Of course all the univariate marginals are standard Pareto. 2. Another simple situation is with {Ws} s R = {Y vs} s R with {vs} s R some positive deterministic continuous function with finite supremum for instance suppose some behaviour described by the curve v subjected to some independent random impact Y. Then we have, for w such that inf s R ws/vs > 1, PW w = P Y inf s R and, for w such that sup s R ws/vs > 1, ws PW > w = P Y > sup s R vs ws ws = 1 sup vs s R xs vs = inf s R ws. The univariate marginal distributions are Pareto vs, i.e. F Ws y = 1 vs/y, y > vs, s R. For example, contour levels ws = cvs for some constant c > ω 0, have exceedance probabilities PW > w also governed by a univariate Pareto distribution. To end this section, we link the finite dimensional distributions of the simple Pareto process with the finite dimensional distributions of simple max-stable processes. For completeness we define max-stable processes and review a representation for these processes. 7

8 Definition 2.2. A process η = {ηs} s R CR with non-degenerate marginals is called max-stable if, for η 1,η 2,..., i.i.d. copies of η, there are real continuous functions c n = {c s n} s R > 0 and d n = {d s n} s R such that, η i d n d max = η for all n = 1,2,... 1 i n c n It is called simple if its marginal distributions are standard Fréchet, and then it will be denoted by η. Proposition 2.2 Penrose1992. All simple max-stable processes in C + R can be generated in the following way. Consider a Poisson point process on 0, ] with mean measure r 2 dr. Let {Z i } i=1 be a realization of this point process. Further consider i.i.d. stochastic processes V 1,V 2,... in C + R with EV 1 s = 1 for all s R and Esup s RVs <. Then η = d max i=1,2,... Z i V i. Conversely each process with this representation is simple max-stable and one can take V such that sup s RVs = c a.s. with c > 0. The spectral measure of the max-stable process is defined as the probability measure associated to V. The process V for max-stable processes plays the same role as the process V for Pareto processes in Theorem 2.1. The connection between finite dimensional distribution functions is discussed next. The finite dimensional distributions of η where computed in de Haan From the given representation we have, for s 1,...,s n R, x 1,...,x n R, for all n N, Gx 1,...,x n = P ηs 1 x 1,..., ηs n x n = exp E max 1 i n Vs i 2.15 Nownotethat2.12 particularizedtofinite-dimensions give, fors 1,...,s n R, x 1,...,x n R, with x i > ω 0, i = 1,...,n, PWs 1 x 1,...,Ws n x n = PYVs 1 x 1,...,YVs n x n x i Vs i = P Y min = 1 E max i n Vs i 1 i n x i MoregenerallyforK 1,K 2,...,K d compactsubsetsofrandx 1,x 2,...,x d > ω 0, P Ws x i, for all s K i,i = 1,2,...,d = 1 E max x 1 i sup Vs. 1 i d s K i Comparing simple Pareto and simple max-stable processes, if the two processes have the same spectral measure, 2.16 equals 1+logG, i.e. the typical distribution-relation between extreme value and Pareto distributions holds in the given region. The definition of a multivariate Pareto distribution in Michel x i. 8

