extremes of multivariate armax processes

Test (2013) 22:606–627DOI 10.1007/s11749-013-0326-6

O R I G I NA L PA P E R

Extremes of multivariate ARMAX processes

Marta Ferreira · Helena Ferreira

Received: 11 December 2012 / Accepted: 17 May 2013 / Published online: 4 June 2013© Sociedad de Estadística e Investigación Operativa 2013

Abstract We define a new multivariate time series model by generalizing the AR-MAX process in a multivariate way. We give conditions on stationarity and analyzelocal dependence and domains of attraction. As a consequence of the obtained results,we derive new multivariate extreme value distributions. We characterize the extremaldependence by computing the multivariate extremal index and bivariate upper tail de-pendence coefficients. An estimation procedure for the multivariate extremal index ispresented. We also address the marginal estimation and propose a new estimator forthe ARMAX autoregressive parameter.

Keywords Multivariate extreme value theory · Maximum autoregressive processes ·Multivariate extremal index · Tail dependence · Asymptotic independence

Mathematics Subject Classification 60G70

1 Introduction

Stationary time series presenting sudden large peaks are usually well modeled byheavy tailed noise ARMA processes. However, models with practical application butsimpler treatment have been studied in literature as an alternative. Davis and Resnick(1989) proposed the MARMA process which is analogous to the ARMA by justreplacing summation by the maximum operation:

Xi = φ1Xi−1 ∨ · · · ∨ φpXi−p ∨ Yi ∨ θ1Yi−1 ∨ · · · ∨ θqYi−q,

M. Ferreira (�)Center of Mathematics, Minho University/DMA, Braga, Portugale-mail: [email protected]

H. FerreiraDepartment of Mathematics, University of Beira Interior, Covilhã, Portugal

mailto:[email protected]

Extremes of multivariate ARMAX processes 607

where 0 ≤ φi, θj ≤ 1, 1 ≤ i ≤ p,1 ≤ j ≤ q , and the innovations Yn, n ≥ 1, are inde-pendent with unit Fréchet distribution. A first-order MARMA type version, i.e.,

Xi = cXi−1 ∨ Yi, 0 < c < 1, (1)

was analyzed in Alpuim (1989) by considering innovations Yn, n ≥ 1, independentand equally distributed, not necessarily unit Fréchet. The model in (1), sometimesdenoted in literature as ARMAX, corresponds to the case α = 0 in the Haslett (1979)model

Xi = βXi−1 ∨ (αβXi−1 + Yi),

0 < β < 1,0 ≤ α ≤ 1, used to describe a solar thermal energy storage system andlater developed in, e.g., Daley and Haslett (1982) and Greenwood and Hooghiemstra(1988). Exponent versions of ARMAX, namely pARMAX and pRARMAX, wereused in the modeling of financial series (Ferreira and Canto e Castro 2010). Furtherapplications of ARMAX processes and their generalizations can be seen, for instance,in Lebedev (2008) and references therein.

The need to model variables that move together can be found in co-movements ofprices, co-movements of wind speeds and water levels, among many other examples.

The multivariate maxima of moving maxima (Smith and Weissman 1996; Zhang2006, 2008), in short M4, are models with applications in finance and climatologywhenever the data exhibit dependence and clustering of high values. The variablesof this model are asymptotically dependent, which has motivated the construction ofthe more general Extended M4 (EM4) process, in Heffernan et al. (2007), in order toobtain asymptotically independent variables.

Our motivation is the construction of new multivariate cross-sectional and serialdependent models, in view of the modeling of extreme events. To the best of ourknowledge, there are no studies about multivariate maximum autoregressive modelswhich, as we will see, may present some variability in what concerns tail dependenceand clustering of high values.

Here we consider a multivariate formulation of the ARMAX model in Alpuim(1989). More precisely, we analyze conditions on stationarity (Sect. 2), local depen-dence conditions (Sect. 3) and domains of attraction (Sect. 4). The relation betweenthe max-attractors of the process and the innovations allows us to evidence new mul-tivariate extreme value distributions (MEV). In computing the multivariate extremalindex, we find that it is possible to have clustering in all or only in some of themarginals, according to their domains of attraction. An estimation procedure for themultivariate extremal index is also stated (Sect. 4). In Sect. 5 we derive the lag-r taildependence coefficient (TDC) (Sibuya 1960; Joe 1997) and the lag-r tail indepen-dence coefficient of Ledford and Tawn (1996, 1997) and we find different types oftail dependence. Some notes on the marginal parameters estimation are given at theend (Sect. 6). In particular, we present a new estimator for the ARMAX parameter c

which is strongly consistent and asymptotically normal.

608 M. Ferreira, H. Ferreira

2 Multivariate model

Let {Xn = (Xn,1, . . . ,Xn,d)}n≥1 be a d-variate sequence, such that

Xn,j = cjXn−1,j ∨ Yn,j , n ≥ 1, j = 1, . . . , d, 0 < cj < 1, (2)

where X0 = (X0,1, . . . ,X0,d ), {Yn = (Yn,1, . . . , Yn,d)}n≥1 is a sequence of indepen-dent random vectors, independent of X0, X0 ∼ F0 and Yn ∼ G. {Xn}n≥1 thus corre-sponds to a d-variate formulation of an ARMAX process given in (1).

It is an immediate consequence of relation (2) that each marginal {Xn,j }n≥1, j ∈D = {1, . . . , d}, of the sequence {Xn}n≥1 can be written in the form

Xn,j = cnj X0,j ∨

n∨

i=1

cn−ij Yi,j , n ≥ 1. (3)

If Ln denotes the distribution of Xn = (Xn,1, . . . ,Xn,d), we also have, for each(x1, . . . , xd) ∈ R

d ,

Ln(x1, . . . , xd) = Ln−1

(x1

c1, . . . ,

xd

cd

)G(x1, . . . , xd) (4)

and

Ln(x1, . . . , xd) = F0

(x1

cn1, . . . ,

xd

cnd

) n∏

i=1

G

(x1

cn−i1

, . . . ,xd

cn−id

). (5)

Observe that for each fixed (x1, . . . , xd) the sequence Ln(x1, . . . , xd) is non-increasing with values in [0,1] and thus there exists limn→∞ Ln(x1, . . . , xd) whichwe denote by F(x1, . . . , xd). Then, based on (4), we have, for each (x1, . . . , xd) ∈ R

d ,

F(x1, . . . , xd) = F

(x1

c1, . . . ,

xd

cd

)G(x1, . . . , xd) (6)

and, by (5),

F(x1, . . . , xd) = limn→∞

n∏

i=0

G

(x1

ci1

, . . . ,xd

cid

)=

∞∏

i=0

G

(x1

ci1

, . . . ,xd

cid

). (7)

Proposition 2.1 {Xn}n≥1 is a strictly stationary sequence with common non-degenerate distribution if and only if there exists (x1, . . . , xd) ∈ R

d+ such that

0 <

∞∑

i=0

− logG

(x1

ci1

, . . . ,xd

cid

)< ∞. (8)

In this case, the common distribution F of {Xn}n≥1 satisfies (6).

