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Author manuscript, published in "Statistics & Probability Letters 78, 11 (2010) 1301"
DOI : 10.1016/j.spl.2007.11.025
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How to compute the extremal index of stationary random fields
peer-00622144, version 1 - 12 Sep 2011
H. Ferreira, L. Pereira
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DOI:
Reference:
S0167-7152(07)00409-9
10.1016/j.spl.2007.11.025
STAPRO 4841
To appear in:
Statistics and Probability Letters
Received date: 16 November 2005
Revised date: 10 July 2007
Accepted date: 6 November 2007
Please cite this article as: Ferreira, H., Pereira, L., How to compute the extremal index of
stationary random fields. Statistics and Probability Letters (2007),
doi:10.1016/j.spl.2007.11.025
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of stationary random fields
H. Ferreira∗ and L. Pereira
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Department of Mathematics
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How to compute the extremal index
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University of Beira Interior
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Abstract: We present local dependence conditions for stationary random fields under
which the extremal index and the asymptotic distribution of the maximum M(n1 ,...,nd ) can
be calculated from the joint distribution of a finite number s1 s2 of variables.
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Introduction
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keywords: Extremal index, local and long range dependence, random field.
Let X = {Xn }n≥1 be a stationary random field on Zd+ , where Z+ is the set of all
positive integers and d ≥ 2. We shall consider the conditions and results for d = 2 since it
is notationally simplest and the proofs for higher dimensions follow analogous arguments.
For i = (i1 , i2 ) and j = (j1 , j2 ), i ≤ j means ik ≤ jk , k = 1, 2.
For a family of real levels {un }n≥1 and a subset I of the rectangle of points Rn =
{1, . . . , n1 } × {1, . . . , n2 }, we will denote the event {Xi ≤ un : i ∈ I} by Mn (I) or simply
by Mn when I = Rn . If I = ∅ then Mn (I) = −∞.
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For each i = 1, 2, we say the pair I and J is in S i (l) if the distance between Πi (I)
and Πi (J) is great or equal to l, where Πi , i = 1, 2 denote the cartesian projections.
∗
Helena Ferreira, Department of Mathematics, University of Beira Interior, 6200 Covilh˜
a, Portugal.
E-mail:[email protected]
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The distance d(I, J) between sets I and J of Zd+ , d ≥ 1, is the minimum of distances
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d(i, j) = max{|is − js |, s = 1, . . . , d}, i ∈ I and j ∈ J.
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Suppose that X satisfies the coordinatewise-mixing ∆(un )-condition introduced in
Leadbetter and Rootz´en (1998) which exploit the past and future separation one coordinate at a time. Therefore, we suppose that for X there exist sequences of integer valued
constants {kni }, {lni }, i = 1, 2, such that, as n = (n1 , n2 ) → ∞, we have (kn1 , kn2 ) → ∞,
(
kn1 ln1 kn2 ln2
n1 , n2 )
→ 0 and (kn1 ∆1 , kn1 kn2 ∆2 ) → 0, where ∆i are the components of the
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mixing coefficient defined as follows:
∆1 = sup|P (Mn (I1 ) ≤ un , Mn (I2 ) ≤ un ) − P (Mn (I1 ) ≤ un )P (Mn (I2 ) ≤ un )|,
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n1
kn1 ,
∆2 = sup|P (Mn (I1 ) ≤ un , Mn (I2 ) ≤ un ) − P (Mn (I1 ) ≤ un )P (Mn (I2 ) ≤ un )|,
where the supremum is taken over pairs of I1 and I2 in S 2 (ln2 ) such that Π1 (I1 ) = Π1 (I2 )
and |Π2 (I2 )| ≤
n2
kn2 .