9 2008 cf. also Falk et al is: any multivariate distribution function that can be represented by 1 + logg, with G any multivariate extreme value distribution, in a neighborhood of the right endpoint of G. Hence we cover this situation. From the finite dimensional distributions one may check that independence in the Pareto process among any points is impossible. For any two points s 1,s 2 R, PWs 1 > c,ws 2 > c = c 1 EVs 1 Vs 2, PWs i > c = EVs i /c > 0, i = 1,2, and EVs 1 Vs 2 = EVs 1 EVs 2 /c > 0, for all c > ω 0 is impossible. All the above results extend in a natural way to R := [,+ ]. 3 The generalized Pareto process Let CS be the space of real continuous functions on S with S R d compact. The more general processes with continuous extreme value index function γ = {γs}, location and scale functions µ = {µs} and σ = {σs} is defined as: Definition 3.1. Let W be a simple Pareto process, µ,σ,γ CS with σ > 0. The generalized Pareto process W µ,σ,γ CS is given by, W µ,σ,γ = µ+σ Wγ 1 γ 3.1 with all operations taken componentwise recall the convention explained in the end of Section 1. The correspondent to Corollary 2.1 is, Corollary 3.1. The random variable sup { 1+γs Wµ,σ,γs µs σs has standard Pareto distribution. Related with stability or homogeneity properties we have: 1/γs } ω 1 0 Proposition 3.1. For any Pareto process W µ,σ,γ, P 1+γ W 1/γ µ,σ,γ µ ra = r 1 P 1+γ W 1/γ µ,σ,γ µ A, σ σ 3.2 for all r > 1 and A B C ω + 0 S. Moreover, there exist normalizing functions ur and sr such that P 1+γ W 1/γ µ,σ,γ ur A sup 1+γ W 1/γ µ,σ,γ ur > ω 0 sr sr = P 1+γ W 1/γ µ,σ,γ µ A, 3.3 σ 9

10 for all r > 1 and A B C + ω 0 S. Conversely, 3.3 and sup { 1+γs Wµ,σ,γs µs σs 1/γs } ω 1 0 being standard Pareto distributed, imply 3.2. Proof. Relation 3.2 is direct from Definition 3.1 and 2.3. Then, with ur = µ+σr γ 1/γ and sr = σr γ and, for all r > 1 and A B C + ω 0 S, P 1+γ W 1/γ µ,σ,γ ur A 1+γ W 1/γ µ,σ,γ ur ω 0 sr sr = P 1+γ W 1/γ µ,σ,γ µ ra 1+γ W 1/γ µ,σ,γ µ rω 0 σ σ 1/γ P 1+γ Wµ,σ,γ µ σ ra = 1/γ = P 1+γ W 1/γ µ,σ,γ µ A P 1+γ Wµ,σ,γ µ σ rω0 σ by 3.2 and Corollary 3.1. Conversely, for all r > 1 and A B C + ω 0 S, 1/γ P 1+γ Wµ,σ,γ ur sr A 1/γ P sup s R 1+γ Wµ,σ,γ ur sr ω 1 0 > 1 1/γ P 1+γ Wµ,σ,γ µ σ ra = r 1 = P 1+γ W 1/γ µ,σ,γ µ A σ { } 1/γs by3.3andsup 1+γs Wµ,σ,γs µs σs ω0 1 being standard Pareto distributed. Example 3.1. Relation 3.3 may be viewed as the correspondent to the stability property in Rootzén and Tajvidi 2006, cf. their relation 6. E.g. in the simple case one checks that P Wsi ω 0 1 t 1 for all x i, i = 1,...,n. t x i ω 0,i = 1,...n Wsi ω 0 Wsi = P ω 0 1 t 1 t 1 x i ω 0,i = 1,...n 0,i = 1,...n, 10

11 The corresponding to Proposition 2.1 on distribution functions is now: PW µ,σ,γ w = E sup Vs 1+γs ws µs 1/γs 1 ω 0 s R σs { E supvs 1+γs ws µs } 1/γs, s R σs for 1 + γw µ/σ C + S and the difference taken as zero if inf {1+γsws µs/σs} = 0. 4 Domain of attraction The maximum domain of attraction of extreme value distributions in infinitedimensional space has been characterized in de Haan and Lin2001. This result leads directly to a characterization of the domain of attraction of a generalized Pareto process. Let CS be the space of real continuous functions in S, with S R d some compact subset, equipped with the supremum norm. The convergences below d denote weak or convergence in distribution. Denote by η = { ηs} any simplemax-stableprocessinc + Scf. Definition2.2. Anymax-stableprocess η = {ηs} in CS can be represented by η = η γ 1/γ, for some η and continuous function γ = {γs}. For simplicity we always take here C 1 + S = {f C+ S : supfs 1}, i.e. consider the constant ω 0 introduced in Section 2 equal to 1. The maximum domain of attraction condition in CS can be stated as: Condition 4.1. For X,X 1,X 2,... i.i.d. random elements of CS, there exists a max-stable stochastic process η CS with continuous index function γ, and a s n > 0 and b s n in CS such that { max 1 i n X i s b s n a s n } d {ηs} 4.1 on CS. The normalizing functions are w.l.g. chosen in such a way that logpηs x = 1+γsx 1/γs for all x with 1+γsx > 0, s S. Next is an equivalent characterization of the domain of attraction condition, in terms of exceedances. Suppose the marginal distribution functions F s x = PXs x are continuous in x, for all S. To simplify notation write the normalized process, η t = 1+γ X bt 1/γ, at + for some normalizing functions in CS, at = {a s t} > 0 and bt = {b s t}. 11