Proof Since {Xn}n≥1 is a tight sequence, then, by Prokhorov (1956), any subse-quence of {Xn}n≥1 has a subsequence that converges in distribution. Therefore, the


limiting function F(x1, . . . , xd) = ∏∞i=0 G(x1

ci1, . . . ,

xd

cid

) in (7) is a distribution func-

tion of some random vector.Now, observe that F is non-degenerate if and only if there exists x = (x1, . . . , xd)

such that 0 < F(x1, . . . , xd) < 1, i.e., such that

0 <

∞∏

i=0

G

(x1

ci1

, . . . ,xd

cid

)< 1.

The assertion in (8) is straightforward by taking logarithms and, if it holds for somex ∈ R

d , then G(x1ci

1, . . . ,

xd

cid

)→i→∞ 1, and thus x ∈ Rd+. �

As a consequence of (8), if any of the marginals Gj , j ∈ D, of G has support withnon-positive right end-point then the corresponding marginal Fn,j has degeneratelimiting distribution and, therefore, F is a d-dimensional degenerate distribution.

Observe that (8) is satisfied by every multivariate distribution with positive depen-dence and marginals Gj satisfying

0 <

∞∑

i=0

− logGj

(x/ci

j

)< ∞, for some x > 0.

This latter condition is satisfied, for instance, by the Generalized Pareto distribution(Alpuim 1989).

Consider the particular case in which G is MEV with Fréchet marginals Gj ,j ∈ D. In this case, the limiting distribution F is also an MEV distributionsince Fj (x) = exp(−xαj /(1 − c

αj

j )), j ∈ D, and its copula CF (u1, . . . , ud) =F(F−1

1 (u1), . . . ,F−1d (ud)), (u1, . . . , ud) ∈ [0,1]d , is max-stable, that is, for each

integer k ≥ 1 and (u1, . . . , ud) ∈ [0,1]d , we have

CkF

(u

1/k

1 , . . . , u1/kd

)

= CkF

(F

1/k

1 (x1), . . . ,F1/kd (xd)

)

= CkF

(F1

(x1k

1/α1), . . . ,Fd

(xdk1/αd

))

= Fk(x1k

1/α1 , . . . , xdk1/αd)

= limn→∞

n∏

i=0

Gk

(x1k

1/α1

ci1

, . . . ,xdk1/αd

cid

)

= limn→∞

n∏

i=0

CkG

(G

1/k

1

(x1

ci1

), . . . ,G

1/kd

(xd

cid

))

= limn→∞

n∏

i=0

G

(x1

ci1

, . . . ,xd

cid

)= F(x1, . . . , xd) = CF (u1, . . . , ud).


However, it is not always easy to find a distribution F(x1, . . . , xd) = ∏∞i=0 G(x1

ci1,

. . . ,xd

cid

). Consider the following choice for G, where we can state clearly F . Assume

that {Ij , j = 1, . . . , k} is a partition of D, ci = cIj, i ∈ Ij , j = 1, . . . , k, RIj

is anMEV with unit Fréchet marginals, and define

G(x1, . . . , xd) =k∏

j=1

RIj(xIj

).

Thus we have

F(x1, . . . , xd) =∞∏

i=0

G

(x1

ci1

, . . . ,xd

cid

)=

∞∏

i=0

k∏

j=1

RIj

(xIj

ciIj

)

=∞∏

i=0

k∏

j=1

RciIj

Ij(xIj

) =k∏

j=1

RbIj

Ij(xIj

)

with bIj= 1/(1 − cIj

), j = 1, . . . , k. In particular, if cj = c, j ∈ D, we find

F(x1, . . . , xd) = G1/(1−c)(x1, . . . , xd).

In the sequel we always assume that {Xn}n≥1 is a stationary d-variate ARMAXsequence, with common distribution F .

3 Long-range and local dependence

As showed in Alpuim (1989) for the univariate case, we prove that the strong-mixingcondition also holds for the multivariate sequence, i.e., for any A ∈ B(X1, . . . ,Xp)

and B ∈ B(Xp+s+1,Xp+s+2, . . .),

∣∣P(A ∩ B) − P(A)P (B)∣∣ ≤ αs

with αs →s→∞ 0, where B(·) denotes the σ -field generated by the indicated randomvectors.

In what follows, all operations and inequalities between vectors are understood tobe component-wise.

Proposition 3.1 {Xn}n≥1 satisfies the strong-mixing condition.

Proof Consider A ∈ B(X1, . . . ,Xp) and B ∈ B(Xp+s+1,Xp+s+2, . . .) and let

Cs = {Yp+1 ≤ cXp, . . . ,Yp+s+1 ≤ cXp+s}.


We have∣∣P(A ∩ B) − P(A)P (B)

∣∣

= ∣∣P(A ∩ B ∩ Cs) + P(A ∩ B ∩ Cs) − P(A)P (B ∩ Cs) − P(A)P (B ∩ Cs)∣∣

≤ ∣∣P(A ∩ B ∩ Cs) − P(A)P (B ∩ Cs)∣∣ + ∣∣P(A ∩ B ∩ Cs) − P(A)P (B ∩ Cs)

∣∣.

Observe that, for the first term,∣∣P(A ∩ B ∩ Cs) − P(A)P (B ∩ Cs)

∣∣

= ∣∣P(Cs)P (B|Cs)P (A|B ∩ Cs) − P(A)P (Cs)P (B|Cs)∣∣

≤ P(Cs)P (B|Cs)∣∣P(A|B ∩ Cs) − P(B|Cs)

∣∣

≤ P(Cs).

