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Multivariate maxima of moving multivariate maxima
Statistics and Probability Letters 82 (2012) 1489–1496
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Multivariate maxima of moving multivariate maxima Helena Ferreira Department of Mathematics, University of Beira Interior, Covilhã, Portugal
article
abstract
info
Article history: Received 12 February 2012 Received in revised form 30 March 2012 Accepted 23 April 2012 Available online 2 May 2012
We define a class of multivariate maxima of moving multivariate maxima, generalising the M4 processes. For these stationary multivariate time series we characterise the joint distribution of extremes and compute the multivariate extremal index. We derive the bivariate upper tail dependence coefficients and the extremal coefficient of the new limiting multivariate extreme value distributions. © 2012 Elsevier B.V. All rights reserved.
Keywords: Moving multivariate maxima Multivariate extremal index Tail dependence Multivariate extreme value distribution
1. Introduction The M4 processes considered by Smith and Weissman (1996) constitute quite a large and flexible class of multivariate time series models exhibiting clustering of extremes. In the following generalisation we will introduce a copula in the array of innovations, rendering the new model more flexible than the previously mentioned M4 processes, to which the new model reduces if the copula is the Fréchet–Hoeffding upper bound. The new family (M5) comprises multivariate time series models exhibiting clustering of extremes across variables and over time. We derive the limiting distribution of the vector of componentwise maxima and the corresponding limit for the sequence of independent and identically distributed variables. Because of the temporal dependence, these two limiting distributions are different, and the difference is quantified by the multivariate extremal index, which will be calculated too. Let {Zl,n = (Zl,n,1 , . . . , Zl,n,d )}l≥1,−∞
max
l≥1 −∞
αl,k,j Zl,n−k,j ,
j = 1, . . . , d, n ≥ 1,
where {αl,k,j , l ≥ 1, −∞ < k < ∞, 1 ≤ j ≤ d} are nonnegative constants satisfying ∞ ∞
αl,k,j = 1,
for j = 1, . . . , d.
l=1 k=−∞
When Zl,n,j = Zl,n , j = 1, . . . , d, the M5 process is the M4 process considered by Smith and Weissman (1996). The common distribution FY of Yn = (Yn,1 , . . . , Yn,d ) satisfies FY (y1 , . . . , yd ) =
∞ ∞ l=1 k=−∞
FZ
y1
αl,k,1
,...,
yd
αl,k,d
,
E-mail address: [email protected]. 0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.04.015
yj > 0, j = 1, . . . , d,
(1)
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H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
and the relation for the corresponding copulas is ∞ ∞
CY (u1 , . . . , ud ) =
αl,k,1
C Z u1
α , . . . , ud l,k,d ,
uj ∈ [0, 1], j = 1, . . . , d.
l=1 k=−∞
This work is concerned with the extreme value properties of the class of stationary processes defined in (1). The M5 processes contribute to the modelling of variables moving together and generalise the important class of moving maxima discussed by several authors (Davis and Resnick, 1989; Deheuvels, 1983; Hall et al., 2002; Smith and Weissman, 1996; Zhang and Smith, 2004, among others). This is the motivation of this work which is organised as follows. By choosing a positive lower orthant dependent distribution function CZ in the domain of attraction of a max-stable copulaC ∗ (Joe, 1997) we will find a new class of limiting multivariate extreme value (MEV) distribution H for the vector Mn = Mn,1 , . . . , Mn,d of componentwise maxima from Y1 , . . . , Yn . In the third section, we derive the multivariate extremal index of the M5 processes and illustrate the result with some choices of C ∗ . Finally, we compare the bivariate upper tail dependence coefficients of FY with the ones for the limiting MEV distribution H. 2. Domains of max-attraction
ˆ n }n ≥ 1 be a sequence of independent random vectors associated with {Yn }n ≥ 1, that is such that F ˆ = FY , and Let {Y Yn
ˆn = M ˆ n ,1 , . . . , M ˆ n,d be the corresponding vector of pointwise maxima. M We shall assume that the copula CZ of the array of innovations in {Yn }n ≥ 1 is positive lower orthant dependent, that is, it satisfies the inequality CZ (u1 , . . . , ud ) ≥
d
ui ,
(u1 , . . . , ud ) ∈ [0, 1]d .
(2)
i=1
Positive lower orthant dependence is implied by several dependence concepts (Joe, 1997, Chapter 2) and, in particular, is satisfied by associated random variables. The multivariate extreme value distributions are associated and therefore the following results can be applied when the array of innovations Zl,n has an MEV distribution. Following Smith and Weissman (1996), we present in this section the limiting distributions of the normalised vectors ˆ n and Mn . M Proposition 2.1. If CZ is positive lower orthant dependent and is in the domain of attraction of a max-stable C ∗ , that is, 1/n
1/n
CZn (u1 , . . . , ud ) −−−→ C ∗ (u1 , . . . , ud ), n→∞
then CY is in the domain of attraction of Cˆ (u1 , . . . , ud ) =
∞ ∞
αl,k,1
C ∗ u1
α , . . . , ud l,k,d .
(3)
l=1 k=−∞
Proof. We have 1/n
1/n
lim CYn (u1 , . . . , ud ) = lim
n→∞
n→∞
∞ ∞
αl,k,1 /n
CZn u1
α
, . . . , ud l,k,d
/n
l=1 k=−∞
and we can take the product of limits since lim
n→∞
∞ ∞ l=1 k=−∞
αl,k,1 /n
CZn u1
αl,k,d /n
, . . . , ud
= exp
lim
n→∞
∞ ∞
αl,k,1 /n
log CZn u1
αl,k,d /n
, . . . , ud
,
l=1 k=−∞
and, by applying the Fréchet lower bound in (2), for each k and l it holds that
n d d αl,k,j /n αl,k,1 /n αl,k,d /n n , . . . , ud uj αl,k,j log uj log CZ u1 ≤ log ≤ j =1 j =1 ∞ ∞ with l=1 k=−∞ αl,k,j log uj summable for each j. It then follows, from the Dominated Convergence Theorem, that ∞ ∞ ∞ ∞ α / n α / n α / n α / n l , k , 1 l , k , d l , k , 1 l , k , d lim CZn u1 , . . . , ud = exp log lim CZn u1 , . . . , ud n→∞
l=1 k=−∞
l=1 k=−∞
= Cˆ (u1 , . . . , ud ) .
n→∞
H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
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We have then, for ∀τ = (τ1 , . . . , τd ) ∈ Rd+ ,
ˆ n,1 ≤ n , . . . , M ˆ n ,d ≤ n P M τ1 τd
CYn e−τ1 /n , . . . , e−τd /n
=
∞ ∞ −−−→ Cˆ e−τ1 , . . . , e−τd = C ∗ e−τ1 αl,k,1 , . . . , e−τd αl,k,d . n→∞
l=1 k=−∞
ˆ denote the multivariate extreme value distribution with standard Fréchet margins and copula Cˆ , i.e. Let H −1 ∞ ∞ −1 ˆ H (x1 , . . . , xd ) = l=1 k=−∞ C ∗ e−x1 αl,k,1 , . . . , e−xd αl,k,d . We will now consider the corresponding limit H for the vector Mn = Mn,1 , . . . , Mn,d of componentwise maxima from Y1 , . . . , Yn . Proposition 2.2. If CZ is a positive lower orthant dependent copula in the domain of attraction of a max-stable C ∗ then
Mn,1 ≤
lim P
n→∞
n
n
, . . . , Mn,d ≤
τ1
=
τd
∞
C ∗ e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd .
