Contributions for the study of high levels that persist over a xed period of time

Contributions for the study of high levels that persist over a xed period of time Marta Ferreira Universidade do Minho, Dep. Matemática, CMAT Luísa Canto e Castro Faculdade de Ciências da Universidade de Lisboa, DEIO-CEAUL Abstract: This work was motivated by a study presented in Draisma [2], in which it is analyzed the extremal behaviour of a series {X i } of daily extreme sea water levels. In its approach, Draisma denes a new series, {Y i }, as the minimum of a xed number of successive high tide water levels. This is obviously a dependent series and we analyze its dependence feature, whether {X i } is an i.i.d. sequence or has some specic dependence structure. We also study its tail behavior in case {X i } is an i.i.d. sequence and assuming that the common distribution of X i is in the domain of attraction of a generalized extreme value distribution. Keywords: Statistic of extremes, Tail index, Extremal index, Stationary sequence. 1. Introduction Extreme Value Theory (EVT has a recognized importance in nowadays society as it is necessary to predict the probability of occurrence of many adverse situations such as oods, wind storms or crash markets. The extremal models for dependent or independent sequences, allow good estimates of the probability of observe great (small values of the variable that we are interested in. Nevertheless, it may happened that the adverse situation comes from the duration of exceedances above (bellow given thresholds in a xed period of time. In Draisma [2], for instance, the main problem is the successive high tide water levels registered on some places of Holland's coast. Its persistence in time can contribute for the sand dunes damage and may give rise to devastating oods. More formally, given a time series of water levels, {X 1,..., X n }, we are interested on the tail behaviour of Y i = min(x i, X i1,..., X is, which is a level that persists for s 1 periods. Our study focus on this new sequence, {Y i }. We start by presenting some basic results of the classical extreme value theory (EVT and some general local and asymptotic dependence conditions that will be used in our study. Section 3 contains the whole study and is divided in two parts. In the rst one (subsection3.1, {X i } is an i.i.d. sequence and we prove that the dependence conditions D(u n and D (u n (Leadbetter [3] both hold for {Y i }. We also show that, if the distribution function (d.f. of X i is in the domain of attraction of a non-degenerate generalized extreme value distribution (GEV then Y i is in the same domain of attraction having a

2 tail index that depends both on the tail index of X i and the xed period of time, s 1. In subsection 3.2, {X i } is a stationary sequence with some specic weak dependence structure. We shall see that these conditions hold for {Y i } whenever they hold for {X i }. In what concerns the tail behaviour of {Y i }, it depends on the d.f. of X i as well as on its dependence feature. 2. Preliminary Results We start with the classical central result in extreme value theory: Theorem 2.1. Let ˆMn = max( ˆX 1,..., ˆX n, where ˆX 1, ˆX 2,... are i.i.d. random variables. If for some constants a n > 0, b n, we have P ( ˆM n a n x b n G(x (1 for some non-degenerate G, then G is one of the three extreme value types distributions Type I (Gumbel : G(x = exp( e x, < x < ; Type II (Fr échet: G(x = Type III (Weibull : G(x = { 0, x 0 exp( x α, for some α > 0, x > 0; { exp( ( x α, for some α > 0, x 0 1, x > 0. Conversely, each d.f. G of extreme value type may appear as a limit in (1. G can be uniquely represented in the following parametric way, known as Generalized Extreme Value function (GEV, G γ (x = exp( (1 γx 1/γ, 1 γx > 0, γ R, which is interpreted as exp( e x if γ = 0. The parameter γ is the tail index with γ > 0 corresponding to the Fréchet case and γ < 0 corresponding to the Weibull case. We say that F is in the domain of attraction of G γ, denoted by F D(G γ, if the limit (1 is veried for some real constants a n > 0 e b n. Another important result follows: Theorem 2.2. Let 0 τ and suppose that {u n } is sequence of real numbers such that n(1 F (u n τ as n. (2 Then, P ( ˆM n u n e τ as n. (3 Conversely, if (3 holds for some τ, 0 τ, then so does (2. Remark 2.1 When (2 holds for some τ 0, u n is said to be a normalized level and it is denoted by u n (τ.