On the other hand, since Xp+s+1 is a function of Xp, . . . ,Xp+s ,Yp+1, . . . ,Yp+s+1,

Xp+s+1 = cp+s+1−kXk ∨p+s+1−k∨

i=1

cp+s+1−k−iYk+i , k = p, . . . ,p + s,

and B ∈ B(Xp+s+1,Yp+s+1,Yp+s+2, . . .), then we can write

A ∩ B ∩ Cs = A ∩ B ′ ∩ Cs and B ∩ Cs = B ′ ∩ Cs,

where B ′ ∈ B(Yp+1,Yp+2, . . .). Thus being, for the second term we have

∣∣P(A ∩ B ∩ Cs) − P(A)P (B ∩ Cs)∣∣ = ∣∣P

(A ∩ B ′ ∩ Cs

) − P(A)P(B ′ ∩ Cs

)∣∣

≤ ∣∣P(A ∩ B ′ ∩ Cs

) − P(A ∩ B ′)∣∣ + ∣∣P

(A ∩ B ′) − P(A)P

(B ′ ∩ Cs

)∣∣

= P(A ∩ B ′)P

(Cs |A ∩ B ′) + ∣∣P(A)P

(B ′) − P(A)P

(B ′ ∩ Cs

)∣∣

≤ 2P(Cs).

Now we just need to prove that P(Cs)→s→∞ 0. Observe that

P(Cs) = P(Yp+1 ≤ cXp,Yp+2 ≤ c2Xp, . . . ,Yp+s+1 ≤ cs+1Xp

)

≤ P(Yp+s+1 ≤ cs+1Xp

) ≤ P(Yp+s+1 �≥ cs+1Xp

)

= 1 − P(Yp+s+1 ≥ cs+1Xp

)

= 1 −∫

Rd

H

(y1

cs+11

, . . . ,yd

cs+1d

)dG(y1, . . . , yd) →

s→∞ 0.�

Therefore, {Xn}n≥1 satisfies condition D(un,αln), for any sequence of real vectors{un}n≥1 and for any sequence {ln}n≥1 such that ln → ∞, corresponding to the multi-variate version of Leadbetter’s D-condition of local dependence (see, e.g., Leadbetteret al. 1983).


Now we see that {Xn}n≥1 also satisfies the multivariate version of D′′ condition ofLeadbetter and Nandagopalan (1989). For a given sequence of real vectors {un}n≥1,we say that condition D′′(un) holds if D(un,αln) also holds and

n

[n/kn]∑

i=2

P(X1 �≤ un,Xi ≤ un,Xi+1 �≤ un) → 0

for some sequence {kn}n≥1 such that, as n → ∞,

kn → ∞,knln

n→ 0 and knαln → 0

(see, e.g., Leadbetter et al. 1983 and Leadbetter and Nandagopalan 1989).

Proposition 3.2 {Xn}n≥1 satisfies condition D′′(u(τ )n ).

Proof Observe that

n

[n/kn]∑

i=2

P(X1 �≤ u(τ )

n ,Xi ≤ u(τ )n ,Xi+1 �≤ u(τ )

n

)

≤d∑

j=1

n

[n/kn]∑

i=2

P(X1,j > u

(τj )

n,j ,Xi,j ≤ u(τj )

n,j < Xi+1,j

)

+∑

1≤s,s′≤d

n

[n/kn]∑

i=2

P(X1,s > u(τs)

n,s ,Xi ≤ u(τ )n ,Xi+1,s′ > u

(τs′ )n,s′

).

Since each marginal sequence {Xn,j }n≥1 satisfies condition D′′(u(τj )

n,j ) (Canto e Cas-tro 1992), the first term above has null limit, as n → ∞. The second term above isupper bound by, successively,

∑

1≤s,s′≤d

n

[n/kn]∑

i=2

P(X1,s > u(τs)

n,s ,Xi,s′ ≤ u(τs′ )n,s′ < cs′Xi,s′ ∨ Yi+1,s′

)

=∑

1≤s,s′≤d

n

[n/kn]∑

i=2

P(X1,s > u(τs)

n,s ,Xi,s′ ≤ u(τs′ )n,s′ < Yi+1,s′

)

≤∑

1≤s,s′≤d

n

[n/kn]∑

i=2

P(X1,s > u(τs)

n,s

)P

(Yi+1,s′ > u

(τs′ )n,s′

)

=∑

1≤s,s′≤d

n

[n

kn

](1 − Fs

(u(τs)

n,s

))(1 − Fs′(u

(τs′ )n,s′ )

Fs′(u(τs′ )n,s′ /cs′)

)


≤ 1

kn

∑

1≤s,s′≤d

n(1 − Fs

(u(τs)

n,s

))(n(1 − Fs′

(u

(τs′ )n,s′

)) − n(1 − Fs′

(u

(τs′ )n,s′ /cs′

)))

× 1

Fs′(u(τs′ )n,s′ /cs′)

,

which also converges to zero for any sequence kn → ∞, since by (9) we have n(1 −Fs′(u

(τs′ )n,s′ /cs′))→ τ ∗

s′ ≥ 0, as n → ∞. �

4 Multivariate extremal index

A phenomenon also noticed in real data is that extreme events often tend to occurin clusters. The measure that is used to capture the clustered extremal dependence isthe extremal index (Leadbetter et al. 1983). More precisely, the extremal index can beinterpreted as the reciprocal of the limiting mean cluster size. A unit extremal indexmeans no serial clustering and is a form of asymptotic independence of extremes.Figure 1(a) and (b), presents the marginal sample paths of a bivariate ARMAX pro-cess with G(x1, x2) = exp(−(x

−1/γ

1 + x−1/γ

2 )γ ), γ = 0.5, c1 = 0.1 and c2 = 0.8. Inthe first marginal, large values tend to occur almost singly and, in the second, we finda visible high values clustering.

For each j ∈ D, suppose that Fj belongs to the max-domain of attraction of Hj ,in short Fj ∈ D(Hj ), i.e., there exists constants {an,j > 0}n≥1 and {bn,j }n≥1, suchthat

n(1 − Fj (an,j x + bn,j )

) →n→∞− logHj(x),

Fig. 1 Marginal sample paths of a bivariate ARMAX process with unit Fréchet innovations: the left plotcorresponds to c = 0.1 where the high values tend to occur almost singly, and the right plot correspondsto c = 0.8 with visible high values clustering


where Hj may be a Gumbel, a Weibull or a Fréchet distribution, respectively,

Λ(x) = exp(−e−x), Ψαj(x) = e−(−x)

αj, x ≤ 0, and Φαj

(x) = e−x−αj

, x > 0, for

some αj > 0. Therefore, a sequence of normalizing levels {u(τj )

n,j }n≥1 for {Xn,j }n≥1,i.e., such that

n(1 − Fj

(u

(τj )

n,j

)) →n→∞ τj ≥ 0

can be written as

u(τj )

n,j = an,jH−1j

(e−τj

) + bn,j

with H−1j (x) = inf{y : F(y) ≥ x} the generalized inverse of Hj . By applying the

Khintchine’s type theorem (see, e.g., Leadbetter et al. 1983), we arrive at the follow-ing property of the normalizing levels for {Xn,j }n≥1 that is used later:

u(τj )

n,j

cj

= u(τ∗

j )

n,j , with τ ∗j =

{0, if Hj ∈ {Λ,Ψαj

},τj c

αj

j , if Hj = Φαj.