(4)
l=1
Proof. It holds that
Mn,1 ≤
lim P
n→∞
n
τ1 ∞ ∞
= lim
n→∞
, . . . , Mn,d ≤
n
τd
CZ e− max1−m≤k≤n−m αl,k,1 τ1 /n , . . . , e− max1−m≤k≤n−m αl,k,d τd /n
l=1 m=−∞
= lim exp n→∞
∞ ∞
log CZ e
− max1−m≤k≤n−m αl,k,1 τ1 /n
− max1−m≤k≤n−m αl,k,d τd /n
,...,e
.
l=1 m=−∞
Now we can interchange the first two limits, by using arguments analogous to those used in Proposition 2.1, since ∞ log CZ e− max1−m≤k≤n−m αl,k,1 τ1 /n , . . . , e− max1−m≤k≤n−m αl,k,d τd /n m=−∞
≤
d ∞ − max α τ /n log e j=1 1−m≤k≤n−m l,k,j j
m=−∞
=
∞ d 1 m=−∞
max
n j=1 1−m≤k≤n−m
m=−∞
and this last sum is bounded for all n by We have therefore
lim P
n→∞
Mn,1 ≤
= exp
∞ l=1
n
τ1
, . . . , Mn,d ≤ ∞ 1
lim
n→∞
m=−∞
n
∞ d
αl,k,j τj ≤ ∞
k=−∞
max max αl,k,j τj , n 1≤j≤d 1−m≤k≤n−m
d max1≤j≤d (αl,k,j τj ), which is summable in l Smith and Weissman (1996).
n
τd
log CZn
e
− max1−m≤k≤n−m αl,k,1 τ1 /n
− max1−m≤k≤n−m αl,k,d τd /n
,...,e
and we have to prove that lim
n→∞
∞ 1 m=−∞
= log C
∗
n
log CZn e− max1−m≤k≤n−m αl,k,1 τ1 /n , . . . , e− max1−m≤k≤n−m αl,k,d τd /n
e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd .
For fixed l, let bl,k,j = al,k,j τj , j = 1, . . . , d, and bl,k = maxj=1,...,d bl,k,j (which are summable in k and l). Suppose that bl,k,j is maximised when k = k∗ (l, j), bl,k is maximised when k = k∗ (l) (not necessarily unique) and assume, without (i)
loss of generality, that k∗ (l, 1) ≤ · · · ≤ k∗ (l, d). Break the above sum into three sums Sn , 1 = 1, 2, 3, corresponding to 1 − min{k∗ (l, 1), k∗ (l)} ≤ m ≤ n − max{k∗ (l, d), k∗ (l)}, m < 1 − min{k∗ (l, 1), k∗ (l)} and m > n − max{k∗ (l, d), k∗ (l)}.
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H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
For large n we have (1)
Sn
n−max{k∗ (l,d),k∗ (l)}
1
=
log CZn e− max1−m≤k≤n−m αl,k,1 τ1 /n , . . . , e− max1−m≤k≤n−m αl,k,d τd /n
n m=1−min{k∗ (l,1),k∗ (l)}
n − max{k∗ (l, d), k∗ (l)} + min{k∗ (l, 1), k∗ (l)}
=
n
log CZn e−αl,k∗ (l,1)1 τ1 /n , . . . , e−αl,k∗ (l,d)d τd /n ,
which converges to log C ∗ e−αl,k∗ (l,1)1 τ1 , . . . , e−αl,k∗ (l,d)d τd = log C ∗ e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd .
Otherwise, by applying (2), we can write
(2) S ≤ 1 n
d
d
max
n m<1−k∗ (l) j=1 1−m≤k≤n−m
≤ ≤
αl,k,j τj +
1
d
max αl,k,j τj n 1−k∗ (l)≤m<1−min{k∗ (l,1),k∗ (l)} j=1 1−m≤k≤n−m d
max
max αl,k,j τj +
n m<1−k∗ (l) 1−m≤k≤n−m j=1,...,d
max max αl,k,j τj n 1−k∗ (l)≤m<1−min{k∗ (l,1),k∗ (l)} 1−m≤k≤n−m j=1,...,d
d
max
max αl,k,j τj +
d
n m<1−k∗ (l) 1−m≤k≤n−m j=1,...,d
n
min{k∗ (l, 1), k∗ (l)} − k∗ (l) bl,k∗ (l)
and
(3) S ≤ 1 n
d
d
max
n m>n−k∗ (l) j=1 1−m≤k≤n−m
≤ ≤
αl,k,j τj +
1
d
max αl,k,j τj n n−max{k∗ (l,d),k∗ (l)}
max
max αl,k,j τj +
n m>n−k∗ (l) 1−m≤k≤n−m j=1,...,d
max max αl,k,j τj n n−max{k∗ (l,d),k∗ (l)}
d
max
max αl,k,j τj +
d
n m>n−k∗ (l)
1−m≤k≤n−m j=1,...,d
n
max{k∗ (l, d), k∗ (l)} − k∗ (l) bl,k∗ (l) .
(i) The second terms in the above upper bounds for Sn , i = 1, 2, tend to zero and the same holds for the first terms, by Lemma 3.2 in Smith and Weissman (1996).
For the particular case of C ∗ (u1 , . . . , ud ) = min1≤j≤d uj , (u1 , . . . , ud ) ∈ [0, 1]d , the above result agrees with (3.5) in Smith and Weissman (1996), that is
Mn,1 ≤
lim P
n→∞
n
τ1
, . . . , Mn,d ≤
If we choose C ∗ (u1 , . . . , ud ) = Martins and Ferreira (2005),
lim P
n→∞
Mn,1 ≤
n
τ1
d
n
= exp −
τd
j =1
, . . . , Mn,d ≤
∞ l =1
max
max αl,k,j τj
−∞
.
uj , (u1 , . . . , ud ) ∈ [0, 1]d , we find again the result in case 2 of the last example in
n
τd
= exp −
d ∞ j =1 l =1
max
−∞
αl,k,j τj .