Now we present the denitions of the weak dependence conditions that will be considered. They have in common the fact that dependence vanishes as r.v.'s become more distant in time. Let us start by the simplest type of dependence considered here (Watson [5]: Denition 2.1. The sequence {X i } is m-dependent, if the r.v.'s X i e X j are independent whenever i j > m. It follows the so-called condition D(u n (Leadbetter, et al. [3]. Let {X i } be a stationary sequence, F the common d.f. of the r.v.'s X i and {u n } a given real sequence. For brevity, consider the notation: P (X i1 u n,..., X im u n = F i1,...,i m (u n,..., u n = F i1,...,i m (u n, and P (X i1 > u n,..., X im > u n = P ((X i1,..., X im > u n = F X i 1,...,i m (u n, for any integers i 1 <... < i m. Denition 2.2. Condition D(u n will be said to hold for {X i }, if for any integers 1 i 1 <... < i p < j 1 <... < j p n with j 1 i p l, then where α n,ln F X i 1,...,i p,j 1,...,j p (u n F X i 1,...,i p (u n F X j 1,...,j p (u n α n,l 0 for some sequence l n = o(n. In a D-dependent context, some limiting results of classical EVT for i.i.d. sequences also hold. Extremal Types Theorem and Theorem 2.2 are some examples as we shall see. Theorem 2.3. Let M n = max(x 1,..., X n and {a n > 0}, {b n R} real sequences such that P ( M n a n x b n converges to a non-degenerate function G(x. If D(an x b n holds for each real x, then G(x is a GEV distribution. Non-independency may give rise to a new parameter, θ, known as extremal index: Theorem 2.4. If for any given positive constant τ there is a normalized levels sequence u n (τ such that D(u n (τ holds and for some τ, P ( M n u n (τ converges then P ( M n u n (τ e θτ, for all τ > 0 with θ [0, 1] constant. Denition 2.3 (Leadbetter et al. [3]. {X i } has extremal index θ (0 θ 1 if for each τ > 0 there is a normalized levels sequence u n (τ such that P ( M n u n (τ e θτ, with θ independent from τ. Hence, i.i.d. sequences whose normalized maximum converges have unit extremal index. 3 Another local dependence condition is considered in Leadbetter et al. [3] which bounds the probability of more than one exceedance of u n, on a time-interval of integers as k.

4 Denition 2.4. Condition D (u n will be said to hold for {X i } if lim sup n P ( X 1 > u n, X j > u n k 0. If condition D (u n holds, the exceedances of level u n tend to come out isolated, similar to an i.i.d. behaviour. Leadbetter et al. [3] consider a generalization of Theorem 2.2 in order to apply to stationary sequences under D(u n and D (u n. Theorem 2.5. Let {u n } be a real sequence such that D(u n e D (u n hold for {X i } and 0 τ <. Then P ( M n u n e τ if and only if n(1 F (u n τ, as n. A sequence verifying the conditions of the latest theorem has an unit extremal index. If condition D (u n doesn't hold, then the exceedances of u n tend to cluster. For such sequences, Leadbetter e Nandagopalan [4] stated another local dependence condition, D (u n, weaker than D (u n, that restricts rapid oscillations near high levels. Denition 2.5. Condition D (u n will be said to hold for {X i } if condition D(u n also holds and k n are integers such that and k n, k n α n,ln 0, k n l n /n 0, k n (1 F (u n 0 (4 [n/kn] 1 lim n P ( X 1 > u n, X j u n < X j1 = 0. Conditions D and D may be of great help in what concerns extremal index computation, especially if the transition probability function is known. Theorem 2.6. Let {u n (τ} be a sequence of normalized levels and assume that D(u n (τ and D (u n (τ hold for each τ > 0. If lim P ( X 2 u n (τ X 1 > u n (τ = θ, for some τ > 0, then convergence to θ occurs for all τ > 0 and {X i } has extremal index θ. 3. High levels persisting in time As we have already mentioned, we have an high levels sequence {X i } and we intend to analyze the distribution of the maximum levels that persist in a previous xed amount of time. More precisely, we continue with Draisma [2] approach and consider the statistics Y i = min(x i, X i1,..., X is, corresponding to high levels that persist throghout s 1 time instants and we study the dependence structure and the extremal behaviour of {Y i }.