(9)

In the sequel we denote by {u(τ )n = (u

(τ1)n,1 , . . . , u

(τd )n,d )}n≥1 the sequence of normalizing

random vectors.The results of the previous section allow us to compute the multivariate extremal

index of {Xn}n≥1 (Nandagopalan 1990). More precisely, if for all τ ∈ Rd+ there exist

normalizing levels {u(τ )n = (u

(τ1)n,1 , . . . , u

(τd )n,d )}n≥1, such that the sequence {nP (X1 �≤

u(τ )n )}n≥1 is convergent and D′′(u(τ )

n ) holds, then {Xn}n≥1 has a multivariate extremalindex function θ if and only if, ∀τ ∈ R

d+, sequence {nP (X1 ≤ u(τ )n ,X2 �≤ u(τ )

n )}n≥1converges too. In this case,

θ(τ1, . . . , τd) = limn→∞

P(X1 ≤ u(τ )n ,X2 �≤ u(τ )

n )

P (X1 �≤ u(τ )n )

, τ ∈ Rd+ (10)

(Ferreira 1994). The marginal extremal index is obtained by θj = limτi→0+i �=j

θ(τ1,

. . . , τd).

Proposition 4.1 If F ∈ D(H) then {Xn}n≥1 has multivariate extremal index func-tion θ with

θ(τ1, . . . , τd) = 1 − logCHI(e

−τj cαjj , j ∈ I )

logCH (e−τ1 , . . . , e−τd ),

for (τ1, . . . , τd) ∈ Rd+, where I is the set of indexes in D for which Hj(x) = Φαj

(x) =e−x

−αj, x > 0. Moreover,

θj ={

1, if Hj ∈ {Λ,Ψαj},

1 − cαj

j , if Hj = Φαj

(11)

is the extremal index of {Xn,j }n≥1, j = 1, . . . , d .


Proof By hypothesis, F ∈ D(H), i.e., Fj ∈ D(Hj , {an,j > 0}, {bn,j }), with Hj of

the extremal type Λ, Ψαjor Φαj

, and CnF (u

1/n

1 , . . . , u1/nd )→n→∞ CH (u1, . . . , ud),

(u1, . . . , ud) ∈ [0,1]d . Thus we guarantee the existence of normalizing levels u(τ )n =

(u(τ1)n,1 , . . . , u

(τd )n,d ), for which the condition D′′(u(τ )

n ) holds.

Moreover, for u(τj )

n,j = an,jH−1j (e−τj ) + bn,j , j = 1, . . . , d , we have

nP(X1 �≤ u(τ )

n

)

= n(1 − F

(an,1H

−11

(e−τ1

) + bn,1, . . . , an,dH−1d

(e−τd

) + bn,d

))

→n→∞− logH

(H−1

1

(e−τ1

), . . . ,H−1

d

(e−τd

)).

On the other hand,

nP(X1 ≤ u(τ )

n ,X2 �≤ u(τ )n

)

= nP(X1 ≤ u(τ )

n , cX1 ∨ Y2 �≤ u(τ )n

)

= nP(X1 ≤ u(τ )

n

)P

(Y2 �≤ u(τ )

n

) = P(X1 ≤ u(τ )

n

)n(1 − G

(u

(τ1)n,1 , . . . , u

(τd )n,d

))

= P(X1 ≤ u(τ )n )

P (X1 ≤ u(τ )n /c)

(n(1 − F

(u

(τ1)n,1 , . . . , u

(τd )n,d

)) − n

(1 − F

(u

(τ1)n,1

c1, . . . ,

u(τd )n,d

cd

)))

→n→∞− logH

(H−1

1

(e−τ1

), . . . ,H−1

d

(e−τd

))

+ logH(H−1

1

(e−τ∗

1), . . . ,H−1

d

(e−τ∗

d))

= − logH(H−1

1

(e−τ1

), . . . ,H−1

d

(e−τd

))

+ logHI

(H−1

1

(e−τ∗

1), . . . ,H−1

d

(e−τ∗

d))

I,

where I is the set of indexes in D for which τ ∗j given in (9) are positive, i.e., for

which Hj(x) = Φαj(x) = e−x

−αj, x > 0, and HI denotes the marginal distribution of

H corresponding to those indexes. Therefore, applying (10), we have

θ(τ1, . . . , τd) = 1 − logHI (H−11 (e−τ∗

1 ), . . . ,H−1d (e−τ∗

d ))I

logH(H−11 (e−τ1), . . . ,H−1

d (e−τd ))

= 1 − logCHI(e−τ∗

1 , . . . , e−τ∗d )I

logCH (e−τ1, . . . , e−τd ).

Observe that if I = ∅ then θ(τ1, . . . , τd) = 1, ∀τ , and if I �= ∅, we have

θj =⎧⎨

⎩

1, if j ∈ D − I,

1 − τ∗j

τj, if j ∈ I

leading to the assertion (11), which corresponds to the univariate marginal extremalindex already derived in Alpuim (1989). �


As a consequence of the previous proposition, we conclude that, if

Fn(an1x1 + bn1, . . . , andxd + bnd) →n→∞H(x1, . . . , xd),

for all continuity points of H , then

P

(n∨

i=1

Xi,1 ≤ an,1x1 + bn,1, . . . ,

n∨

i=1

Xi,d ≤ an,dxd + bn,d

)

→n→∞T (x1, . . . , xd)

= {H(x1, . . . , xd)

}1− logCH (H1(x1)1−θ1 ,...,Hd (xd )1−θd )

logCH (H1(x1),...,Hd (xd ))

= {H(x1, . . . , xd)

}1− logHI (x1c1

,...,xdcd

)I

logH(x1,...,xd ) ,

where I is the set of indexes j in D for which Hj(x) = exp(−x−αj ), x > 0.If we consider that F is an MEV with Fréchet marginals, then F ∈ D(F ) and

G(x1, . . . , xd) = F(x1, . . . , xd)

F (x1c1

, . . . ,xd

cd)

=exp

(− ∫Sd

∨dj=1

wj

xαjj

dW(w))

exp(− ∫

Sd

∨dj=1

wj cαjj

xαjj

dW(w)) ,

where W is a measure on Sd = {w : ∑dj=1 wj = 1} such that

∫Sd

wj dW(w) = 1.Therefore

G(x1, . . . , xd) = F(x1, . . . , xd)1− logF(

x1c1

,...,xdcd

)

logF(x1,...,xd )

and, since I = D, we have T (x1, . . . , xd) = G(x1, . . . , xd).