12 Condition 4.2. For X CS suppose, for some at > 0 and bt in CS, lim tp Xs bs t > x = 1+γsx 1/γs, 1+γsx > 0, 4.2 t a s t uniformly in s, and P sup lim η t s > x t P sup η t s > 1 = 1, for all x > 1, 4.3 x lim P t η t sup η t s B sup η t s > 1 = QB, 4.4 for each B B C+ 1 S with Q B = 0, with Q some probability measure on C + 1 S. Note that this is the same as for max-stable processes; cf. Theorem in de Haan and Ferreira Note also that specify a simple Pareto process in the limit. Theorem 4.1. Conditions 4.1 and 4.2 are equivalent and the corresponding limiting processes share the same spectral measure. Proof. Cf. Theorem in de Haan and Ferreira The normalization there is { } 1 { η t s} =, t1 F s Xs but the results are the same. The following are direct consequences: Corollary 4.1. Any max-stable process verifies Condition 4.2. Corollary 4.2. Relations imply lim P t η t A supη t s > 1 = PW A, with A B C + 1 S, P A = 0 and W some simple Pareto process. The converse statement of Corollary 4.2 is as follows: Theorem 4.2. Suppose that there exists a continuous function bu = { b s u} that is increasing and with the property that PXs > b s u for some s S 0 as u, and a positive continuous function ãu = {ã s u} > 0 such that lim P u X bu A Xs b s u > 0 for some s S = ãu PA, for all A BCS and P A = 0. Then Conditions 4.1 and 4.2 are fulfilled. 12

13 Proof. Bytheconditionson bu, wecandetermineq = qtsuchthatpxs > bs qt for some s S = 1/t. Then with b s t = b s qt and a s t = ã s qt, X bt lim tp C and Xs > b s t for some s S = t at PC, for all C BCS and P C = 0. In particular, if inf{sup fs : f C} > 0 we have X bt lim tp C = t at PC. 4.5 We proceed as usual in extreme value theory. Fix for the moment s S. It follows that for x > 0 lim tp X st > b s t+xa s t = P{f : fs > x}. t Let U s be the inverse function of 1/PXs > x and V s be the inverse function of 1/ P {f : fs > x}. Then U s tx b s t lim = V s x, for x > 0. t a s t It follows that for some real γs and all x > 0 b s tx b s t lim = xγs 1 t a s t γs and a s tx lim t a s t = xγs. 4.6 Since the limit process has continuous paths, the function γ must be continuous on S. Now replace t in 4.5 by ct where c > 0. Then lim tp t hence, by 4.6 and by 4.5 lim tp t lim tp t bs t b s tc a s tc + a st X bt C a s tc at = 1 c PC 1+γ X bt 1/γ c1+γc 1/γ = 1 at c PC Write PA = P A γ 1/γ. Then 1+γ X bt 1/γ 1+γC 1/γ = at PC. lim t tp η t A = PA, withpca = c 1 PA,forallc > 0andA BCSsuchthatinf{sup fs : f A} > 1 and P A = 0. The rest is exactly like the proof of the equivalence between 2b and 2c of Theorem in de Haan and Ferreira