In this last work only CZ = C ∗ is considered and the above two choices for C ∗ are treated directly. By calculating the limits in (3) and (4) for a given initial copula C ∗ , the above propositions enable us to obtain a larger class of MEV models. Some choices for C ∗ are available in the literature. For instance, Capéraà et al. (2000) gives sufficient conditions for an Archimedean copula CZ to be in the domain of attraction of the Gumbel–Hougaard or logistic copula C ∗ (u1 , . . . , ud ) = exp −
d j =1
α 1/α − log uj . More examples of CZ in the domain of attraction of an MEV copula C ∗ can be found in
Demarta and McNeil (2005) and Hüsler and Reiss (1989). 3. The multivariate extremal index of the M5 process In this section we will extend the results about the multivariate extremal index of the M4 process, as a corollary of Propositions 2.1 and 2.2.
H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
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We first recall the definition of the multivariate extremal index function θ (τ) = θ (τ1 , . . . , τd ), τ ∈ Rd+ , that relates the ˆ and which was introduced by Nandagopalan (1990). MEV distribution functions H and H A d-dimensional stationary sequence {Yn }n≥1 is said to have a multivariate extremal index θ (τ) ∈ [0, 1], τ ∈ Rd+ , if for (τ)
each τ = (τ1 , . . . , τd ) in Rd+ , there exists un (τj )
nP (Y1j > un,j ) −−−→ τj , n→∞
(τ )
(τ )
= (un,11 , . . . , un,dd ), n ≥ 1, satisfying
j = 1, . . . , d,
ˆ n ≤ u(τ) P (M −−→ γˆ (τ) ∈ (0, 1] n ) −
and P (Mn ≤ u(τ) −−→ γˆ (τ)θ (τ) . n ) −
n→∞
n→∞
Proposition 3.1. If CZ is a positive lower orthant dependent copula in the domain of attraction of a max-stable C ∗ then the multivariate extremal index of the M5 process {Yn } defined in (1) is given by ∞
θ (τ1 , . . . , τd ) =
log C ∗ e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd
l =1
∞ ∞
log C ∗ e−αl,k,1 τ1 , . . . , e−αl,k,d τd
l=1 k=−∞
and the extremal index of {Yn,j }n≥1 is
θj =
∞ l=1
max
−∞
αl,k,j ,
j = 1, . . . , d. (τj )
Proof. By applying Propositions 2.1 and 2.2 with un,j =
γˆ (τ) =
∞ ∞
C ∗ e−αl,k,1 τ1 , . . . , e−αl,k,d τd
τj n
, j = 1, . . . , d, we find
l=1 k=−∞
and
γ (τ) = γˆ (τ)θ(τ) =
∞
C ∗ e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd ,
l =1 log γ (τ) log γˆ (τ)
which leads to the results for θ (τ1 , . . . , τd ) =
and θj = limτi →0+ , i∈{1,...,d}−{j} θ (τ1 , . . . , τd ).
Therefore the copulas C ∗ , Cˆ and the copula C of the limiting MEV distribution H are related by a multivariate extremal index θ (τ1 , . . . , τd ) through C ( u1 , . . . , ud ) =
=
1/θ u1 1
Cˆ (
,...,
∞ ∞
1/θ ud d
θ )
αl,k,1 /θ1
C (u1 ∗
log u
log u
− θ 1 ,...,− θ d d 1
αl,k,d /θd
, . . . , ud
)
θ
log u
log u
− θ 1 ,...,− θ d d 1
.
l=1 k=−∞
Proposition 3.1 leads to the results of Smith and Weissman (1996) and Martins and Ferreira (2005) when C ∗ is the copula of the minimum and product, respectively. Otherwise, if we take for instance the Logistic copula C ∗ we find
∞ d θ (τ1 , . . . , τd ) =
l =1
j =1
max
−∞≤k≤+∞
∞ ∞ l=1 k=∞
d
αl,k,j τj
αl,k,j τj
α
α 1/α
1/α
.
j =1
4. The tail dependence of the M5 process For a random vector X = (X1 , . . . , Xd ) with continuous margins F1 , . . . , Fd and copula C , let the bivariate (upper) tail dependence coefficients parameters be defined by
λj(,Cj′) = lim P (Fj (Xj ) > u|Fj′ (Xj′ ) > u), u ↑1
1 ≤ j < j ′ ≤ d.
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H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
(C )
The tail dependence coefficient characterises the dependence in the tail of a random pair (Xj , Xj′ ), i.e., λj,j′ > 0 (C ) corresponds to tail dependence and λj,j′ = 0 means tail independence, and can be defined via the copula of the random vector which refers to their dependence structure independently of their marginal distributions. It holds that log Cj,j′ (u, u)
λ(j,Cj′) = 2 − lim
log u
u↑1
,
where Cj,j′ is the copula of the sub-vector (Xj , Xj′ ). (C )
(Cˆ )
ˆ We first remark that, for each In this section we will relate λj,j′ to λj,j′ , where C is copula of H and Cˆ is the copula of H.
(uj , uj′ ) ∈ [0, 1]2 , we have Cj,j′ (uj , uj′ ) =
1/θj Cj,j′ uj
ˆ (
,
1/θj′ uj ′
)
θ
log u ′
log u
− θ j ,− θ j j j′
,
(5)
where θ (τj , τj′ ) is the bivariate extremal index of {(Yn,j , Yn,j′ )}n≥1 . This relation enables us to compare the tail dependence (C )
(Cˆ )
parameters λj,j′ with λj,j′ through the function θ (τj , τj′ ). Proposition 4.1. If C and Cˆ satisfy (5) then: (C )
(a) λj,j′ = 2 + θ ( θ1 , θ1′ ) log Cˆ j,j′ (e−1/θj , e j j (C )
(Cˆ )
(b) λj,j′ = λj,j′ + log
−1/θj′
),
−θ( θ1 , θ1 )/θj −θ( θ1 , θ1 )/θj′ j j′ j j′ ,e ) Cˆ j,j′ (e
Cˆ j,j′ (e−1 ,e−1 )
.