Example 1 Consider F with F1,F2 ∈ D(Λ) and Fj ∈ D(Φ1), j = 3, . . . , d . If wehave CF (u1, . . . , ud) = exp(−(

∑dj=1(− loguj )

1/γ )γ ), 0 < γ ≤ 1, then F ∈ D(H),with CH = CF , H1 = H2 = Λ and Hj = Φ1, j = 3, . . . , d . Therefore, we have

θ(τ1, . . . , τd) = 1 − (∑d

j=3(τj cj )γ )1/γ

(∑d

j=1 τγ

j )1/γ, (τ1, . . . , τd) ∈ R

d+,

θ1 = θ2 = 1 and θj = 1 − cj , j = 3, . . . , d .

Example 2 Consider F with Fj ∈ D(Φ1), j = 1, . . . , d and CF (u1, . . . , ud) =∧dj=1 uj . Then F ∈ D(H), with CH = CF , Hj = Φ1, j = 1, . . . , d . Therefore, we

have

θ(τ1, . . . , τd) = 1 −∨d

j=1 τj cj∨d

j=1 τj

, (τ1, . . . , τd) ∈ Rd+.


The next result relates the domain of attraction of F with the one of G.

Proposition 4.2 If F ∈ D(H) then G ∈ D(V ) with Vj = Hθj

j and θj given in (11),j ∈ D, and

CV (u1, . . . , ud) = CH (u1/θ11 , . . . , u

1/θd

d )

CH (u1/θ1−11 , . . . , u

1/θd−1d )

. (12)

Proof By hypothesis, Fj ∈ D(Hj , {an,j > 0}, {bn,j }), j ∈ D, i.e., Fnj (an,j xj +

bn,j )→n→∞ Hj(xj ) and CnF (u

1/n

1 , . . . , u1/nd )→n→∞ CH (u1, . . . , ud), (u1, . . . ,

ud) ∈ [0,1]d . In addition,

Fnj

(an,j xj + bn,j

cj

)→

n→∞

⎧⎨

⎩

1, if Hj ∈ {Λ,Ψαj},

Hcαjj

j (xj ), if Hj = Φαj.

From the stationarity relation in (6), we have

Fnj (an,j xj + bn,j ) = Fn

j

(an,j xj + bn,j

cj

)Gn

j (an,j xj + bn,j ).

Therefore,

Gnj (an,j xj + bn,j ) →

n→∞

⎧⎨

⎩

Hj(xj ), if Hj ∈ {Λ,Ψαj},

H1−c

αjj

j (xj ), if Hj = Φαj

and thus Gj ∈ D(Hθj

j ), j ∈ D, with θj given in (11).Now we look at the copula of G. We have

Fn(an,1x1 + bn,1, . . . , an,dxd + bn,d)

→n→∞H(x1, . . . , xd) = CH

(H1(x1), . . . ,Hd(xd)

)

and

Fn

(an,1x1 + bn,1

c1, . . . ,

an,dx1 + bn,d

cd

)

= CnF

((Fn

1

(an,1x1 + bn,1

c1

))1/n

, . . . ,

(Fn

d

(an,dx1 + bn,d

cd

))1/n)

→n→∞CHI

(H

cαjj

j (xj ), j ∈ I) = CHI

(Hj

(xj

cj

), j ∈ I

)

= HI

(xj

cj

, j ∈ I

).


Again, from the relation between F and G in (6), we obtain

Gn(an,1x1 + bn,1, . . . , an,dxd + bn,d) →n→∞

CH (H1(x1), . . . ,Hd(xd))

CHI(H

cαjj

j (xj ), j ∈ I )

.

Thus we can say that G ∈ D(V ), where Vj = Hθj

j and

H(x1, . . . , xd) = HI

(xj

cj

, j ∈ I

)V (x1, . . . , xd),

or equivalently,

CH (u1, . . . , ud) = CH

(u

1−θ11 , . . . , u

1−θd

d

)CV

(u

θ11 , . . . , u

θd

d

),

given θj , j ∈ D, stated in (11). �

Observe that if θj = θ , j = 1, . . . , d , or if CH is the product copula then CV = CH .The expression obtained for the multivariate extremal index function has the ad-

vantage that, once known/estimated the constants cj and the marginal domains ofattraction, we are only dependent on the attractor copula of {Xn}n≥1 correspondingto the i.i.d. sequence with the same distribution F .

Since we have CnF (u

1/n

1 , . . . , u1/nd )→n→∞ CH (u1, . . . , ud) uniformly in [0,1]d

and CH is continuous, we can replace the discrete variable n by a continuous variablet and equivalently state t (1 −CF (1 − x1/t, . . . ,1 − xd/t))→t→∞ logCH (e−x1 , . . . ,

e−xd ), x ∈ [0,∞)d . If we rewrite the result of the previous proposition as

θ(τ1, . . . , τd) = 1 − limt→∞

t (1 − CFI(1 − τ1c

α11 /t, . . . ,1 − τdc

αd

d /t)I )

t (1 − CF (1 − τ1/t, . . . ,1 − τd/t))

= 1 − limt→∞

tP (⋃

j∈I {FX1,j(X1,j ) > 1 − τ1c

α11 /t})

tP (⋃d

j=1{FX1,j(X1,j ) > 1 − τ1/t}) ,

we can then estimate the multivariate extremal index through tail dependence func-tions estimators concerning FI and F . For this issue see, e.g., Huang (1992), Schmidtand Stadtmüller (2006), Einmahl et al. (2012) and references therein.

5 Coefficients of tail dependence and tail independence

Loosely speaking, tail dependence describes the limiting proportion of exceedancesof a high threshold by one margin given that the other margin has already exceededthat threshold. The most used definition of tail dependence, provided in the mono-graph of Joe (1997), is the tail dependence coefficient (TDC):

λ = limt↓0

P(FY (Y ) > 1 − t |FX(X) > 1 − t

). (13)


We say that the random pair (X,Y ) is tail dependent whenever λ > 0 and tail inde-pendent if λ = 0.