14 By Corollary 3.1 and Proposition 3.1 it follows: Corollary 4.3. Any Pareto process is in the domain of attraction of a maxstable process with the same spectral measure. Example 4.1. The finite dimensional distributions of the moving maximum processes obtained in de Haan and Pereira 2006 can be applied to obtain the finite dimensional distributions of the correspondent Pareto process, in the appropriate region. Example 4.2 Regular variation de Haan and Lin 2001, Hult and Lindskog A stochastic process X in CS is regularly varying if and only if there exists an α > 0 and a probability measure Q such that, P sup Xs > tx, X/sup Xs d x α Q, x > 0, t, P sup Xs > t 4.7 on {f CS : sup fs = 1}. Hence, a regularly varying process such that 4.2 holds for the marginals, verifies Condition 4.2 with γ = 1/α, bt = t and at = t/α; note that the index function is constant in this case. On the other hand, the normalized process tη t or t η t with η t verifying , verifies regular variation with α = 1 and spectral measure Q on C 1 + S. Remark 4.1. Our analysis is also valid - mutatis mutandis - in the finitedimensional set-up. The definition in that case is the same as in Rootzén and Tajvidi The difference is that their analysis is entirely based on distribution functions whereas ours is more structural. Here are some remarks. Let F = 1 F with F some d variate distribution function, x = x 1,x 2,...,x d R d, and u = u 1,u 2,...,u d and σ the normalizing functions considered in Rootzén and Tajvidi 2006 see e.g. their definition of X u. By using σxt/σt x γ1,x γ2,...,x γ d and uxt ut/σt x γ 1 1 γ, xγ γ t, for some reals γ 1,γ 2,...,γ d cf. proof of Theorem 2.1ii and by F x := F u 1 x 1,u 2 x 2,...,u d x d, one simplifies their relation 19 to x γ 1 t F 1 tx logg 1 γ 1 and one simplifies their relation 6 to 2 1,..., xγd d 1, xγ2 γ 2 P X tx X t1 = P X x t for t 1. Hence one can take ut := γ 1 1 γ 1, tγ 2 1 γ 2,..., tγ d 1 γ d in Theorem 2.2 of that paper. γ d,,..., xγ d 1 2 γ d, 14

15 In Theorem 2.2 ii it is not sufficient to require 6 for x > 0. For example the probability distribution in R 2 given by PX > x or Y > y = 1 2 e 2x /2 2 e 2y 0, x y 0, satisfies 6 for x > 0 but not for all x and it is not a generalized Pareto distribution. 5 Application Deltares is an advisory organization of the Dutch government concerningamong others the safety of the coastal defenses against severe wind storms. One studies the impact of severe storms on the coast, storms that are so severe that they have never been observed. In order to see how these storms look like it is planned to produce wind fields on and around the North Sea using certain climate models. These climate models produce independent and identically distributed i.i.d. wind fields similar to the ones that could be observed but that are only partially observed. Since the model runs during a limited time, some of the wind fields will be connected with storms of a certain severity but we do not expect to see really disastrous storms that could endanger the coastal defenses. The question put forward by Deltares is: can we get an idea how the really disastrous wind fields look like on the basis of the observed wind fields? We want to show that this can be done using the generalized Pareto process. Consider i.i.d. continuous stochastic processes {X i s} where S is a compact subset of R d. Suppose that the probability distribution of the process is in the domain of attraction of some max-stable process i.e., there exist functions a s n > 0 and b s n s S such that the sequence of processes { max 1 i n X i s b s n a s n converges to a continuous process, say Y, in distribution in CS. Then Y is a max-stable process. According to Theorem 4.1 the process X 1 is then in the domain of attraction of the corresponding generalized Pareto process i.e. from relations /γs, with R X t := sup lim P t 1+γs Xs bst a st } 1+γ X bt 1/γ R X tb and R X t > x RX t > 1 at = x 1 QB 5.1 for x > 1 and B B C+ 1 S with Q B = 0. This statement leads to a peaks-over-threshold method in this framework: 15