ˆ , we get Proof. From the spectral measure representation (Resnick, 1987) of the copula Cˆ , with measure W λ(j,Cj′) = 2 − lim ˆ
log Cˆ j,j′ (u, u) log u
u↑1
ˆ (w1 , . . . , wd ) = 2 + log Cˆ j,j′ (e−1 , e−1 ). max{wj , wj′ } dW
=2−
(6)
Sd
By using (5) and the homogeneity of order 0 of the multivariate extremal index, it follows that
λ(j,Cj′) = 2 − lim
log Cj,j′ (u, u) log u
u ↑1
= 2−θ = 2−θ = 2+θ
1
,
1
,
1
lim
θj θj′
1 1
1
,
j
− log u
u↑1
max
θj θj′
1/θj′ 1/θj − log u − log u log Cˆ j,j′ (uj , uj′ ) , θj θj′ log u − log uwj − log uwj′ ˆ (w1 , . . . , wd ) , θ′ dW max Sd θj
= 2 − lim θ u ↑1
Sd
w j w j′ , θj θj′
ˆ (w1 , . . . , wd ) dW
−1/θj′
log Cˆ j,j′ (e−1/θj , e
θj θj′
),
which has (6) as a particular case. To obtain the second statement we combine the first with (6) and use the max-stability of Cˆ j,j′ , as follows:
λ(j,Cj′)
= 2 + log Cˆ j,j′ (e−1 , e−1 ) − log Cˆ j,j′ (e−1 , e−1 ) + θ = λj(,Cj′) + log ˆ
Cˆ j,j′ (e−1/θj , e
Cˆ j,j′
=
+ log
)
θ
1
1
θj , θj ′
1
1
, θj θj′
−1/θj′
log Cˆ j,j′ (e−1/θj , e
)
Cˆ j,j′ (e−1 , e−1 )
ˆ λj(,Cj′)
−1/θj′
e
−θ θ1 , θ1 /θj ′ j j
,e
−θ θ1 , θ1 /θj′ ′ j j
Cˆ j,j′ (e−1 , e−1 )
.
We will now apply these relations to the particular case of M5 processes, enhancing the effect of the multivariate extremal index in the tail dependence.
H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
1495
Proposition 4.2. Let {Yn }n ≥ 1 be an M5 process defined as in (1) and such that CZ is a positive lower orthant dependent copula in the domain of attraction of C ∗ . Then, for any 1 ≤ j < j′ ≤ d it holds that: (Cˆ )
(a) λj,j′ = 2 + (b)
λ(j,Cj′)
∞ ∞ l =1
= 2 + θ ( θj , 1
1 θj′
)
log Cj∗,j′ e−αl,k,j , e
k=−∞
∞ ∞
∗ k=−∞ log Cj,j′
l =1
−αl,k,j′
.
−αl,k,j /θj
e
, e−αl,k,j′ /θj′ .
Proof. By using (6) and Proposition 2.1, we obtain (a); by using Proposition 4.1-(a) and Proposition 2.1, we get (b).
If C (u1 , . . . , ud ) = min1≤j≤d uj , (u1 , . . . , ud ) ∈ [0, 1] , we obtain ∗
d
λj(,Cj′) = 2 − ˆ
∞ ∞
max{αl,k,j , αl,k,j′ }
(7)
l=1 k=−∞
which is in general greater than zero, which agrees with the result (2.10) presented in Heffernan et al. (2007). We can also say from (b) that, for this copula,
λj(,Cj′)
=2−
∞
max
max −∞≤k≤+∞
l =1
(C )
αl,k,j /θj ,
max −∞≤k≤+∞
αl,k,j′ /θj′
(Cˆ )
and it holds that λj,j′ > λj,j′ if and only if ∞ ∞
max{αl,k,j , αl,k,j′ } −
l=1 k=−∞
where θj =
∞
max
−∞≤k≤+∞
l=1
∞
l =1
αl,k,j /θj ,
max
max−∞
max −∞≤k≤+∞
(C )
αl,k,j′ /θj′ > 0,
(Cˆ )
(C )
(Cˆ )
We can easily construct examples of M4 processes for which λj,j′ > λj,j′ or λj,j′ < λj,j′ . The results for this class of moving multivariate maxima show us that if we obtain or estimate a tail dependence parameter of the common distribution of the variables in a stationary sequence we do not have necessarily the corresponding parameter in the limiting MEV model C . The multivariate extremal index of the stationary sequence can increase or decrease the tail dependence and the extremal coefficients of the limiting MEV model Cˆ arising from the i.i.d. sequence. (C )
(Cˆ )
Since λj,j′ = λj,j′Y , the result in (7) says that in the M4 processes the variables Yn,j , j = 1, . . . , d, are in general asymptotically dependent. For the M5 processes we can choose C ∗ in order to produce variables that are asymptotically (C ∗ )
(Cˆ )
(C ∗ )
independent. Take for instance C ∗ such that λj,j′ = 0 and αl,k,j = αl,k,j′ . We then find λj,j′ = λj,j′ = 0. Proposition 4.2 also points out that even for a choice of a copula C ∗ with symmetric tail dependences, the values of the signatures αl,k,j can lead to asymmetric tail dependences in both copulas C and Cˆ , that is, to different values of λj,j′ for different choices of (j, j′ ). The results in the above proposition are translations of the classical result λ = 2 − ϵ for Cj,j′ and Cˆ j,j′ (Nelsen, 2006). In fact, if we define the extremal coefficient of the MEV copula C as the constant ϵC such that C (u, . . . , u) = uϵC for all u ∈ [0, 1], then from
C (u1 , . . . , ud ) = exp −θ
log u1
θ1
,...,
log ud
− log uj wj ˆ dW (w1 , . . . , wd ) , θj Sd 1≤j≤d
max
θd
we find
ϵC = −θ
1
θ1
,...,
1
θd
log Cˆ e−1/θ1 , . . . , e−1/θd
and, in particular,
ϵCˆ = − log Cˆ e−1 , . . . , e−1 . (Cˆ )
The relation between Cˆ j,j′ and Cj∗,j′ enables us now to obtain Proposition 4.2 from λj,j′ = 2 − ϵCˆ
(C )
j,j′
and λj,j′ = 2 − ϵCj,j′ .
Other features of the M5 processes such as a directory of tail dependence coefficients for different C ∗ and signatures αl,k,j , illustrating the range of dependence structures, the model selection and estimation, are key directions for future research. Acknowledgements The final form of this work had a significant contribution from the referees and an associate editor. They pointed out several corrections needed in the manuscript. I am very grateful to them for their help. This research was partially supported by FCT/POCI2010/FEDER/CMUBI (project EVT) and FCT/PTDC/MAT108575/2008.
