On the Multivariate Extremal Index

The exceedance point process approach of Hsing et al. is extended to multivariate stationary sequences and some weak convergence results are obtained. It is well known that under general mixing assumptions, high level exceedances typically have a limiting Compound Poisson structure where multiple events are caused by the clustering of exceedances. In this paper we explore (a) the precise effect of such clustering on the limit, and (b) the relationship between point process convergence and the limiting behavior of maxima. Following this, the notion of multivariate extremal index is introduced which is shown to have properties analogous to its univariate counterpart. Two examples of bivariate moving average sequences are presented for which the extremal index is calculated in some special cases.


Introduction
Extreme value theory for multivariate iid sequences has been studied for quite some time now but attention to the dependent case has been relatively recent.For univariate sequence.^it is known that local dcpcrKience causes extreme values to occur in clusters, which in turn results in a stocha-stically smaller distribution for the maximum than if the observations were indepeivdcnt.We begin with a brief review of these results, which we shall later extend to the multivariate case.

H(t)=Hitf
(1) The extremal index is thus a measure of the effect of dependence on the limiting distribution ol' A/,,.The slochasticully smaller limiting distribution ot A/" is in fact a direct result of the clustering of extremes, as explained below.Sec Ref. f5| for detatls.
For fixed 7>0 let the excevdance poinl process Ar.-AC' be defined by where I A denotes the indicator function of the event A. Then tor a broad class of weakly dependent sequences, the iiinit in distribution of N^, if it exists, is a Compouixi Poisson process with intensity tl-rand multiplicity distribution iron {1,2,...}, The Poisson events may in fact be regarded as the positions of "cxceedancc clusters" while the multiplicities coiTespond to cluster sizes.More expliciliy, one may divide the n obsei"vations into k" blocks of roughly equal size and regard exccedances within each block as forming a single "'cluster", so that the cluster sizes are given by N"(J,), i=i,...Ji".where ■/r(^.jMT7' ^f ^^^ ^ suitable choice of k" depending on the mixing rale of {^ >, one then has lim P {N" (7,)-y | /V,(./,)>0}-n(/)-J^ 1, and lim P'"{/V,(J,M)}=lim />{.V,[0,n=0} =lim/'{A/"i=«,(T))=c ■", SO that in particular, lim"_," k" f {Af"(>/iJ>0H^7.Hence lim t A'" 10.1 Him Jt./i:" N,t7,; =lim it.£(^,(i,)I N.iJ,)>0)P {N.U,)>^} while on the other hand, iini,_^ t A^"lO,r|=iini"^, nP{(i>u"iT)}=T.The cluster size distribution and the extremal index are therefore related by Now let {^"=(^|....,£^,), nt.2 > be a mullivariatc sta tionary sequence where d^l is a fixed integer, and write A/"-(W,i,...,A/,"/) where M"j=max{^i;,...,^,}, j=l,...,(/.The study of multivariatc extremes began in the early I95(ts, ftxrusing mainly on the limiting behavior of M" under a linear normalization, when the observations arc iid.The resulting class of limiting distribu tions was characterized m Ref. (6] and domains of attraction criteria were given in Ref [7J. See also Ref [8J,Chapter 5, for an account of the literature surrounding this theory, l-'or stationary sequences satisfying a general mi.xing assumption, it is known (see Rcfs. 19.101, and Theorem 1.1 below) that the class of limiting distributions of .M" is the same as for iid sequences.In this paper we explore the precise effect of dependence on the limiting distribution by extending the univariatc theory described above to the multivariate ciLsc, Essentially, this involves studying the intcr-rclationsbip between the two dependence structures present, one due to dependence over time and the otlicr due to the dependence between the various components of the multivariate observations.Tlie ideas become most transparent when presented in terms of so-called dependence functions [HJ.Here we adopt the slightly modified definition found in Ref. [9] FAxj)) could be defined to be a dependence function of F, alth(Mjgh the present choice is a natural one. Write 7'-{0,l)A{l} where l=(l DeiR'', and for t=(r tj)^T, let v"(t)=(i',.(fi),-,i'»rf(fj)) where i-,,(/j) satisfies lim,^^ /(f{f,p'i'"^(r,)}--log/,.Ixt //, denote the distribution function of M" (i.e., H^ixy^P {M"^x », with marginals H",.. j=\,...J.Then (.see Refs. [8,11}).
'ihc limiting bcha^'iof of M" can therefore be separated into two parts, one pertaining to the convergence of the niargmals (a univariate problem) and the other to the convergence of the dependence functions.Here we focus attention cxclusiviely on the latter, ll should be noted thai the choice of normalising constants does not affect the dependence funclion of the limit distribution H, but only alters the marginals (sec Ref. 191, Lemma 3.2).
Since our main interest is in the dependence function, the present choice of normalising con.stants is appropriate in view of the fact that it results in Uniform[0.1]'marginals for the limit distribution when {^, > is iid, so that in particular /->«=■//.According to Theorem 3..^ of Ref.
[91, the class of all possible limits H in Eq. ( 3) (for lid {^")} is precisely the class of extreme dependence functions, tlial is those that satisfy