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Under the coordinatewise-mixing ∆(un )-condition we have the following asymptotic
independence for maxima over {(i − 1)[ knn1 ] + 1, . . . , i[ knn1 ]} × {(j − 1)[ knn2 ] + 1, . . . , j[ knn2 ]},
1
1
2
2
i = 1, . . . , kn1 , j = 1, . . . , kn2 :
P (Mn ≤ un ) − P kn1 kn2 (M([
n1
kn1
n2
n2
],[ k
])
≤ un ) −→n→∞ 0.
(1.1)
Accordingly Choi (2002), we shall say that X has extremal index θ if for each τ > 0
(τ )
(τ )
(τ )
there exists {un }n≥1 such that, as n → ∞ , n1 n2 P (X1 > un ) → τ and P (Mn ≤ un ) →
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e−θτ .
This paper is concerned with describing how to compute the extremal index of the
stationary random field under local mixing conditions analogous to those considered in
Chernick et al. (1991).
Those local restrictions on the oscillations of the values of the random field enable to
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where the supremum is taken over pairs of I1 and I2 in S 1 (ln1 ) such that |Π1 (I2 )| ≤
compute the extremal index from the joint distribution of a finite number s1 s2 of variables.
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Condition D(s) (un )
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In the following consider R∗i,j = {i1 , i1 + 1, . . . , j1 } × {i2 , i2 + 1, . . . , j2 } − {i}. In par-
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ticular, for i = 1 we write simply R∗j . For sake of simplicity we write [n/k] for ([ knn1 ], [ knn2 ]).
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Definition The random field X satisfies the condition D(s) (un ), for some s ∈ Z2+ ,
if there exist sequences of integer valued constants {kni }, {lni }, i = 1, 2, such that, as
X
n1 n2 max
i≤[n/k]
kn1 ln1 kn2 ln2
n1 , n2 )
→ 0 and
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n → ∞, we have (kn1 , kn2 ) → ∞, (
P (Xi > un , Mn (R∗i,i+s−1 ) ≤ un , Xj > un ) → 0.
j∈R[n/k] \Ri
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conditions considered in Pereira and Ferreira (2005) and Pereira and Ferreira (2006). We
shall pursue the direction of this dependence conditions and extend to spatial processes
some known formulas to obtain the extremal index of time series.
By applying stationarity and the D(s) (un )-condition, the asymptotic distribution of
the maximum Mn can be calculated from the joint distribution of the s1 s2 variables X1
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and Xi , i ∈ R∗s . Different local dependence conditions can be considered by changing the
definition of the region R∗s . Therefore the parameter ν in the following proposition could
be computed from a different set of variables.
Proposition 2.1 Let X be a stationary random field satisfying the ∆(un )-condition and
D(s) (un )-condition for some s. As n → ∞, we have
n1 n2 P (X1 > un , Mn (R∗s ) ≤ un ) → ν > 0
if only if
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P (Mn ≤ un ) → e−ν , ν > 0.
Proof: From (1.1) and the stationarity we get
P (Mn ≤ un ) − P (M[n/k]
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For the cases s = 1 = (1, 1) and s = 2 = (2, 2) we find in the above definition the local
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kn kn
2
1
≤ un )
=
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P (Mn ≤ un ) − exp −(1 + o(1))kn1 kn2 P (M[n/k] > un ) =
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X
∗
P (Mn ≤ un ) − exp −(1 + o(1))kn1 kn2
P (Xi > un , Mn (Ri,[n/k] ) ≤ un ) + o(1) =
i≤[n/k]
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X
∗
P (Mn ≤ un )−exp −(1+o(1))kn1 kn2
P (X1 > un , Mn (R[n/k]−i+1 ) ≤ un )+o(1) = o(1).
i≤[n/k]
The result follows now by applying the D(s) (un )-condition, since for [ knn1 ] − i1 + 1 ≥ s1
or [ knn2 ] − i2 + 1 ≥ s2 it holds
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P (X1 > un , Mn (R∗s ) ≤ un ) − P (X1 > un , Mn (R∗[n/k]−i+1 ) ≤ un ) ≤
X
P (X1 > un , Mn (R∗s ) ≤ un , Xj > un ) = o(n1 n2 ).
j≤[n/k]
d(1,j)≥min{s1 ,s2 }
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Therefore
∗
P (Mn ≤ un ) − exp −(1 + o(1))n1 n2 P (X1 > un , Mn (Rs ) ≤ un ) + o(1) = o(1).
