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Tail inference for a law in a max-semistable domain of attraction Abstract Max-semistables were introduced as the limit laws for the maximum term in samples with sizes growing geometrically with ratio r ≥ 1. They have a log-periodic component and can be characterized by the period therein (which is ln r), by a tail index parameter and by a real function y representing a repetitive pattern. The applicability of max-semistable laws to real data has been argued with the fact that these distribution functions can appear as the maximum of an independent and identically distributed sequence of a Poissonian number of “local events” (that is, a clustering of “microevents”) with different scales. On the other side, log-periodicity can be related with discrete scale invariance and seems to be present in physical data associated with turbulence and seismology, and in financial data. Statistical inference in max-semistable domains of attraction can be performed through convenient sequences of generalized Pickands statistics, {Zs }, where s is a tuning parameter for the choice of the number of upper order statistics. More precisely, we can use the fact that these kind of sequences converge in probability when s equals r (or any of its integer powers) and has an oscilatory behavior otherwise, in order to obtain an estimate of the period. Estimators for the tail index and for the function y are, then, easily derived. Simulation studies indicate that the estimators perform well in most cases but there is a need of larger sample sizes than in max-stable contexts. The methodology presented is applied to a real data set consisting in seismic moments of major earthquakes in the Pacific Region. 1 1 Introduction We say that a d.f. F belongs to the domain of attraction of a max-stable d.f. G, and we write F ∈ M S(G), if and only if there exist real sequences {an } e {bn }, with an > 0, such that lim F n (an x + bn ) = G(x), x ∈ R. n→+∞ The most common continuous d.f.’s belong to this class, although the same doesn’t happen to a wide variety of multimodal continuous d.f.’s as well as to discrete d.f.’s. However, some of those d.f.’s belong to a new class the one of max-semistable d.f.’s. We say that a d.f. F belongs to the domain of attraction of a maxsemistable d.f. G, and we write F ∈ M SS(G), if and only if there exist real sequences {an } e {bn }, with an > 0, such that lim F kn (an x + bn ) = G(x), x ∈ CG n→+∞ (1) where CG is the set of continuity points of the function G and {kn } is a non decreasing sequence that verifies the following geometric growing condition kn+1 = r ≥ 1 (< ∞). n→+∞ kn lim (2) Notice that in the case of r = 1 we are in the particular case of the maxstable laws. The pionieres in the study of max-semistable laws were Grinevich [5], [6] and Pancheva [9]. Grinevich proved that the limit d.f. G is max-semistable if and only if there exist r > 1, a > 0 and b ∈ R such that G is solution of the functional equation G(x) = Gr (ax + b), x ∈ R. (3) This author obtained explicit expressions for the max-semistable d.f.’s G, proving that if a d.f. G is max-semistable then it is of the same type as some element of the following family of distribution functions: 2 ³ ´ ³ ´ −1/γ −1/γ exp − (1 + γx) ν ln (1 + γx) 1 + γx > 0, γ = 6 0 Gγ,ν (x) = 1I]−∞,0[ (γ) 1 + γx ≤ 0, γ = 6 0 , exp (−e−x ν(x)) γ = 0, x ∈ R where ν is a positive, bounded and periodic function with period p = log r. The parameters r and γ are related with parameters a and b in (3), the following way: a = rγ and b = log r if γ = 0. In current language, we can say that a max-semistable d.f. is a maxsable d.f. perturbed by a log-periodic function. This relation can be see easily seen when we construct a QQ-plot of a max-semistable d.f. against a max-stable one, both with the same parameter γ. In fact, as it is easily derived, the respective quantiles, y and x, are related through the equations y = x [ν(log x)]−γ , γ 6= 0, and y = x ν(x), γ = 0, meaning that its graph is a log-periodic (periodic) curve along the straight line y = x, when γ 6= 0 (γ = 0). Figure 1 shows the QQ-plot of the max-semistable d.f. F (x) = exp{−x−2 (8 + cos(4π log x))}, x > 0, against the max-stable d.f. G(x) = exp(−x−2 ), x > 0. 5 10 15 20 25 30 Figure 1: QQ-Plot of the d.f. F (x) = exp{−x−2 (8 + cos(4π log x))} against the d.f. G(x) = exp(−x−2 ). 