5 3.1. {X i } i.i.d. If {X i } is an i.i.d. sequence, {Y i } is obviously a stationary and (s 1-dependent sequence. Therefore, condition D(u n holds for all {u n }. Lets see that D (u n also holds for a certain levels choice {u n }. Our assumptions lead us to the following development: lim sup = lim sup = lim sup n i=2 P ( Y 1 > u n, Y i > u n = n P ( X 1 > u n,..., X 1s > u n, X i > u n,..., X is > u n = i=2 s1 n (1 F (u n is n i=2 Taking {u n } such that we have i=s2 (1 F (u n 22s. n 1/(s1 (1 F (u n τ, as n, (6 ( s1 lim sup n (1 F (u n is n i=2 [ ( n 1 i=s2 (1 F (u n 22s = (5 s1(1 = lim sup s1 (1 F (un F 1 (1 F (uns (un F (u n ] s1( (n 1 s1 (1 F (un s 1(1 F (un 1s = (7 = τ 22s k 0, as k. The conclusion is immediate by (5, (6 and (7. Therefore, if we choose levels u n such that (6 holds or, equivalently, such that n(1 F Y (u n τ, as n with F Y the common d.f. of Y 1, Y 2,..., then the maximum of {Y i } has a similar asymptotic behaviour as the one considered in a independent context and θ = 1. Now we focus on the tail index computation. If we consider again u n satisfying (6 and such that u n u n (x = a n x b n, with {a n > 0} and {b n } real sequences, then, taking τ τ(x, we have n 1/(s1( 1 F (a n x b n τ(x, 0 < τ(x <. (8 The i.i.d. assumption for {X i } combined ( with (8 leads to n(1 F Y (a n x b n = n 1/(s1( 1 F (a n x b n s1 τ(x s1. (9 Assume Mn Y = max(y 1,..., Y n. As F D(G γ, there are real sequences {a n > 0} and {b n} such that P ( M n a nx b n G(x as n or, equivalently, n(1 F (a nx b n ln G(x, (10 where G(x is a GEV. We have that conditions D(a n x b n and D (a n x b n both hold for {Y i }. Applying Theorem 2.5 and relation (9 then, as n,

6 P ( M Y n a n x b n e τ(x s1, or, P ( M Y n a n x b n e ( ln G(x s1, (11 since (10 also holds replacing n by n 1/(s1 and considering a [n 1/(s1 ] = a n and b [n 1/(s1 ] = b n. Now, observe that e ( ln G(xs1 = e (1γx (s1/γ = e (1 γ s1 x(s1 1/(γ/(s1. (12 From (11 and (12 we conclude ( P (Mn Y ã n x b n exp ( 1 γ x 1/γ, as n, with γ = γ/(s 1 and ã n = a n /(s 1. Hence F Y D(G γ. 3.2. {X i } stationary with a weak dependence structure In this section we prove that the weak local dependence conditions D, D and D hold for {Y i }, whenever they hold for {X i }, respectively. {Y i } is obviously a stationary sequence due to the denition of Y i and the stationarity of {X i }. We begin by stating the following lemma which is an important tool to prove the above assertion with respect to condition D. Lemma 3.1. Suppose that D(u n holds for {X i } for a given real numbers sequence {u n }, i.e., for any integer sets I = { i 1,..., i p } and J = { j1,..., j p } such that 1 i 1 <... < i p < j 1 <... < j p n and j 1 i p l, (13 we have F X i 1,...,i p,j 1,...,j p (u n F X i 1,...,i p (u n F X j 1,...,j p (u n α n,l with α n,ln 0, (14 for some sequence l n = o(n. Then F X i 1,...,i p,j 1,...,j p (u n F X i 1,...,i p (u n F X j 1,...,j p (u n α n,l (15 with α n,l n 0.

7 Proof. Writing the rst member of (15 using the complementary and after some calculations, we have F X i 1,...,ip,j 1,...,j p (un F X i 1,...,ip (unf X j 1,...,j p (un i I,j J... F X i,j (un F X i (unf X j (un i I i,i I; j J; i<i... j J F X i,j 1,...,j p (un F X i i I; j,j J; j<j (unf X j 1,...,j p (un F X i,i,j (un F X i,i (unf X j i,i I; i<i F i X 1,...,ip,j (un F i X 1,...,ip (unf j X j,j J; j<j... (un F X i,j,j (un F X i F X i,i,j 1,...,j p (un F X i,i (unf X j 1,...,j p (un... (un F X i 1,...,ip,j,j (un F X i 1,...,ip (unf X j,j (un F i X 1,...,ip,j 1,...,j p (un F i X 1,...,ip (unf j X 1,...,j p (un (unf X j,j (un Note that all parcels have r.v.'s that are distant (in time, at least, l integers. So, using (13, we can bound each one of them by α n,l with α n,l verifying the conditions in (14. We have as much parcels as the number of dierent possible combinations of, at least, one element from I with, at least, one element from J. Denoting this number by c p,p we have then F X i 1,...,i pj 1...j (u p n F X i 1,...,i p (u n F X j 1,...,j (u p n αn,l, (16 with α n,l = c p,p α n,l 0, as n. Proposition 3.2. Under the assertions of Lemma 3.1, condition D(u n holds for {Y i }. Proof. Considering the notation stated in Section 2, we have that F Y i 1,...,i m (u n = P ((X i1,..., X i1 s,..., X im,..., X ims > u n = F X i 1,...,i 1 s,...,i m,...,i ms(u n (17 As D(u n holds for {X i } then (13 also holds for some l (X = o(n. Consider any integer sets I = {i 1,...i p } and J = {j 1,...j p } such that, for a given n, we have 1 i 1 <... < i p < j 1 <... < j p n, and j 1 (i p s l (X (i.e., j 1 i p l (X s =: l (Y. An analogous reasoning used in the proof of Lemma 3.1 and an application of (17, lead us to