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References Capéraà, P., Fougères, A.L., Genest, C., 2000. Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72, 30–49. Davis, R.A., Resnick, S.I., 1989. Basic properties and prediction of max-ARMA processes. Adv. in Appl. Probab. 21, 781–803. Deheuvels, P., 1983. Point processes and multivariate extreme values. J. Multivariate Anal. 13, 252–257. Demarta, S., McNeil, A., 2005. The t-copula and related copulas. Internat. Statist. Rev. 73, 111–129. Hall, P., Peng, L., Yao, Q., 2002. Moving-maximum models for extrema of time series. J. Statist. Plann. Inference 103, 51–63. Heffernan, J.E., Tawn, J.A., Zhang, Z., 2007. Asymptotically (in)dependent multivariate maxima of moving maxima processes. Extremes 10, 57–82. Hüsler, J., Reiss, R., 1989. Maxima of normal random vectors: between independence and complete dependence. Statist. Probab. Lett. 76, 283–286. Joe, H., 1997. Multivariate Models and Dependence Concepts. Chapman & Hall, London. Martins, A.P., Ferreira, H., 2005. The extremal index and the dependence structure of a multivariate extreme value distribution. TEST 14 (2), 433–448. Nandagopalan, S., 1990. Multivariate extremes and estimation of the extremal index. Ph.D. Dissertation. Department of Statistics, University of North Carolina, USA. Nelsen, R.B., 2006. An Introduction to Copulas, second ed. Springer, New York. Resnick, S., 1987. Extreme Values, Regular Variation and Point Processes. Springer-Verlag, New York. Smith, R.L., Weissman, I., 1996. Characterization and estimation of the multivariate extremal index. Technical Report. Univ. North Carolina. Zhang, Z., Smith, R.L., 2004. The behavior of multivariate maxima of moving maxima processes. J. Appl. Probab. 41, 1113–1123.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Multivariate maxima of moving multivariate maxima Helena Ferreira Department of Mathematics, University of Beira Interior, Covilhã, Portugal
article
abstract
info
Article history: Received 12 February 2012 Received in revised form 30 March 2012 Accepted 23 April 2012 Available online 2 May 2012
We define a class of multivariate maxima of moving multivariate maxima, generalising the M4 processes. For these stationary multivariate time series we characterise the joint distribution of extremes and compute the multivariate extremal index. We derive the bivariate upper tail dependence coefficients and the extremal coefficient of the new limiting multivariate extreme value distributions. © 2012 Elsevier B.V. All rights reserved.
Keywords: Moving multivariate maxima Multivariate extremal index Tail dependence Multivariate extreme value distribution
1. Introduction The M4 processes considered by Smith and Weissman (1996) constitute quite a large and flexible class of multivariate time series models exhibiting clustering of extremes. In the following generalisation we will introduce a copula in the array of innovations, rendering the new model more flexible than the previously mentioned M4 processes, to which the new model reduces if the copula is the Fréchet–Hoeffding upper bound. The new family (M5) comprises multivariate time series models exhibiting clustering of extremes across variables and over time. We derive the limiting distribution of the vector of componentwise maxima and the corresponding limit for the sequence of independent and identically distributed variables. Because of the temporal dependence, these two limiting distributions are different, and the difference is quantified by the multivariate extremal index, which will be calculated too. Let {Zl,n = (Zl,n,1 , . . . , Zl,n,d )}l≥1,−∞
max
l≥1 −∞
αl,k,j Zl,n−k,j ,
j = 1, . . . , d, n ≥ 1,
where {αl,k,j , l ≥ 1, −∞ < k < ∞, 1 ≤ j ≤ d} are nonnegative constants satisfying ∞ ∞
αl,k,j = 1,
for j = 1, . . . , d.
l=1 k=−∞
When Zl,n,j = Zl,n , j = 1, . . . , d, the M5 process is the M4 process considered by Smith and Weissman (1996). The common distribution FY of Yn = (Yn,1 , . . . , Yn,d ) satisfies FY (y1 , . . . , yd ) =
∞ ∞ l=1 k=−∞
FZ
y1
αl,k,1
,...,
yd
αl,k,d
,
E-mail address: [email protected]. 0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.04.015
yj > 0, j = 1, . . . , d,
(1)
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H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
and the relation for the corresponding copulas is ∞ ∞
CY (u1 , . . . , ud ) =
αl,k,1
C Z u1
α , . . . , ud l,k,d ,
uj ∈ [0, 1], j = 1, . . . , d.
l=1 k=−∞
This work is concerned with the extreme value properties of the class of stationary processes defined in (1). The M5 processes contribute to the modelling of variables moving together and generalise the important class of moving maxima discussed by several authors (Davis and Resnick, 1989; Deheuvels, 1983; Hall et al., 2002; Smith and Weissman, 1996; Zhang and Smith, 2004, among others). This is the motivation of this work which is organised as follows. By choosing a positive lower orthant dependent distribution function CZ in the domain of attraction of a max-stable copulaC ∗ (Joe, 1997) we will find a new class of limiting multivariate extreme value (MEV) distribution H for the vector Mn = Mn,1 , . . . , Mn,d of componentwise maxima from Y1 , . . . , Yn . In the third section, we derive the multivariate extremal index of the M5 processes and illustrate the result with some choices of C ∗ . Finally, we compare the bivariate upper tail dependence coefficients of FY with the ones for the limiting MEV distribution H. 2. Domains of max-attraction
ˆ n }n ≥ 1 be a sequence of independent random vectors associated with {Yn }n ≥ 1, that is such that F ˆ = FY , and Let {Y Yn
ˆn = M ˆ n ,1 , . . . , M ˆ n,d be the corresponding vector of pointwise maxima. M We shall assume that the copula CZ of the array of innovations in {Yn }n ≥ 1 is positive lower orthant dependent, that is, it satisfies the inequality CZ (u1 , . . . , ud ) ≥
d
ui ,
(u1 , . . . , ud ) ∈ [0, 1]d .
(2)
i=1
Positive lower orthant dependence is implied by several dependence concepts (Joe, 1997, Chapter 2) and, in particular, is satisfied by associated random variables. The multivariate extreme value distributions are associated and therefore the following results can be applied when the array of innovations Zl,n has an MEV distribution. Following Smith and Weissman (1996), we present in this section the limiting distributions of the normalised vectors ˆ n and Mn . M Proposition 2.1. If CZ is positive lower orthant dependent and is in the domain of attraction of a max-stable C ∗ , that is, 1/n
1/n
CZn (u1 , . . . , ud ) −−−→ C ∗ (u1 , . . . , ud ), n→∞
then CY is in the domain of attraction of Cˆ (u1 , . . . , ud ) =
∞ ∞
αl,k,1
C ∗ u1
α , . . . , ud l,k,d .
(3)
l=1 k=−∞
Proof. We have 1/n
1/n
lim CYn (u1 , . . . , ud ) = lim
n→∞
n→∞
∞ ∞
αl,k,1 /n
CZn u1
α
, . . . , ud l,k,d
/n
l=1 k=−∞
and we can take the product of limits since lim
n→∞
∞ ∞ l=1 k=−∞
αl,k,1 /n
CZn u1
αl,k,d /n
, . . . , ud
= exp
lim
n→∞
∞ ∞
αl,k,1 /n
log CZn u1
αl,k,d /n
, . . . , ud
,
l=1 k=−∞
and, by applying the Fréchet lower bound in (2), for each k and l it holds that
n d d αl,k,j /n αl,k,1 /n αl,k,d /n n , . . . , ud uj αl,k,j log uj log CZ u1 ≤ log ≤ j =1 j =1 ∞ ∞ with l=1 k=−∞ αl,k,j log uj summable for each j. It then follows, from the Dominated Convergence Theorem, that ∞ ∞ ∞ ∞ α / n α / n α / n α / n l , k , 1 l , k , d l , k , 1 l , k , d lim CZn u1 , . . . , ud = exp log lim CZn u1 , . . . , ud n→∞
l=1 k=−∞
l=1 k=−∞
= Cˆ (u1 , . . . , ud ) .