3 A new characterization of the limit functions G was proposed by Canto e Castro et al. [2]. Assuming, without lost of generality, that G(0) = e−1 G(1) = exp(−r−1 ) , (4) G continuous in x = 0 this authors obtained the following representation − log(− log G(an x + sn )) = n log r + y(x), ∀ x ∈ [0, 1], n ∈ Z, (5) where the function y : [0, 1] → [0, log r] is non decreasing, right continuous and continuous at x = 1, and sn = (an − 1)/(a − 1) if a 6= 1 and a > 0 or sn = n if a = 1. ESTE É O TEOREMA QUE FALTAVA ACRESCENTAR. With this representation those authors obtained a new characterization of the max-semistable domains of attraction not dependent from the existence of some distribution function F0 in the max-stable domain of attraction. Theorem 1. Let Gγ,ν be a max-semistable satisfying (4). In order that () holds for some distribution function F and some sequence {kn } satisfying (2), it is necessary and sufficient that there exists a real sequence {bn } such that bn+1 − bn lim = a, n→+∞ bn − bn−1 lim U (bn+1 ) − U (bn ) = log r, n→+∞ (6) and lim U (bn + x(bn+1 − bn )) − U (bn ) = y(x), n→+∞ for all continuity points x of y in [0, 1]. Furthermore, the convergence (1) is verified with an := bn+1 − bn and kn = [eU (bn ) ]. This characterization was used in the development of parameter estimators by Canto e Castro and Dias [4]. In fact, those estimators were based on the sequence of statistics, suggested by Temido [18], Zs (m) := X(m/s2 ) − X(m/s) , X(m/s) − X(m) 4 where X(m) := XN −[m]+1,N are the order statistics of an independent and identically distributed random sample of size N drawn from a population distributed as X and m := mN is a intermediate sequence (that is, mN is a integer sequence verifying mN → +∞ and mN /N → 0). The analysis of the asymptotic behavior of this sequence when the d.f. of X is in a maxsemistable domain of attraction was done in Dias and Canto e Castro [3]. It was proved that it converges in probability to ac if and only if s = rc , c ∈ N. Furthermore, if s 6= rc , c ∈ N, then Zs (m) as an oscillatory behaviour. This results allows to prove that if s = rc , c ∈ N, then the following sequences of statistics converge in probability to some constant, namely: • Rs (m) := Zs2 (m) P −→ 1, n → +∞; (Zs (m))2 • γ bs (m) := log (Zs (m)) P −→ γ, n → +∞. log s Based on this results, to estimate r, the authors proposed a heuristic method which uses the sequence of statistics Rs (m), considering as an estimate of r the value ( ) rb = mode arg max Bs (²), ² = 0.01, (0.01), 0.1 s=1.1,(0.1),3.0 where k 1X Bs (²) := 1I (i) (m(i) ) (i) 2 k i=1 {m ∈An :(Rs (m )−1) <²} (that is, Bs (²) is the percentage of time that the sequence of statistics Rs spends in a ² neighborhood of 1), An = {m(1) , m(2) , ..., m(k) } is a set of suitable values of m and k = #An . Having obtained an estimate of r, γ can also be estimated: γ b= 1 1 X log Zrb(m). log rb k m∈A n 5 2 The estimation of the function y In this section we propose a method to estimate the function y. This method is based in the characterization of the limit d.f. G obtained by Canto e Castro et al. [2]. Combining the estimator of y with the estimators of the parameters r and γ we can characterize completely the d.f. G. Let Xi,n be the i th ascending order statistic from a i.i.d. sample of size n with common d.f. F and consider the empirical distribution function Fn . Notice that it is possible to establish a characterization similar to (5) for the general case of a function G, continuous at 0 but not satisfying the other two conditions in (4), taking into account that, by (3), we have − log(− log G)(ax + b)) = − log(− log G(x)) + log r. With z0 = − log(− log G(0)) we obtain − log(− log G(b)) = z0 + log r and, more generally, − log(− log G(an x + s∗n )) = y ∗ (x) + n log r, ∀x ∈ [0, b], where y ∗ : [0, b] → [z0 , z0 +log r] is the restriction of the function − log(− log G) to [0, b] and s∗n = sn b. In fact, this type of characterization can be achieved starting the process in any continuity point x0 . To simplify the notation we denote by y any of the functions y ∗ that can appear in the previous characterization. Due to (5) if we decompose the image set of the function − log(− log G) into disjoint intervals of length log r, after a suitable change of location and scale in each correspondent segment of the function’s domain, the function y is obtained. Then, in a first step, we apply