(τ )
As a corollary we provide a relation between the limit of n1 n2 P (X1 > un , Mn (R∗s ) ≤
(τ )
un ) and the extremal index.
(τ )
Proposition 2.2 Let X be a stationary random field satisfying the ∆(un )-condition and
(τ )
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D(s) (un )-condition for each τ > 0. The extremal index of X exists and is θ if and only
if, as n → ∞, it holds
n1 n2 P (Mn (R∗s ) ≤ un |X1 > un ) → θ.
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From the Normal comparison lemma we can prove that the stationary normal random
(τ )
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fields such that ρn = E(Xj Xj+n ) satisfies ρn log(n1 n2 ) → 0, as n → ∞, satisfy ∆(un )(τ )
condition and D(1) (un )-condition (Pereira e Ferreira (2005)).
(τ )
(τ )
some s.
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The m-dependent random fields satisfy ∆(un )-condition and D(s) (un )-condition for
It is an open and interesting question to know if some autoregressive random fields can
(τ )
satisfy D(s) (un )-condition for some s with different coordinates.
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Here we present a simple example to apply the results.
2 , let b (i), s = 1, 2, . . . , 8, be the neighbors of i defined
Example: For each i = (i1 , i2 ) ∈ Z+
s
as b1 (i) = (i1 + 1, i2 ) , b2 (i) = i + 1, b3 (i) = (i1 , i2 + 1) , b4 (i) = (i1 − 1, i2 + 1) , b5 (i) =
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2.
bs (i) ∈
/ Z+
Let Y = {Yn }n≥0 be an i.i.d. random field with common distribution function FY and
define
Xn = max Yn , Yb5 (n) , Yb6 (n) , Yb7 (n) , n ≥ 1.
(τ )
Let {un }n≥1 be such that n1 n2 (1 − FY (un )) −−−→ τ > 0. Then, we have un ≡ un 1 for
X with τ1 = 4τ .
(τ )
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n→∞
(τ )
X satisfies the ∆(un )-condition and D(2) (un )-condition for each τ > 0 and n1 n2 P (Mn (R∗2 ) ≤
un |X1 > un ) → 41 . Therefore X has extremal index θ = 14 .
We present a simulation of this random field to illustrate its predisposition to form
clusters of high values. In the figure 1 we present the values of Xi and in figure 2 the
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contours of values higher than the 97-percentile.
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(i1 − 1, i2 ) , b6 (i) = i − 1, b7 (i) = (i1 , i2 − 1), b8 (i) = (i1 + 1, i2 − 1) and Xbs (i) = −∞ if
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figure 1
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Acknowledgements:
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We are grateful to the referee for his corrections. Work
partially supported by FCT/POCI2010/FEDER (EXMIXMOD project).
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References
Chernick, M.R., Hsing, T. and McCormick, W.P. (1991). Calculating the extremal index
for a class of stationary sequences. Adv. Appl. Prob. 23, 835-850.
Choi, H. (2002) Central limit theory and extremes of random fields. PhD Dissertation,
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Univ. of North Carolina at Chapel Hill.
Leadbetter and Rootz´en, H. (1998) On extreme values in stationary random fields Sto-
Preprint. Univ. of Lisbon.
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Pereira, L. and Ferreira, H. (2005) On extreme values in non stationary random fields.
Pereira, L. and Ferreira, H. (2006) Limiting crossing probabilities of random fields. Journal
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of Applied Probability 43, 3.
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chastic processes and related topics, 275-285, Trends Math. Birkhauser Boston, Boston.
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