n→∞
H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
1491
We have then, for ∀τ = (τ1 , . . . , τd ) ∈ Rd+ ,
ˆ n,1 ≤ n , . . . , M ˆ n ,d ≤ n P M τ1 τd
CYn e−τ1 /n , . . . , e−τd /n
=
∞ ∞ −−−→ Cˆ e−τ1 , . . . , e−τd = C ∗ e−τ1 αl,k,1 , . . . , e−τd αl,k,d . n→∞
l=1 k=−∞
ˆ denote the multivariate extreme value distribution with standard Fréchet margins and copula Cˆ , i.e. Let H −1 ∞ ∞ −1 ˆ H (x1 , . . . , xd ) = l=1 k=−∞ C ∗ e−x1 αl,k,1 , . . . , e−xd αl,k,d . We will now consider the corresponding limit H for the vector Mn = Mn,1 , . . . , Mn,d of componentwise maxima from Y1 , . . . , Yn . Proposition 2.2. If CZ is a positive lower orthant dependent copula in the domain of attraction of a max-stable C ∗ then
Mn,1 ≤
lim P
n→∞
n
n
, . . . , Mn,d ≤
τ1
=
τd
∞
C ∗ e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd .
(4)
l=1
Proof. It holds that
Mn,1 ≤
lim P
n→∞
n
τ1 ∞ ∞
= lim
n→∞
, . . . , Mn,d ≤
n
τd
CZ e− max1−m≤k≤n−m αl,k,1 τ1 /n , . . . , e− max1−m≤k≤n−m αl,k,d τd /n
l=1 m=−∞
= lim exp n→∞
∞ ∞
log CZ e
− max1−m≤k≤n−m αl,k,1 τ1 /n
− max1−m≤k≤n−m αl,k,d τd /n
,...,e
.
l=1 m=−∞
Now we can interchange the first two limits, by using arguments analogous to those used in Proposition 2.1, since ∞ log CZ e− max1−m≤k≤n−m αl,k,1 τ1 /n , . . . , e− max1−m≤k≤n−m αl,k,d τd /n m=−∞
≤
d ∞ − max α τ /n log e j=1 1−m≤k≤n−m l,k,j j
m=−∞
=
∞ d 1 m=−∞
max
n j=1 1−m≤k≤n−m
m=−∞
and this last sum is bounded for all n by We have therefore
lim P
n→∞
Mn,1 ≤
= exp
∞ l=1
n
τ1
, . . . , Mn,d ≤ ∞ 1
lim
n→∞
m=−∞
n
∞ d
αl,k,j τj ≤ ∞
k=−∞
max max αl,k,j τj , n 1≤j≤d 1−m≤k≤n−m
d max1≤j≤d (αl,k,j τj ), which is summable in l Smith and Weissman (1996).
n
τd
log CZn
e
− max1−m≤k≤n−m αl,k,1 τ1 /n
− max1−m≤k≤n−m αl,k,d τd /n
,...,e
and we have to prove that lim
n→∞
∞ 1 m=−∞
= log C
∗
n
log CZn e− max1−m≤k≤n−m αl,k,1 τ1 /n , . . . , e− max1−m≤k≤n−m αl,k,d τd /n
e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd .
For fixed l, let bl,k,j = al,k,j τj , j = 1, . . . , d, and bl,k = maxj=1,...,d bl,k,j (which are summable in k and l). Suppose that bl,k,j is maximised when k = k∗ (l, j), bl,k is maximised when k = k∗ (l) (not necessarily unique) and assume, without (i)
loss of generality, that k∗ (l, 1) ≤ · · · ≤ k∗ (l, d). Break the above sum into three sums Sn , 1 = 1, 2, 3, corresponding to 1 − min{k∗ (l, 1), k∗ (l)} ≤ m ≤ n − max{k∗ (l, d), k∗ (l)}, m < 1 − min{k∗ (l, 1), k∗ (l)} and m > n − max{k∗ (l, d), k∗ (l)}.
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H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
For large n we have (1)
Sn
n−max{k∗ (l,d),k∗ (l)}
1
=
log CZn e− max1−m≤k≤n−m αl,k,1 τ1 /n , . . . , e− max1−m≤k≤n−m αl,k,d τd /n
n m=1−min{k∗ (l,1),k∗ (l)}
n − max{k∗ (l, d), k∗ (l)} + min{k∗ (l, 1), k∗ (l)}
=
n
log CZn e−αl,k∗ (l,1)1 τ1 /n , . . . , e−αl,k∗ (l,d)d τd /n ,
which converges to log C ∗ e−αl,k∗ (l,1)1 τ1 , . . . , e−αl,k∗ (l,d)d τd = log C ∗ e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd .
Otherwise, by applying (2), we can write
(2) S ≤ 1 n
d
d
max
n m<1−k∗ (l) j=1 1−m≤k≤n−m
≤ ≤
αl,k,j τj +
1
d
max αl,k,j τj n 1−k∗ (l)≤m<1−min{k∗ (l,1),k∗ (l)} j=1 1−m≤k≤n−m d
max
max αl,k,j τj +
n m<1−k∗ (l) 1−m≤k≤n−m j=1,...,d
max max αl,k,j τj n 1−k∗ (l)≤m<1−min{k∗ (l,1),k∗ (l)} 1−m≤k≤n−m j=1,...,d
d
max
max αl,k,j τj +
d
n m<1−k∗ (l) 1−m≤k≤n−m j=1,...,d
n
min{k∗ (l, 1), k∗ (l)} − k∗ (l) bl,k∗ (l)
and
(3) S ≤ 1 n
d
d
max
n m>n−k∗ (l) j=1 1−m≤k≤n−m
≤ ≤
αl,k,j τj +
1
d
max αl,k,j τj n n−max{k∗ (l,d),k∗ (l)}
max
max αl,k,j τj +
n m>n−k∗ (l) 1−m≤k≤n−m j=1,...,d
max max αl,k,j τj n n−max{k∗ (l,d),k∗ (l)}
d
max
max αl,k,j τj +
d
n m>n−k∗ (l)
1−m≤k≤n−m j=1,...,d
n
max{k∗ (l, d), k∗ (l)} − k∗ (l) bl,k∗ (l) .
(i) The second terms in the above upper bounds for Sn , i = 1, 2, tend to zero and the same holds for the first terms, by Lemma 3.2 in Smith and Weissman (1996).