the same decomposition to the empirical function Un = − log(− log Fn ), starting the construction in the right tail. Indeed we use the maximum, Xn,n , as the upper bound of the first segment. Therefore, the first segment has domain [Xm1 ,n , Xn,n ] where Xm1 ,n := max{Xl,n : Un (Xn,n ) − Un (Xl,n ) ≥ log r}. Recursively, we obtain the domain of the i-th segment, [Xmi ,n , Xmi+1 ,n ], i = 1, ..., k where k is the total number of segments, with Xmi ,n := max{Xl,n : Un (Xmi−1 ,n ) − Un (Xl,n ) ≥ log r}, 6 log r log r log r X 3 (k/r ) X X 2 (k/r) (k/r ) X(k) Figure 2: Decomposition of the right tail of the empirical function Un = − log(− log Fn ). (see Figure 2 for an illustration) In a second step, we obtain an equivalent representation of Un (changing scale and location) which will be called empirical versions of y. In order to unify the methodology, the change of location and scale is such that each empirical version of y has domain [0, 1] and image set [0, log r]. Consequently, for each Xl,n ∈ [Xmi ,n , Xmi−1 ,n ], with l ∈ {1, ..., n}, we consider the jump point of the i th empirical version of y given by (i) tl−mi +1 = Xl,n − Xmi ,n ∈ [0, 1], Xmi−1 ,n − Xmi ,n whose images are (i) yl−mi +1 = Un (Xl,n ) − Un (Xmi ,n ) ∈ [0, log r]. Thus, the i-th empirical version of y is defined by y (i) (x) = ni X j=1 (i) yj 1I[t(i) , t(i) [ (x), x ∈ [0, 1], j j+1 (i) where ni is the number of jump points tj (see Figure 3). The k empirical versions of the function y obtained in the second step (i) allow us to estimate y in the grid {tj , i = 1, 2, ..., k, j = 1, 2, ..., ni }, using the following method: 7 1 (5) y (6) y (7) y y(8) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Figure 3: Empirical versions of the function y. (i) • for each tj ∈ [0, 1], with i = 1, 2, ..., k and j = 1, 2, ..., ni , we calculate (i) the values y (l) (tj ), with l = 1, ..., k; (i) • the estimates of y in tj are equal to the sum, for each l, of the values obtained in the previous step weighted by the percentage of jump points that each empirical versions has (taking into account that the empirical version of the function y will be closer to the function y as the number of jump points increases, so that points from empirical versions close to y have more weight in the construction of the estimated function y). Namely, considering the total number of points P (i) tj given by nt = kl=1 nl , we define (i) yb(tj ) = k X nl l=1 nt (i) y (l) (tj ). Using these points we apply linear interpolation taking into account that the function yb must be non decreasing. Eu, LUISA, ACHO QUE ESTE PARAGRAFO É DESNECESSÁRIO (Next, we need to evaluate the goodness of fit of yb. As previously said, the function y is not unique since it depends on the point used to obtain a characterization analogous to (5). Then to choose the best version of y to be compared with yb is another problem in this context. Moreover, since 8 we can obtain an estimate of r or an integer power of r, the analysis of the goodness of fit of yb becomes a difficult task. We suggest a way to contour this problem, using hight quantiles, based on the fact that when the fitting is not good it is expected that the same happens with the estimates for the quantiles.) 3 High quantiles estimation After the estimation of the parameters r and γ as well as the function y we can characterize completely the limit d.f. G and then proceed to the estimation of high quantiles. ATENÇÃO: OS HIGHT QUANTILIES NÃO SÃO ÚNICOS PORQUE F PODE NÃO SER INJECTIVA, A MENOS QUE SE DEFINAM COM a INVERSA GENERALIZADA Given q close to zero, let us consider a hight quantile of probability 1 − q of F , that is, a real number xq that satisfies F (xq ) = 1 − q. Assuming that (1) holds, we obtain µ ¶ xq − b n 1/kn ∼ F (xq ) = G , an or equivalently µ µ ¶¶ xq − b n ∼ − log − log G = − log kn − log(− log(1 − q)). an (7) Taking into account all the possible specifications of the sequences {an }, {bn } and {kn } presented in Theorem 1, we can estimate these sequences from the empirical function of F . In order to do that we will use the order statistics Xη1 ,n and Xη2 ,n that are, respectively, the lower and upper bounds of any of the intervals [Xmi ,n , Xmi−1 ,n ] construted in the last section Thus, using these order statistics we consider b kn = exp{Un (Xη1 ,n )}, bbn = Xη ,n 1 and, in view of Un (Xη2 ,n ) − Un (Xη1 ,n ) ≈ log rb, from (6) we obtain bbn+1 = Xη2 ,n and then b an = Xη2 ,n − Xη1 ,n . 