Download Tail inference for a law in a max

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
Ã
!
b
x − bn
y1 (x) = yb
+ Un (bbn ),
bbn+1 − bbn
and so we also have
Ã
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
Ã
Ã
!!
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
Ã
!
(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
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(Rodrigues, P. M. M., Rebelo, E. L. e Rosado, F., eds.), p. 177-187.
Edições SPE.
17
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domı́nios de atracção max-semiestáveis. Em Estatı́stica Jubilar (Braumann, C. A., Infante, P., Oliveira, M. M., Alpizar-Jara, R. e Rosado,
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earthquake, J. Geophys. Res. 105, 25451-25471.
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17, 147-158.
[9] Pancheva, Z. (1992). Multivariate max-semistable distribuitions. Theory of Probab. Appl., Vol. 37, p. 731-732.
[10] Pisarenko, V. F., Sornette, D. (2003). Characterization of the frequency
of extreme events by generalized Pareto distribution, Pure and Geophysics, Vol. 160(12), p. 2343-2364.
[11] Pisarenko, V. F., Sornette, D. and Rodkin, M. (2004). Deviations of
the distribution of seismic energies from the Gutenberg-Richer law,
Computational Seismology, Vol. 35, p. 138-159.
[12] Saleur, H., Sammis, C.G. e Sornette, D. (1996). Discrete scale invariance, complex fractal dimensions and log-periodic corrections in earthquakes, Journal of Geophysical Research-Solid Earth 101, NB8, 1766117677.
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18
[14] Saleur, H. e Sornette, D. (1996). Complex exponents and log-periodic
corrections in frustrated systems, J.Phys.I France 6, n3, 327-355.
[15] Sornette, D. e Sammis, C.G. (1995). Complex Critical Exponents
from Renormalization Group Theory of Earthquakes: Implications for
Earthquake Predictions, J. Phys. I France 5, 607-619.
[16] Sornette, D. e Knopoff, L. (1997). The paradox of the expected time
until the next earthquake, Bull. Seism. Soc. Am. 87, 789-798.
[17] Sornette, D. (1998). Discrete scale invariance and complex dimensions,
Physics Reports 297, 239-270.
[18] Temido, M. G., Gomes, M.I. e Canto e Castro, L. (2001). Inferência
Estatı́stica em Modelos Max-semiestáveis. Em Um olhar sobre a estatı́stica (Oliveira, P., Athayde, E., eds.), p. 291-305. Edições SPE.
19