In case of tail independence, Ledford and Tawn (1996, 1997) proposed to modelthe null limit in (13) by introducing a coefficient (η) to rule the decay rate of the jointbivariate survival function:

P(FY (Y ) > 1 − t |FX(X) > 1 − t

) ∼ L(t)t1/η−1, as t ↓ 0,

where L is a slowly varying function at 0, i.e. L(tx)/L(t) → 1 as t ↓ 0, for anyfixed x > 0 and η ∈ (0,1] is a constant. Coefficient η measures the degree of tailindependence between r.v.’s X and Y . Observe that tail dependence occurs if η = 1and L(t) �→ 0, as t ↓ 0, and tail independence otherwise. The r.v.’s X and Y arecalled positively associated when 1/2 < η < 1, nearly independent when η = 1/2and negatively associated when 0 < η < 1/2.

Both concepts can be naturally extended to a lag-r (r ∈ N0) formulation of {Xn =(Xn,1, . . . ,Xn,d)}n≥1. More precisely, the lag-r TDC is defined as

λ(r)

jj ′(X) = limt↓0

P(Fj ′(X1+r,j ′) > 1 − t |Fj (X1,j ) > 1 − t

)

and the lag-r (r ∈ N0) Ledford and Tawn coefficient η(r)

jj ′(X) defined by

P(Fj ′(X1+r,j ′) > 1 − t |Fj (X1,j ) > 1 − t

) ∼ L(t)t1/η

(r)

jj ′ (X)−1, as t ↓ 0,

where L is a slowly varying function at 0. We denote λjj ′(X) ≡ λ(0)

jj ′(X) as the TDC

between the j th and the j ′th components, λ(r)j (X) ≡ λ

(r)jj (X) is the lag-r TDC within

the j th sequence and λ(r)

jj ′(X) is the lag-r cross-sectional TDC between the j th andthe j ′th sequences. An analogous description holds for the Ledford and Tawn coeffi-cients, respectively, ηjj ′(X), η

(r)j (X) and η

(r)

jj ′(X).

In the sequel we denote wj ′t = F−1j ′ (1 − t).

Proposition 5.1 If Cjj ′ is the common copula of (Xn,j ,Xn,j ′), n ≥ 1, then

λ(r)

jj ′(X) = 2 − limt↓0

1

t

(1 − Cjj ′

(1 − t,Fj ′

(c−rj ′ wj ′t

)) 1 − t

Fj ′(c−rj ′ wj ′t )

).

Moreover,

1 − 1 − t


≤ 2 − 1

t

(1 − Cjj ′

(1 − t,Fj ′

(c−rj ′ wj ′t

)) 1 − t


)

≤ 2 − 1

t

(1 − (1 − t)2


). (14)


Proof We have that

limt↓0

P(Fj (X1,j ) > 1 − t,Fj ′(X1+r,j ′) > 1 − t)

P (Fj (X1,j ) > 1 − t)

= 2 − limt↓0

1

t

(1 − P

(Fj (X1,j ) ≤ 1 − t,Fj ′(X1+r,j ′) ≤ 1 − t

))

= 2 − limt↓0

1

t

(1 − P

(Fj (X1,j ) ≤ 1 − t,Fj ′(X1,j ′)

≤ Fj ′(c−rj ′ wj ′t

)) r∏

i=1

Gj ′(wj ′t /c

r−ij ′

))

.

Now the first result follows from (6). The second assertion is a consequence ofthe Fréchet–Hoeffding copula bounds, i.e., max(u1 + u2 − 1,0) ≤ C(u1, u2) ≤min(u1, u2), for all (u1, u2) ∈ [0,1]2. �

In the following we state some consequences of this result, considering differentsituations for the domains of attraction of Fj ′ .

Corollary 5.2 Under the condition of Proposition 5.1, we have

λ(r)

jj ′(X) = 2 − limt↓0

1

t

(1 − Cjj ′

(1 − t,1 − tc

rαj ′j ′

) 1 − t

1 − tcrαj ′j ′

)

if Fj ′ ∈ D(Φαj ′ ). Moreover, 0 ≤ λ(r)

jj ′(X) ≤ cαj ′ rj ′ .

Proof The result is straightforward since we have Fj ′(c−rj ′ wj ′t )∼t↓0 1 − tc

rαj ′j ′ (see,

for instance, Proposition 3.3 in Ferreira and Canto e Castro 2008). �

Corollary 5.3 Under the condition of Proposition 5.1, we have λ(r)

jj ′(X) = 0 and

η(r)

jj ′(X) = 1/2, whenever Fj ′ has positive finite right end-point, with r ∈ N.

Proof Observe that Fj ′(F−1j ′ (1 − t)c−r

j ′ )∼t↓0 1, and thus

2 − 1

t

(1 − Cjj ′(1 − t,1)(1 − t)

) ∼t↓0

2 − 1

t

(1 − (1 − t)2) ∼

t↓0t. �

Corollary 5.4 Under the condition of Proposition 5.1, we have λ(r)

jj ′(X) = 0 and

1/2 ≤ η(r)

jj ′(X) ≤ max(1/2, crkj ′ ) whenever Fj ′(c−r

j ′ wj ′t ) ∼ 1 − tc−rk

j ′ , for k > 0 andr ∈ N.

Proof Just observe that the left and right hand-side of (14) approximates, respec-

tively, t and t + tc−rk

j ′ −1, as t ↓ 0. �


Fig. 2 Scatter-plots for model (X1,X2,X3) having marginals ARMAX, respectively, c = 0.8,0.1,0.1with innovations distributed as unit Fréchet, Exponential and Uniform, where G has Gumbel copula,CG(u1, u2, u3) = exp(−(

∑3j=1(− loguj )1/γ )γ ), 0 < γ ≤ 1; Left to right: points of (X1,X2), (X1,X3)

and (X2,X3) with Gumbel’s copula dependence parameter γ = 0.1 (strong dependence) on the top andGumbel’s copula dependence parameter γ = 0.9 (weak dependence) on the bottom

Examples of d.f.’s satisfying Fj ′(c−rj ′ wj ′t ) ∼ 1− t

c−rk

j ′ , k > 0, include, e.g., Weibull

(with d.f. F(x) = 1 − exp(−xk)) and Exponential (k = 1).In a max-autoregressive context, the non-negative associated tail independence

(1/2 ≤ η < 1) can also be described through a pARMAX process (Ferreira and Cantoe Castro 2008, 2010), whose logarithm corresponds to ARMAX.