9 b be the limit distribution function estimated using the arguments Let G b is max-semistable, due to (5) a presented before. Since the function G function y1 : [bbn , bbn+1 ] → [Un (bbn ), Un (bbn+1 )] exists such that b am x + sbm )) = y1 (x) + m log rb, − log(− log G(b holds. EU, GRAÇA, PENSO QUE A FUNÇÃO y1 NÃO DEVE LEVAR CHAPÈU PORQUE NÃO RESULTA DE UM PROCESSO DE ESTIMAÇÃO DIRECTO. The function y1 is related with the estimated function yb as it follows à ! b x − bn y1 (x) = yb + Un (bbn ), bbn+1 − bbn and so we also have à x − bbn b am x + sbm )) = yb − log(− log G(b bbn+1 − bbn ! + Un (bbn ) + m log rb. (8) On the other side, from (7), an estimate of xq , denoted by x bq , should satisfy à à !! bbn x b − q b − log − log G = −Un (bbn ) − log(− log(1 − q)). (9) bbn+1 − bbn (m) For any m ∈ Z take tq ∈ [bbn , bbn+1 ] such that the left hand side of (8) and (9) are equal. Thus, we deduce à ! (m) tq − bbn yb = − log(− log(1 − q)) − m log rb − 2Un (b̂n ). bbn+1 − bbn Now, choosing a particular value of m, say m0 , such that the right hand side of the previous expression belongs to the image set of yb (that is [0, log rb]) (m ) we obtain a value b tq = tq 0 , since the function yb is injective. Finally, we get x bq = (b am0 b tq + sbm0 )(bbn+1 − bbn ) + bbn . 10 Observe that, if we do not reject the hypothesis γ = 0, we can replace b a by 1 and sbn by m0 , obtaining x bq = (b tq + m0 )(bbn+1 − bbn ) + bbn . A simulation study was designed to evaluate the procedure here proposed to estimate high quantiles. In its implementation the same models were selected as in the simulation study concerning the estimation of the parameters r and γ ([4]), that is, max-semistable models and models that are convenient “perturbations” of Pareto, Burr and Weibull models. More precisely, we consider distribution functions F in a max-semistable domain of attraction obtained in the following way: 1 − F (x) = (1 − F0 (x))θ(x) where F0 is in a max-stable domain of attraction and θ is a positive, bounded, and periodic function such that F is a distribution function (see Grinevich (1993)). In the specific choice of the distribution functions were combined several orders of magnitude for the underlying parameters. For each distribution function 1000 sample replicas were simulated with sample sizes 1000, 2000, 5000, 10000. The first aspect to point out respects the choice of the interval [Xη1 ,n , Xη2 ,n ] to use in the quantile estimation. Clearly we must use high order statistics but, in fact, not too high because we need empirical versions of y that are good representations of the true function. After some preliminary simulation studies we chose [Xη1 ,n , Xη2 ,n ] as the first interval that contains more than 30 order statistics in the case γ = 0, and more than 50 order statistics otherwise (recall that we have different estimators according to γ = 0 or γ 6= 0). More details on the design and results of the simulation study can be seen in TESE DA SANDRA. As an overall conclusion, we can say that the procedure performs as expected (giving better results for lower values of r and moderately high quantiles) and the precision of the estimates are comparable with the ones obtained in analogous studies in max-stable contexts. 4 Aplication to seismic data The role performed that the max-semistable law can have in modeling observables of physical phenomenons appeared for the first time incite11 Huil. Nesse trabalho as leis surgem como leis marginais em processos maxsemi-autossimilares, havendo um relacionamento análogo ao que se verifica entre as leis e os processos max-autossimilares (“max-selfsimilar”). Por outro lado, no estudo de escalas em fı́sica, têm-se revelado de interesse os fenómenos que apresentam invariância de escala. Estes apresentam o mesmo tipo de leis independentemente da escala de observação. Prova-se que as densidades dos modelos associados a medições destes fenómenos têm a forma de uma lei potência, sendo por isso leis paretianas. Quando se admite invariância de escala discreta (isto é, válida para alguma mudança de escala de observação mas não para todas) em [?] prova-se que as leis associadas a estes fenómenos têm uma componente log-periódica. A invariância de escala discreta tem sido postulada no estudo de fenómenos envolvendo turbulência ou ruptura, de entre os quais se destacam os fenómenos