16 Let X 1,X 2,...,X n be i.i.d. and let the underlying distribution satisfy the conditions above. Select from the normalized processes { 1+γ X bt } 1/γ n at i=1 those that satisfy X i s > b s t for some s S i.e. for which R Xi t > 1. Here t is a large value and of course b s t can be taken as inf{y : PX 1 s > y 1/t}. Let us denote the selected normalized processes as {X 1 j } r j=1 They are still i.i.d. and follow approximately the distribution given by 5.1: P X 1 j R 1 X tb and R 1 j X t > x = x 1 QB. j This means that the processes {X 1 j } r j=1 are approximately generalized Pareto processes. Next, in order to use Proposition 3.1, we multiply these processes by alarge factor t 0. Define for j = 1,2,...,r, X 2 j s := t 0 X 1 j s for s S. This brings the processes to a higher level without changing the distribution essentially by the homogeneity property. Finally we undo the normalization and define for j = 1,2,...,r γs X 2 X 3 j s 1 j s := a s t +b s t γs for s S. We claim that the processes {X 3 j } r j=1 are peaks-over-threshold processes with respect to a much higher threshold namely b s tt 0 than the processes {X 1 j } r j=1 with threshold b st. In order to prove this we need the following properties cf. relation page 312 of the Haan and Ferreira 2006, b s tx b s t lim = xγs 1 t a s t γs 5.2 and a s tx lim t a s t = xγs 5.3 uniformly for s S and locally uniformly in x. The derivation is straightfor- 16

17 ward: X 3 btt 0 P att 0 [t 0 at = P att 0 [ t γ 0 P t γ 0 A 1+γ X1 bt at γ 1+γ X1 bt at 1+γ X1 bt at = P γ X 1 bt = P A. at γ t γ 0 1/γ ] γ 1 ] 1 + t γ 0 1 γ + t γ 0 1 γ + bt btt 0 att 0 A A A Finally we comment that this is the third application of the peaks-overthreshold method. The other two are: estimation of the exceedance probability of a high level and high quantile estimation Coles, Simulations We exemplify the lifting procedure with the process Xs = Zs γs, with γs = 1 s1 s 2, s [0,1], and Z is the moving maximum process with standard gaussian density. The Z process can be easily simulated in R-package due to Ribatet In Figures 1 2 are represented the 11 out of 20 of these processes for which R Xi t > 1 i.e. {X 1 j } 11 j=1 processes {X 3 j } 11 j=1 with t 0 = 10. and the corresponding lifted Acknowledgements. Research partially supported by FCT Project PTDC /MAT / /2009. References [1] Balkema, A.A. and de Haan, L Residual life time at great age. Ann. Probab. 2, [2] Coles, S.2001 An Introduction to Statistical Modeling of Extreme Values, Springer. [3] Falk, M., Hüsler, J. and Reiss, R.-D Laws of Small Numbers: Extremes and Rare Events, Birkhäuser, Springer. 17

18 x x s s Figure 1: a {X 1 j } 11 j=1 obtained from moving maximum process with standard gaussian density; b the corresponding lifted processes {X 3 j } 11 j=1 with t 0 = 10 x1 and x s Figure 2: All processes: {X 1 j } 11 j=1 - slashed lines - and {X3 j } 11 j=1 - continuous lines 18

19 [4] de Haan, L.1984 A spectral representation for max-stable processes. Ann. Prob. 12, [5] de Haan, L. e Pickands, J Stationary min-stable stochastic processes. P robab. Th. Rel. Fields 72, [6] de Haan, L. and Ferreira, A Extreme Value Theory: An Introduction. Springer, Boston. [7] Hult, H. and Lindskog, F Extremal behaviour of regularly varying stochastic processes. Stoch. Proc. Appl. 1, [8] Pickands, J. III 1975 Statistical inference using extreme order statistics. Ann. Statist. 3, [9] de Haan, L. e Pereira, T.T Spatial Extremes: the stationary case. Ann. Statist. 34, [10] de Haan, L. and Lin, T On convergence toward an extreme value distribution in C[0, 1]. Ann. Prob. 29, [11] Michel, R Some notes on multivariate generalized Pareto distributions. J. Multiv. Analysis 99, [12] Penrose, M.D Semi-min-stable processes. Ann. Probab. 20, [13] Ribatet, M Package SpatialExtremes [14] H. Rootzén and N. Tajvidi 2006 Multivariate generalized Pareto distributions. Bernoulli 12,