For the particular case of C ∗ (u1 , . . . , ud ) = min1≤j≤d uj , (u1 , . . . , ud ) ∈ [0, 1]d , the above result agrees with (3.5) in Smith and Weissman (1996), that is
Mn,1 ≤
lim P
n→∞
n
τ1
, . . . , Mn,d ≤
If we choose C ∗ (u1 , . . . , ud ) = Martins and Ferreira (2005),
lim P
n→∞
Mn,1 ≤
n
τ1
d
n
= exp −
τd
j =1
, . . . , Mn,d ≤
∞ l =1
max
max αl,k,j τj
−∞
.
uj , (u1 , . . . , ud ) ∈ [0, 1]d , we find again the result in case 2 of the last example in
n
τd
= exp −
d ∞ j =1 l =1
max
−∞
αl,k,j τj .
In this last work only CZ = C ∗ is considered and the above two choices for C ∗ are treated directly. By calculating the limits in (3) and (4) for a given initial copula C ∗ , the above propositions enable us to obtain a larger class of MEV models. Some choices for C ∗ are available in the literature. For instance, Capéraà et al. (2000) gives sufficient conditions for an Archimedean copula CZ to be in the domain of attraction of the Gumbel–Hougaard or logistic copula C ∗ (u1 , . . . , ud ) = exp −
d j =1
α 1/α − log uj . More examples of CZ in the domain of attraction of an MEV copula C ∗ can be found in
Demarta and McNeil (2005) and Hüsler and Reiss (1989). 3. The multivariate extremal index of the M5 process In this section we will extend the results about the multivariate extremal index of the M4 process, as a corollary of Propositions 2.1 and 2.2.
H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
1493
We first recall the definition of the multivariate extremal index function θ (τ) = θ (τ1 , . . . , τd ), τ ∈ Rd+ , that relates the ˆ and which was introduced by Nandagopalan (1990). MEV distribution functions H and H A d-dimensional stationary sequence {Yn }n≥1 is said to have a multivariate extremal index θ (τ) ∈ [0, 1], τ ∈ Rd+ , if for (τ)
each τ = (τ1 , . . . , τd ) in Rd+ , there exists un (τj )
nP (Y1j > un,j ) −−−→ τj , n→∞
(τ )
(τ )
= (un,11 , . . . , un,dd ), n ≥ 1, satisfying
j = 1, . . . , d,
ˆ n ≤ u(τ) P (M −−→ γˆ (τ) ∈ (0, 1] n ) −
and P (Mn ≤ u(τ) −−→ γˆ (τ)θ (τ) . n ) −
n→∞
n→∞
Proposition 3.1. If CZ is a positive lower orthant dependent copula in the domain of attraction of a max-stable C ∗ then the multivariate extremal index of the M5 process {Yn } defined in (1) is given by ∞
θ (τ1 , . . . , τd ) =
log C ∗ e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd
l =1
∞ ∞
log C ∗ e−αl,k,1 τ1 , . . . , e−αl,k,d τd
l=1 k=−∞
and the extremal index of {Yn,j }n≥1 is
θj =
∞ l=1
max
−∞
αl,k,j ,
j = 1, . . . , d. (τj )
Proof. By applying Propositions 2.1 and 2.2 with un,j =
γˆ (τ) =
∞ ∞
C ∗ e−αl,k,1 τ1 , . . . , e−αl,k,d τd
τj n
, j = 1, . . . , d, we find
l=1 k=−∞
and
γ (τ) = γˆ (τ)θ(τ) =
∞
C ∗ e− max−∞≤k≤+∞ αl,k,1 τ1 , . . . , e− max−∞≤k≤+∞ αl,k,d τd ,
l =1 log γ (τ) log γˆ (τ)
which leads to the results for θ (τ1 , . . . , τd ) =
and θj = limτi →0+ , i∈{1,...,d}−{j} θ (τ1 , . . . , τd ).
Therefore the copulas C ∗ , Cˆ and the copula C of the limiting MEV distribution H are related by a multivariate extremal index θ (τ1 , . . . , τd ) through C ( u1 , . . . , ud ) =
=
1/θ u1 1
Cˆ (
,...,
∞ ∞
1/θ ud d
θ )
αl,k,1 /θ1
C (u1 ∗
log u
log u
− θ 1 ,...,− θ d d 1
αl,k,d /θd
, . . . , ud
)
θ
log u
log u
− θ 1 ,...,− θ d d 1
.
l=1 k=−∞
Proposition 3.1 leads to the results of Smith and Weissman (1996) and Martins and Ferreira (2005) when C ∗ is the copula of the minimum and product, respectively. Otherwise, if we take for instance the Logistic copula C ∗ we find
∞ d θ (τ1 , . . . , τd ) =
l =1
j =1
max
−∞≤k≤+∞
∞ ∞ l=1 k=∞
d
αl,k,j τj
αl,k,j τj
α
α 1/α
1/α
.
j =1
4. The tail dependence of the M5 process For a random vector X = (X1 , . . . , Xd ) with continuous margins F1 , . . . , Fd and copula C , let the bivariate (upper) tail dependence coefficients parameters be defined by
λj(,Cj′) = lim P (Fj (Xj ) > u|Fj′ (Xj′ ) > u), u ↑1
1 ≤ j < j ′ ≤ d.
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H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
(C )
The tail dependence coefficient characterises the dependence in the tail of a random pair (Xj , Xj′ ), i.e., λj,j′ > 0 (C ) corresponds to tail dependence and λj,j′ = 0 means tail independence, and can be defined via the copula of the random vector which refers to their dependence structure independently of their marginal distributions. It holds that log Cj,j′ (u, u)
λ(j,Cj′) = 2 − lim
log u
u↑1
,
where Cj,j′ is the copula of the sub-vector (Xj , Xj′ ). (C )
(Cˆ )
ˆ We first remark that, for each In this section we will relate λj,j′ to λj,j′ , where C is copula of H and Cˆ is the copula of H.
(uj , uj′ ) ∈ [0, 1]2 , we have Cj,j′ (uj , uj′ ) =
1/θj Cj,j′ uj
ˆ (
,
1/θj′ uj ′
)
θ
log u ′
log u
− θ j ,− θ j j j′
,
(5)
where θ (τj , τj′ ) is the bivariate extremal index of {(Yn,j , Yn,j′ )}n≥1 . This relation enables us to compare the tail dependence (C )
(Cˆ )
parameters λj,j′ with λj,j′ through the function θ (τj , τj′ ). Proposition 4.1. If C and Cˆ satisfy (5) then: (C )
(a) λj,j′ = 2 + θ ( θ1 , θ1′ ) log Cˆ j,j′ (e−1/θj , e j j (C )
(Cˆ )
(b) λj,j′ = λj,j′ + log
−1/θj′
),
−θ( θ1 , θ1 )/θj −θ( θ1 , θ1 )/θj′ j j′ j j′ ,e ) Cˆ j,j′ (e
Cˆ j,j′ (e−1 ,e−1 )
.