sı́smicos (ver [?] e [?]). Given the impact that earthquakes have socially and economically, the study of extreme observations is very important, especially in areas such as seismic risk, seismic hazard, insurance, etc. In literature there exist several studies about earthquakes where a distribution function is fitted to the earthquakes magnitudes or its seismic moments. From the proposed laws the more well known is the Gutenberg-Richter law, that establishes a relation between the magnitude, Mw , and the frequency of earthquakes with magnitude larger than m, a value previously fixed, in a given geographic area and in a large time interval. According to this law the number of earthquakes with magnitude larger than m can be modeled by a exponential law, that is, the relative frequency of the number of earthquakes with magnitude larger than m has a negative exponential behaviour. More precisely, conditionally to the magnitudes larger than a given value a, the distribution function of the magnitudes is given by F (x) = 1 − exp((a − x)/b), for x > a. Although usually the size of an earthquake is defined using the magnitude, we can also use the seismic moment, M0 , that are related with the magnitude in the following way Mw = 2 log10 M0 − 10.7, 3 12 if the seismic moments are defined in dyne-cm. Therefore, the GutenbergRichter law can be rewritten using the seismic moments, obtaining the distribution function of the seismic moments defined by F (x) = 1 − (cx)−β , para x > 1/c, where β = 2/3b and b is approximately 1. Although the Gutenberg-Richter law is a good approximation for a large interval of magnitudes that range from 4 (M0 = 1024 ) until near 7 (M0 = 1026.5 ), this law do not seems a good approximation for the remaining magnitudes, in particular for the case that is more interesting, the one of the large magnitudes. So, a large number of investigators studies the possibility of variations of this law. In [10] the Generalized Pareto distribution (GP D) was fitted to the seismic moments (larger than 1024 dyce-cm) of shallow earthquakes (deep ≤ 70 km) occurring in subduction zones in the Pacific region (using the seismic moments in the Harvard catalog from 1977 to 2000). Furthermore, a change point was determinated so that values smaller than that point were modeled with a GP D and larger values were modeled with a Pareto distribution with a different parameter γ (shape parameter??). The fitting of both distributions to the data was considered quite satisfactory and none could be preferred. As it happens with the Gutenberg-Richter law, this models also presented large deviation in the tail of the distribution, although the were well adjusted for intermediate values. The study of this data was continued in [11], where was proposed a test statistic to test deviations from the Gutenberg-Richter law. Those authors applied the test statistic to the seismic moments of earthquakes with seismic moments larger than 1024 dyce-cm in subduction zones. The data showed large deviations from the Pareto distribution not only on the tail, but elsewhere in the distribution. Furthermore, the behaviour of the test statistic was similar to the one presented by a mixture of two Pareto models with log-periodic oscillations. In [11] is also referred a physical explanation to justify the existence of log-periodic oscillations. Since a Pareto distribution seems to fit reasonable the data, although not completely, and the data showed the existence of log-periodic oscillations, it seems reasonable to assume a good fitting could be provided by a LogP erP areto distribution (a Pareto distribution perturbed by periodic 13 function, that is, a max-semistable distribution). The refereed data was not online so we used the data from ftp://element.ess.ucla.edu/2003107-esupp/index.htm obtained by Peter Bird e Yan Y. Kagan. Notice that this data is from the Harvard catalog and includes the earthquakes in subduction zones on a larger period that the one in [10] and [11]. From this data we selected earthquakes that occurred from 1977/01/01 to 2000/05/31 in the zones refereed by [10], with seismic moment larger than 1024 dyce-cm and deep inferior to 70 km, obtaining a sample of 3865 seismic moments. To observe the fitting of a Pareto distribution with the parameters determinated in [10], in Figures 4 and 5, we have the QQ-plot of the seismic moments against the estimated Pareto distribution and the survival function, in a logarithmic scale, of the seismic moments and the estimated Pareto distribution, respectively. 