Journal of Research of the National Institute of Standards and Technology
D"(t)=/J(r,"....,f;) (4) A^"(B)=SA,"6fl,5,, BB[OM (5) for each n>l andt-(( /j)£[0,11''.Theorem 1-1 below shows that the same is true also if {^"} is a slalionary sequence satisfying the following mixing tondition.and for 1 ^l^n-1 The mixing condition A(v"(t)) is then said to hold if Q(,;"-*0 for some sequence {/"} satisfying iJn^fO.This is the multivariate version of the mixing condition used in Ref. J51 and is slightly stronger than the D(u") condition in Ref. [91.Henceforth {^" ) will be assumed to satisfy A(v"(t)), for some or all t, as required.
PROOF: The first part is an immediate consequence of Theorem 4.2 of Ref, [91 while the second part follows from the definition of extreme dependence functions upon noting that (by the univariatc theory described above), the miu-ginals of fi are of the form Hj(i^)=tp where Oj is the extremal index of {£,j}, the jth component sequence of {f"}.
In the next section we apply the cxcccdance point process approach to multivariate extremes and obtain some weak convergence results.The multivariate extremal index is then defined (in Sec. 3).ba.scd on the multivariate analogue of Eq. ( 1).It is seen to be a function of only d 1 variables and its properties naturally extend those of the univariatc extremal index.Finally in Sec. 4 v/Q consider two examples of bivariatc moving average sequences for which the computation of the extremal index is demonstrated.
Let {k"} be any sequence of positive integers satisfying % V .
The following theorem which gives a useful characterizaibn of the convergence of M is an immediate consequence of the results in Sec. 5 of Ref. [12].
Next we consider the iid case in some detail and obtain an interesting connection with Theorem 5.3.1 of Ref. (8).PROI-OSITION 2.2.Let {^ } be iid and for fixed iE.T let N" he defined by Eq. ( 5).lfl^"->'' M> then the multiplicity distribution IT in Eq. ( 6) is supported on the set 5={0.1 }'\{0}.
Hence by TTicorem 2.1 K'^'' N^" where M'" has the specified parameters.8) holds for each t^T.Note that //(t)=c '"' so that f/ and the ^''''s can be obtained from each other.Also the 7r""s can be obtained from the v*"'s by first inverting Eq. ( 9) to get the /ij(t)(t)"s and then inverting Eq. (iO).(The inversion of Eq. ( 9) is carried out inductively using the fact that the wealt convergence of W"(i'"(t)) implies that of all lower dimensional marginals.)D Analogous results for the dependent case take on a -slightly different form.Let {^,} be a stationary sequence satisfying A(v,(t)) for each tGT.As before let r,=[n/A', ] where {k"} is any sequence satisfying Eq. ( 7), and define G,^dv,{ty)=P{Mr^,>v,j,itj,),...Mr^,>v"j,{tj,)}.
CatOLLARY 2.7.Let {^"} be as in CvroUary 2.6 and suppose that P {A/,:£v"(tJ}-*" //(t).Then the Jolluwing are equivalent: (i} H is independent, (it) Hit)~n%,H,{t,)for some (£(0,1)^ (Hi) k^GrM^.m^Oforeach j(2)./ar.^omet£(0,i/. It should be noted that Refs.[9, 10) give some iiiteresting sufficient conditions for H to be independent when {£,} is a stationary sequence.A natural question to ask in the present context is whether h is irxlependent whenever H is. Proposition 3.4 gives a necessary and sufficient condition for this in terms of the extremal index, but the answer in general is negative and a coun ter-examplc can be found in [10].It seems more plausible that the converse may be true, i.e., that H is intlepen dent whenever H is. In fact however, this tot> is not the case, as shown by an interesting counter-example in [15].
■ Wfc conclude diis section by stating a result which extends Theorem 5.1 of [5| and is proved similarly.

3.)
While the above results illustrate some of the basic properties of the multivariate extremal index, they are far from complete.For instance; it would be useful to identify the set of all "admissible" B(-) for a given R, that is the set of all 6{-) such that D"() defined by Eq. ( 14) is a probability distribution on [0,1 J"*.It wiHjId also be of interest to study the properties of d{-) when one or both of ft and H are independent.In this context we have the following simple result.
In particular, if both ft and H are independent then d{i) is a convex combination of the d/s.
The extremal index can be given the following more general tbrmulaiion.Let /I and /a.be the probability measures on (0,1)'' corresponding to ft and H. respectively.Thus for in.stance.
Note that 6(t)-ti((0,/,)X-X(0,/j)) for tGT.Thus if {0(1): tST} is known along widi either of//or W. then it is possible at least in theory to obtain {d{A) : /1C(0,I)''}.In practice, however, it may not be possible to obtain 9{A) m a tractable form, but frequently one is only interested in certain special sets, typically rectangles of the form lll., (aj,bj), and for such sets the computation is easy.
The definition of M" as the vector of componentwise maxima actually corresponds to regarding ^; as an extreme observation if ^ip>v"j(tj) for some j.More generally, one may define ^ to be an extreme value if ^G.v"(A) for some AC(0,1 j"*, in which case 8(A) has an interpretation as a measure of the clustering of such extremes.Note that the original definition of extremes corresponds to letting A-{{0^t)X-..X(0,tj)Y.Alter nately one may consider taking A-{ti,\)X-.X{t^,\) which corresponds to defining £ as an extreme observa-

For
Proposition 3.2 i.s simply the multivariate version of Eq. (i) and shows how the extremal index is related to the clustering of "cxceedances."Indeed,according to the present viewpoint, an excccdance occurs at time I if ^^v"{t\ i.e., if £,>>v,j(/j) for at least one j.Thus Propositions 3.1 and 3.2 show that while the degree of clustering may depend on t, it is constant on each L,.Note also the connection to Theorem 2.S.The next result gives the relation between the dependence functions of H and H, which is seen to involve the extremal index in an intricate manner.