An illustration of the tail dependence between the marginals of {Xn}n≥1, for thethree cases of domains of attraction can be seen in Fig. 2. Observe that the copula’sdependence is determinant: dependence is evident whenever a strong dependent cop-ula is used, whilst a weak dependent copula leads to an almost random scatter-plot.Figure 3 illustrates cross-sectional tail dependence of {Xn}n≥1, considering againthe three domains. Observe the presence of some tail dependence for random pairs(Xj ,X

(r)

j ′ ) whenever the lag-r apart j ′th marginal is Fréchet (first column plots) cor-roborating Proposition 5.2. An almost randomness can be seen in the other scatter-plots which is consistent with Propositions 5.3 and 5.4.

6 Marginal parameters estimation

In this section we focus on the marginal ARMAX autoregressive parameter. In whatconcerns the tail index αj of each marginal j ∈ D, it can be estimated through es-timators already stated in literature as Hill (in case αj > 0), Pickand’s, maximumlikelihood, moments or generalized moments estimator, whose asymptotic proper-


Fig. 3 Cross-sectional scatter-plots for model (X1,X2,X3) having marginals ARMAX (0.5) with inno-vations distributed as, respectively, unit Fréchet, Exponential and Uniform, where G has Gumbel copula

(γ = 0.5); Left to right and top to bottom: points of (X1,X(2)1 ), (X1,X

(2)2 ), (X1,X

(2)3 ), (X2,X

(2)1 ),

(X2,X(2)2 ), (X2,X

(2)3 ), (X3,X

(2)1 ), (X3,X

(2)2 ) and (X3,X

(2)3 ), where X

(r)j

denotes the j th marginallag-r apart

ties of consistency and normality still hold under an ARMAX dependence structure(Ferreira and Canto e Castro 2008; Proposition 3.4).

The following result allows to introduce an estimator for the ARMAX parame-ter cj , j ∈ D.

Proposition 6.1 If F0,j and Gj are unit Fréchet d.f.’s, j ∈ D, then

cj = 2 − 1

E(e−X−1

j )

, (15)

where Xj is a r.v. with the distribution of {Xn,j }n≥1.

Proof We use the result of Proposition 3.1 in Ferreira and Ferreira (2012). Moreprecisely, considering F(x) = e−x−1

and s ∈ N, we have

E(F(Xn,j )

s) = E

(F

(cnj X0,j ∨

n∨

i=1

cn−ij Yi,j

)s)


= − logF(X0,j ,Y1,j ,...,Yn,j )(c−nj , c−n+1

j , . . . , c−1j ,1)

s − logF(X0,j ,Y1,j ,...,Yn,j )(c−nj , c−n+1

j , . . . , c−1j ,1)

=∑n

i=0 cn−ij

s + ∑ni=0 cn−i

j

. (16)

Taking limits in both of the members with s = 1, we have

E(e−X−1

j) =

11−cj

1 + 11−cj

,

leading to the assertion. �

As a consequence of this result, we verify that if {Xn,j }n≥1 is stationary than

E(e−X−1

j ) ∈ (1/2,1).From (15) we derive the estimator

cj = 2 − 1

Uj

, (17)

where Uj = 1n

∑ni=1 e

−X−1i,j . Observe that no definite result can be obtained for Uj ≤

1/2, which may be an indication of an unsuitable model choice.Based on (3), we have that Xn,j = ∨∞

i=1 cij Yn−i,j is the unique stationary solution

of recursion (1) (Davis and Resnick 1989; Proposition 2.2). Therefore, an ARMAX

process is ergodic (Stout 1974; Theorem 3.5.8) and, since E(|e−X−1j |) < ∞, we have

Uj → E(e−X−1

j ) almost surely (see, e.g. ergodic theory in Billingsley 1995). Thusestimator cj is strongly consistent. The asymptotic normality is stated in the nextresult.

Proposition 6.2 Under the conditions of Proposition 6.1, we have√

n(Uj −E(e

−X−1j )) → N(0, σ 2) where

σ 2 = 1

3 − 2cj

−(

1

2 − cj

)2

+ 2∞∑

r=1

( 1 − crj

(2 − cj )(2 − cj − crj − cr+1

j )−

(1

2 − cj

)2).

Moreover,√

n(cj − cj ) → N(0, σ 2(3 − 2cj )).

Proof The asymptotic normality also holds given the strong-mixing dependencestructure with variance given by Billingsley (1995)

σ 2 = var(e−X−1

j) + 2

∞∑

r=1

cov((

e−X−1

j)(

e−X−1

j+r))

.


Since Gj and F0,j are unit Fréchet, according to the stationarity relation in (6),

Fj (x) = e−x−1 1

1−cj , x > 0, j ∈ D.

For each r ∈ N, the joint d.f. of (Xn,j ,Xn+r,j ) is given by

F(Xn,j ,Xn+r,j )(x, y)

= P(Xn,j ≤ x ∧ yc−r

j

)P

(Yn+1,j ≤ yc−r+1

j , . . . , Yn+r,j ≤ ycj

), x > 0, j ∈ D,

whose joint density, for y > xc−rj > 0, is derived as

∂

∂x

∂

∂yF(Xn,j ,Xn+r,j )(x, y) = ∂

∂x

∂

∂y

(e−x−1 1

1−cj

r∏

i=1

e−y−1cr−i

j

)

= 1

x2(1 − cj )e−x−1 1

1−cj1 − cr

j

y2(1 − cj )e−y−1

1−crj

1−cj , j ∈ D.

Therefore, and after some calculations, we obtain

E(e−X−1

j e−X−1

j+r) =

∫ ∞

0

∫ yc−rj

0e−x−1

e−y−1 ∂

∂x

∂

∂yF(Xn,j ,Xn+r,j )(x, y) dx dy

= 1 − crj

(2 − cj )(2 − cj − crj − cr+1

j ).

Now just observe that, by (16),

E((

e−X−1

j)2) =

11−cj

2 + 11−cj

= 1

3 − 2cj

.