29 x 10 0 momentos sísmicos G1.51 9 −0.5 7 −1 Log10 (1−Fn(x)) Quantis da distribuição Pareto 8 6 5 4 −1.5 −2 −2.5 3 2 −3 1 −3.5 0 0 0.5 1 1.5 2 2.5 Quantis dos momentos sísmicos 3 24 3.5 25 26 27 28 29 Log10 (x) 28 x 10 Figure 4: QQ-plot of the seismic moments against the estimated Pareto distribution. Figure 5: Survival function, in a logarithmic scale, of the seismic moments and the estimated Pareto distribution. To proceed to the estimation of the limit law, we have to begin with the estimation of the parameter r, since all the other parameters and also the function y are dependent of r. So, we determinate the sample trajectories of Rs for s = 1.1 : 0.1 : 3 to verify intuitively which value should be a good estimate of r. In Figure 6 we can observe the behaviour of some trajectories of Rs . To see more easily which of the trajectories spends more time in a neighbourhood of 1, we draw a band around 1 with range 0.15. Using the method proposed for the estimation of r we obtained as an estimate of r 14 the value 1.3, that is the value of s for which the trajectory of Rs seems to stabilize around 1. 3 s s s s 2.5 = = = = 1.1 1.3 1.5 2.6 2 1.5 1 0.5 0 0 500 1000 1500 2000 2500 3000 3500 m Figure 6: Sample trajectories of Rs (m) for s = 1.1, 1.3, 1.5 and 2.6. Having an estimate of r we can proceed to the estimation of the parameter γ. The proposed estimator of γ allowed us to obtain the estimate γ b = 1.5248. Finally, we only need to estimate the function y, so using the method proposed we determinate the estimated function y showed in Figure 7. Since, now the estimated limit law is completely determinated and b is showed in Figure 8, as well as the estimated function − log(− log G). 0.25 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 Figure 7: Estimated function y. 15 1 5 0.9 4 0.8 0.7 3 0.6 0.5 2 0.4 1 0.3 0.2 0 0.1 0 0 500 1000 1500 2000 2500 3000 3500 −1 4000 x 0 500 1000 1500 2000 2500 3000 3500 4000 x b γ,ν and − log(− log G b γ,ν ). Figure 8: Estimated functions G In Figures 9 and 10 we have the QQ-plot of the seismic moments against the estimated max-semistable distribution function G1.52,ν and the survival function of the seismic moments and the estimated function G1.52,ν , in a logarithmic scale. It seems that this fitting does not shows significantly improvements to the fitting of a Pareto distribution to the data. Therefore is difficult to decide which distribution models better the data. Nevertheless, observing some empirical versions of y in Figure 11, we notice some oscillations with distinct behaviour. This behaviour can be related with the fact that the log-periodicity associated to these phenomenons only has the same behaviour in smaller regions, appearing a mixture of these behaviours when we consider larger regions. Given the models suggested in literature to fit the magnitudes, we can not finish without referring the possibility of the underlying model to be a truncated LogP erP areto (since there should be a maximum magnitude for earthquakes), or a model that is a mixture of a max-stable with a maxsemistable model, or a mixture of two max-semistable models with different parameters, similar to the models suggested in [11]. References [1] Ben Alaya, M. e Huillet, T. (2004). On Max-Multiscaling Distributions as Extended Max-Semistable Ones, Stochastic Models, Vol. 20, N4, 49316 0 momentos sísmicos G1.52,ν 3.5 −0.5 3 −1 log10(1−Fn(x)) Quantis da distribuição max−semiestável estimada 29 x 10 2.5 2 1.5 −3 0.5 −3.5 0.5 1 1.5 2 2.5 3 Quantis dos momentos sísmicos 24 3.5 25 26 27 28 29 log10(x) 28 x 10 Figure 9: QQ-plot of the seismic moments against the estimated maxsemistable G1.52,ν . Figure 10: Survival function, in a logarithmic scale, of the seismic moments and the estimated max-semistable function G1.52,ν . (17) (21) y y(18) (19) y y(20) 0.25 −2 −2.5 1 0 0 −1.5 y y(22) (23) y y(23) 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 11: Empirical versions of y. 512. [2] Canto e Castro, L., de Haan, L. e Temido, M. G. (2002). Rarely observed sample maxima. Theory of Probab. Appl., Vol. 45, p. 779-782. [3] Dias, S. e Canto e Castro, L. (2004). Contribuições para a estimação em modelos max-semiestáveis. 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