ˆ , we get Proof. From the spectral measure representation (Resnick, 1987) of the copula Cˆ , with measure W λ(j,Cj′) = 2 − lim ˆ
log Cˆ j,j′ (u, u) log u
u↑1
ˆ (w1 , . . . , wd ) = 2 + log Cˆ j,j′ (e−1 , e−1 ). max{wj , wj′ } dW
=2−
(6)
Sd
By using (5) and the homogeneity of order 0 of the multivariate extremal index, it follows that
λ(j,Cj′) = 2 − lim
log Cj,j′ (u, u) log u
u ↑1
= 2−θ = 2−θ = 2+θ
1
,
1
,
1
lim
θj θj′
1 1
1
,
j
− log u
u↑1
max
θj θj′
1/θj′ 1/θj − log u − log u log Cˆ j,j′ (uj , uj′ ) , θj θj′ log u − log uwj − log uwj′ ˆ (w1 , . . . , wd ) , θ′ dW max Sd θj
= 2 − lim θ u ↑1
Sd
w j w j′ , θj θj′
ˆ (w1 , . . . , wd ) dW
−1/θj′
log Cˆ j,j′ (e−1/θj , e
θj θj′
),
which has (6) as a particular case. To obtain the second statement we combine the first with (6) and use the max-stability of Cˆ j,j′ , as follows:
λ(j,Cj′)
= 2 + log Cˆ j,j′ (e−1 , e−1 ) − log Cˆ j,j′ (e−1 , e−1 ) + θ = λj(,Cj′) + log ˆ
Cˆ j,j′ (e−1/θj , e
Cˆ j,j′
=
+ log
)
θ
1
1
θj , θj ′
1
1
, θj θj′
−1/θj′
log Cˆ j,j′ (e−1/θj , e
)
Cˆ j,j′ (e−1 , e−1 )
ˆ λj(,Cj′)
−1/θj′
e
−θ θ1 , θ1 /θj ′ j j
,e
−θ θ1 , θ1 /θj′ ′ j j
Cˆ j,j′ (e−1 , e−1 )
.
We will now apply these relations to the particular case of M5 processes, enhancing the effect of the multivariate extremal index in the tail dependence.
H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
1495
Proposition 4.2. Let {Yn }n ≥ 1 be an M5 process defined as in (1) and such that CZ is a positive lower orthant dependent copula in the domain of attraction of C ∗ . Then, for any 1 ≤ j < j′ ≤ d it holds that: (Cˆ )
(a) λj,j′ = 2 + (b)
λ(j,Cj′)
∞ ∞ l =1
= 2 + θ ( θj , 1
1 θj′
)
log Cj∗,j′ e−αl,k,j , e
k=−∞
∞ ∞
∗ k=−∞ log Cj,j′
l =1
−αl,k,j′
.
−αl,k,j /θj
e
, e−αl,k,j′ /θj′ .
Proof. By using (6) and Proposition 2.1, we obtain (a); by using Proposition 4.1-(a) and Proposition 2.1, we get (b).
If C (u1 , . . . , ud ) = min1≤j≤d uj , (u1 , . . . , ud ) ∈ [0, 1] , we obtain ∗
d
λj(,Cj′) = 2 − ˆ
∞ ∞
max{αl,k,j , αl,k,j′ }
(7)
l=1 k=−∞
which is in general greater than zero, which agrees with the result (2.10) presented in Heffernan et al. (2007). We can also say from (b) that, for this copula,
λj(,Cj′)
=2−
∞
max
max −∞≤k≤+∞
l =1
(C )
αl,k,j /θj ,
max −∞≤k≤+∞
αl,k,j′ /θj′
(Cˆ )
and it holds that λj,j′ > λj,j′ if and only if ∞ ∞
max{αl,k,j , αl,k,j′ } −
l=1 k=−∞
where θj =
∞
max
−∞≤k≤+∞
l=1
∞
l =1
αl,k,j /θj ,
max
max−∞
max −∞≤k≤+∞
(C )
αl,k,j′ /θj′ > 0,
(Cˆ )
(C )
(Cˆ )
We can easily construct examples of M4 processes for which λj,j′ > λj,j′ or λj,j′ < λj,j′ . The results for this class of moving multivariate maxima show us that if we obtain or estimate a tail dependence parameter of the common distribution of the variables in a stationary sequence we do not have necessarily the corresponding parameter in the limiting MEV model C . The multivariate extremal index of the stationary sequence can increase or decrease the tail dependence and the extremal coefficients of the limiting MEV model Cˆ arising from the i.i.d. sequence. (C )
(Cˆ )
Since λj,j′ = λj,j′Y , the result in (7) says that in the M4 processes the variables Yn,j , j = 1, . . . , d, are in general asymptotically dependent. For the M5 processes we can choose C ∗ in order to produce variables that are asymptotically (C ∗ )
(Cˆ )
(C ∗ )
independent. Take for instance C ∗ such that λj,j′ = 0 and αl,k,j = αl,k,j′ . We then find λj,j′ = λj,j′ = 0. Proposition 4.2 also points out that even for a choice of a copula C ∗ with symmetric tail dependences, the values of the signatures αl,k,j can lead to asymmetric tail dependences in both copulas C and Cˆ , that is, to different values of λj,j′ for different choices of (j, j′ ). The results in the above proposition are translations of the classical result λ = 2 − ϵ for Cj,j′ and Cˆ j,j′ (Nelsen, 2006). In fact, if we define the extremal coefficient of the MEV copula C as the constant ϵC such that C (u, . . . , u) = uϵC for all u ∈ [0, 1], then from
C (u1 , . . . , ud ) = exp −θ
log u1
θ1
,...,
log ud
− log uj wj ˆ dW (w1 , . . . , wd ) , θj Sd 1≤j≤d
max
θd
we find
ϵC = −θ
1
θ1
,...,
1
θd
log Cˆ e−1/θ1 , . . . , e−1/θd
and, in particular,
ϵCˆ = − log Cˆ e−1 , . . . , e−1 . (Cˆ )
The relation between Cˆ j,j′ and Cj∗,j′ enables us now to obtain Proposition 4.2 from λj,j′ = 2 − ϵCˆ
(C )
j,j′
and λj,j′ = 2 − ϵCj,j′ .
Other features of the M5 processes such as a directory of tail dependence coefficients for different C ∗ and signatures αl,k,j , illustrating the range of dependence structures, the model selection and estimation, are key directions for future research. Acknowledgements The final form of this work had a significant contribution from the referees and an associate editor. They pointed out several corrections needed in the manuscript. I am very grateful to them for their help. This research was partially supported by FCT/POCI2010/FEDER/CMUBI (project EVT) and FCT/PTDC/MAT108575/2008.
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H. Ferreira / Statistics and Probability Letters 82 (2012) 1489–1496
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