8 F i Y 1,...,ip,j 1,...,j p (un F i Y 1,...,ip (unf j Y 1,...,j p (un F X i,...,is,j,...,js (un F X i,...,is (unf X j,...,js (un i I,j J i I; j<j J... F X i,...,is,j,...,js,j,...,j s (un F X i,...,is (unf X j,...,js,j,...,j s (un F X i,...,is,j 1,...,j p s (un F X i,...,is (unf X j 1,...,j p s (un i I i<i I; j J...... F X i,...,is,i,...,i s,j,...,js (un F X i,...,is,i,...,i s (unf X j,...,js (un F X i,...,is,i,...,i s,j 1,...,j p s (un F X i,...,is,i,...,i s (unf X j 1,...,j p s (un i<i I F X i 1,...,ips,j,...,js (un F X i 1,,...,ips (unf X j,...,js (un j J j<j J... F X i 1,...,ips,j,...,js,j,...,j s (un F X i 1,...,ips (unf X j,...,js,j,...,j s (un F X i 1,...,ips,j 1,...,j 1 s,...,j p s (un F X i 1,...,ips (unf X. j 1,...,j p s (un Note that, in each parcel, r.v.'s distant, at least, l (X integers from each other and so, we can bound them by α. Note also that the number of parcels is exactly the same n,l (X obtained in Lemma 3.1 proof, i.e., c p,p. Therefore, Fi Y 1,...,i p,j 1,...,j (u p n Fi Y 1,...,i p (u n Fj Y 1,...,j (u p n α where, by (16, we have then with l (Y = l (X s = o(n. n,l (Y, α = c n,l (Y p,p α = c 2 n,l (X p,p α n,l (X 0, (18 Proposition 3.3. If condition D (u n holds for {X i }, then it also holds for {Y i }. Proof. Assuming that condition D (u n holds for {X i }, we have that lim sup n P ( Y 1 > u n, Y j > u n = = lim sup n P ( (X 1,..., X 1s, X j,..., X js > u n [ ( = lim sup n P (X2,..., X 1s, X j1,..., X js > u n ( ] (X 1, X j > u n P (X1, X j > u n lim sup n P ( X 1 > u n, X j > u n = 0, as k and so, it also holds for {Y i }.

Proposition 3.4. Suppose that conditions of Lemma 3.1 and condition D (u n hold for {X i }, for some integers k n satisfying (4. Then condition D (u n also holds for {Y i }. Proof. By Proposition 3.2 we conclude that {Y i } veries condition D(u n too and, by (18 and (4 it is immediate that k n, k n α n,l (Y n 0, k n l n (Y /n 0. Considering again (4, then k n (1 F Yi (u n = k n P ( (X i,..., X is > u n = k n P ( (X i1,..., X is > u n X i > u n P ( Xi > u n k n P ( X i > u n 0. An analogous reasoning used in Proposition 3.3 lead us to lim 1 n Hence D (u n holds for {Y i }. P ( Y 1 > u n, Y j u n < Y j1 = 0 9 Remark 3.1 The tail behaviour of {Y i } depends on the distribution of X i as well as on its dependence structure.

References [1] Canto e Castro, L. (1992. Sobre a Teoria Assintótica de Extremos. Ph. D. Thesis, FCUL. [2] Draisma, G. (2001. Duration of extremes at sea. Em Parametric and semi-parametric methods in E.V.T., p. 137-143. Ph. D. Thesis, Erasmus University. [3] Leadbetter, M.R., Lindgren, G. e Rootzén, H. (1983.. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New-York, Heidelberg-Berlin. [4] Leadbetter, M.R. e Nandagopalan, S. (1989.. On Exceedance Point Processes for Stationary Sequences under Mild Oscillation Restrictions. Em Extreme Value Theory. (J. Hüsler and R.-D. Reiss, eds., p. 69-80. Springer-Verlag. [5] Watson, G.S. (1954.. Extreme Values for Approximate Independence of Largest and Smallest Order Statistics. J.Amer.Statist.Assoc., Vol.65, p. 860-863. 11