Considering g(x) = 2 − 1/x, we have [g′(E(e−X−1

j ))]2 = E(e−X−1

j )−2 and, by theDelta Method, the second assertion holds. �

Other estimators can be found in literature. A strongly consistent estimator wasearlier proposed in Davis and Resnick (1989):

c∗j =

n∧

i=2

Xi

Xi−1. (18)

Another estimator, with a quite similar expression to our proposal, was derivedin Lebedev (2008). More precisely, for unit Fréchet marginals Fj and F0,j (x) =Gj(x) = e

−x−1 11−cj ,

cj = 2 − 1

pj

,


Table 1 Absolute bias and root mean squared error (in brackets) of the simulation results; Symbols ∗, ∗∗,∗ ∗ ∗ and ∗ ∗ ∗∗ mean that the number of simulations where calculations failed were, respectively, 152,405, 558 and 1000

c = 0.25 δ = 0 δ = 0.01 δ = 0.1 δ = 1

n = 100

c 0.0057 (0.1199) 0.0100 (0.1149) 0.0123 (0.1189) 0.0343 (0.1300)

c 0.0112 (0.0969) 0.0054 (0.0975) 0.0066 (0.0958) 0.0710 (0.1235)

c∗ 0.0000 (0.0000) 0.0061 (0.0073) 0.0764 (0.0894) ****

n = 500

c 0.0024 (0.0516) 0.0038 (0.0411) 0.0031 (0.0512) 0.0357 (0.0664)

c 0.0014 (0.0403) 0.0016 (0.0402) 0.0033 (0.0403) 0.0697 (0.0840)

c∗ 0.0000 (0.0000) 0.0114 (0.0125) 0.1366 (0.1461) ****

n = 2000

c 0.0000 (0.0246) 0.0019 (0.0259) 0.0030 (0.0262) 0.0308 (0.0411)

c 0.0003 (0.0197) 0.0005 (0.0203) 0.0019 (0.0210) 0.0679 (0.0717)

c∗ 0.0000 (0.0000) 0.0163 (0.0172) 0.0015 (0.1965) ****

c = 0.5 δ = 0 δ = 0.01 δ = 0.1 δ = 1

n = 100

c 0.0097 (0.1047) 0.0116 (0.1005) 0.0154 (0.1038) 0.0456 (0.1202)

c 0.0119 (0.0757) 0.0117 (0.0797) 0.0090 (0.0746) 0.1041 (0.1410)

c∗ 0.0000 (0.0000) 0.0108 (0.0120) 0.1177 (0.1318) 0.4975 (0.4979)*

n = 500

c 0.0028 (0.0443) 0.0029 (0.0440) 0.0039 (0.0415) 0.0398 (0.0637)

c 0.0022 (0.0316) 0.0025 (0.0322) 0.0030 (0.0338) 0.0947 (0.1035)

c∗ 0.0000 (0.0000) 0.0184 (0.0196) 0.1950 (0.2069) ****

n = 2000

c 0.0002 (0.0207) 0.0021 (0.0225) 0.0027 (0.0224) 0.0339 (0.0415)

c 0.0006 (0.0155) 0.0010 (0.0154) 0.0027 (0.0166) 0.0900 (0.0923)

c∗ 0.0000 (0.0000) 0.0252 (0.0262) 0.2742 (0.2852) ****

c = 0.75 δ = 0 δ = 0.01 δ = 0.1 δ = 1

n = 100

c 0.0154 (0.0773) 0.0162 (0.0752) 0.0171 (0.0757) 0.0411 (0.0965)

c 0.0148 (0.0539) 0.0158 (0.0555) 0.0154 (0.0554) 0.1403 (0.1705)

c∗ 0.0000 (0.0000) 0.0094 (0.0107) 0.0981 (0.1116) 0.6906 (0.6997)***

n = 500

c 0.0026 (0.0286) 0.0040 (0.0319) 0.0055 (0.0326) 0.0271 (0.0460)

c 0.0022 (0.0230) 0.0034 (0.0223) 0.0048 (0.0231) 0.1265 (0.1332)

c∗ 0.0000 (0.0000) 0.0148 (0.0157) 0.1511 (0.1602) 0.7495 (0.7496)**

n = 2000

c 0.0002 (0.0150) 0.0017 (0.0161) 0.0016 (0.0163) 0.0218 (0.0286)

c 0.0005 (0.0114) 0.0006 (0.0143) 0.0015 (0.0115) 0.1198 (0.1216)

c∗ 0.0000 (0.0000) 0.0201 (0.0208) 0.2105 (0.2201) ****


with pj = P(X2,j ≤ X1,j ) ∈ (1/2,1), and thus

cj = 2 − 1

pj

, (19)

where pj = (n − 1)−1 ∑n−1i=1 1{Xi+1,j ≤Xi,j }. Note that a similar restriction to our

method must be considered, i.e., 1/2 < p1 < 1. The consistency and asymptotic nor-mality of this estimator can be seen in Ferreira (2012).

Although the Davis and Resnick estimator c∗ in (18) is strongly consistent, it istoo sensitive to the assumptions of the process formulation. As observed in Davisand Resnick (1989), it seems unrealistic that non-simulated data follow exactly anARMAX model. In practice, we may have, for example, noisy processes of the formX

(δ)n = (Xn + δζn)∨0, with {ζn}n≥1 independent standard normal r.v.’s, which are ar-

bitrarily close to those obeying an ARMAX recursion. In the following, we illustratethrough simulation that our estimator c in (17) and the Lebedev estimator c in (19)are more robust in this context.

We consider 1000 independent random samples of sizes n = 100,500,2000 gen-erated from a noisy ARMAX model X

(δ)n , by taking c = 0.25,0.5,0.75 and δ = 0,

0.01,0.1,1 (δ = 0 corresponds to the non-noisy case, i.e., the ordinary ARMAX).The absolute bias and root mean squared error (in brackets) of the simulation resultsare in Table 1 (notations “∗”, “∗∗”, “∗ ∗ ∗” and “∗ ∗ ∗∗” mean the number of simula-tions in which the estimates could not be calculated, respectively, 152, 405, 558 and1000). Observe that c∗ has an excellent performance for the non-noisy ARMAX buthas the worst one once a small disturbance is considered. Observe also that the Lebe-dev estimator c performs better than our estimator c in almost all cases. However, ourproposal seems to be the best method for very noisy ARMAX processes (δ = 1).

Acknowledgements We are very grateful to the referees for their valuable corrections and suggestions.Helena Ferreira was partially supported by the research unit “Centro de Matemática” of the University ofBeira Interior and the research project PTDC/MAT/108575/2008 through the Foundation for Science andTechnology (FCT) co-financed by FEDER/COMPETE. Marta Ferreira was financed by FEDER Fundsthrough “Programa Operacional Factores de Competitividade—COMPETE” and by Portuguese Fundsthrough FCT—“Fundação para a Ciência e a Tecnologia”, within the Project Est-C/MAT/UI0013/2011.

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