CENTER FOR STOCHASTIC PROCESSES

r%. Department of Statistics10 University of North Carolina

Cy Chapel Hill, North Carolina

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ON A BASIS FOR "PEAKS OVER THRESHOLD" MODELING

by

M.R. Leadbetter

ODTICEECTE

Technical Report No. 285

March 1990

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On a basis for "peaks over threshold" modeling

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19. Abstract-. . . &

Abs.t. cj. "Peaks over Threshold" ("POT") models commonly used e.g. in

hydrology, assume that peak values of an lid or st- lonary sequence Xi above a

high value u, occur at Poisson points, and the ex.: , values of the peak above

u are independent with an arbitrary common d.f. G. Motivation for these models

has been provided by R.L. Smith (cf. [7],[83), by using Pareto-type

approximations of Pickands ([6]) for distributions of such excess values.

These works strongly suggest that the Pareto family provides the appropriate

class of distributions C for the POT model.

In the present paper we consider the point process of excess values of

peaks above a high level u and demonstrate that this converges in distribution

to a Compound Poisson Process as u -# a under appropriate assumptions. It is

shown that the multiplicity distribution of this limit (i.e. the limiting

distribution of excess values of peaks) must belong to the Pareto family and

detailed forms are given for the normalizing constants involved. This exhibits

the POT model specifically as a limit for the point process of excesses of

peaks and delineates the distributions involved.

ON A BASIS FOR "PEAKS OVER THRESHOLD" MODELING

by

M.R. Leadbetter

University of North Carolina

Abstract. "'Peaks over Threshold". F' models commonly used e.g. inhydrology, assume that peak values of an iid or stationary sequence Xi above a

high value u, occur at Poisson points, and the excess values of the peak above

u are independent with an arbitrary common d.f. G. Motivation for these models

has been provided by *t6O Smith-4cf-:E by using Pareto-type

approximations of Pickands4E6-7 for distributions of such excess values.

These works strongly suggest that the Pareto family provides the appropriate

class of distributions G for the POT model.

In Q- prs m-t paper - considereithe point process of excess values of

peaks above a high level u and demonstrate that this converges in distributionraa -L ,, , ; l r , '

to a Compound Poisson Process as u under- opriate assumptions. It is

shown that the multiplicity distribution of this limit (i.e. the limiting

distribution of excess values of peaks) must belong to the Pareto family and

detailed forms are given for the normalizing constants involved. This exhibits

the POT model specifically as a limit for the point process of excesses of

peaks and delineates the distributions involved. )

Research sponsored by the Air Force Office of Scientific Research Contract No.F49260 &9C 0144.

1. Introduction.

In what are sometimes called "Peaks over Threshold" (POT) models (cf.

[7]), the excess values over a high level u by an observed time series are

assumed to occur at Poisson points and to have arbitrary common distributions.

That is if {Xi: i=1.2,...} is e.g. a stationary (or iid) sequence, the

exceedance points {i: X. > u} are assumed to be Poisson and the corresponding1

excess values (Xi-u)+ to be independent with an arbitrary distribution.

The Poisson nature of the occurrence of exceedances is intuitively clear

since e.g. if Xi are iid with d.f. F, the number of exceedances of a high

level un by X 1.... n , is binomial with parameters (n, 1-F(u n)) and hence

approximately Poisson if n(l-F(un)) converges to some value T > 0. This will

be made more precise below by a time normalization. Further motivation for the

model is provided by R.L. Smith ([7],[8]) based on theory of Pickands [6],

restricting the distribution for excess values to a "generalized Pareto" (GP)

form

(1.1) Ga1 3 (x) = 1 - (1 + ax/p)-1/a P > 0, a A 0

= 1 - e-x /P P > 0, a = 0

where the range of x is (O,w) if a>O and (0,-a- 1P) if a<O.

This class is a flexible 2-parameter family, but more importantly for a

wide class of F's the excess distribution

(1.2) F (x) = P{X - u xIX > u} Accession For

NTIS GRA&Iis approximately GP in a sense shown by Pickands [6]. viz. DTIC TAB i

Unanmnowiced 0Justificitin

(1.3) inf sup IFu(X) - G (x) - 0 as u -- _

x ByDi~tribution/

Availability Codes

fAvail and/oMe pca

2

for some fixed a. That is for high levels u. 13 may be chosen (depending on u)

so that G a(x) is uniformly close to Fu(x).

Write xF for the right endpoint (sup{x: F(x) < 1) of the d.f. F and

F(x) = 1-F(x), the tail of F. Then the class of d.f.'s F for which the above

GP approximation holds includes all those satisfying

(1.4) F(u+xg(u))F(u) -- U(x) as u -- xF

for some function g(u) > 0. some d.f. C and all 0 < x < xF ( w). It is known

([6]) that any such G in (1.4) must be G.P. as in (1.1) for some a,13.

Note that (1.4) holds for all d.f.'s F of interest in extreme value theory

i.e. such that if Mn =............Xn), P a(M-b n x} (= n(x/an +b )) has a

non degenerate limit A(x). For example if G(x) = e-x, (1.1) is a classical

domain of attraction criterion for a "Type I" extreme value distribution

A(x) = exp(-e-x). If instead F has a regulary varying tail (l-F(ux))/(l-F(u))

-- a , a>O, as u -- , each x > 0, then (1.1) holds with G(x) = (l+x)-a .

g(u) = u, and A(x) is then "Type II" i.e. A(x) = exp(-x-a), x > 0.

Hence in a wide variety of cases of interest the distribution Fu(X) of

excesses (given by (1.2)) of the level u is approximately GP, C (x) in the

sense stated, where a is fixed but P can change with u. As discussed in [7]

this provides significant intuitive support for the POT model. Our purpose

here is to further justify the model by exhibiting it as the limit of point

processes of excesses of high levels, the limiting distribution of excess

values being shown to be GP, C (x) where now P as well as a, is independenta ,1

of u.

For clarity this will be shown for iid sequences in Section 2 and extended

to dependent (mixing) situations in Section 3. In the latter case high serial

correlation can cause clustering of exceedances of high levels and the peak

3

values are then defined to be the largest in each cluster. Two points worth

noting are (i) the dependence modifies the theory via the introduction of a

single parameter, the "extremal index" (essentially the inverse of mean cluster

size) and (ii) the CP form applies to the peak values but not necessarily to

other cluster properties such as their lengths. A corresponding theory will be

indicated in Section 3 for other such cases.

2. The iid case.

Let funI be a sequence of levels such that

(2.1) n(l-F(un) ) --+ T > 0.

Define point processes Nn to consist of the points i/n for which Xi > u i.e.

the exceedance points normalized by 1/n. N is thus defined on the positiven

real line but it will be convenient to restrict attention to the unit interval,

corresponding to exceedances among the n sample values X ....,X n . Hence Nn(B)

is defined for (Borel) subsets B C (0,1] by Nn(B) = #{i/n C B: Xi > u }.

It is trivial to show that under (2.1) that N d-+ N where N is a Poissonn

Process on (0,1] with intensity T. For if I = (a.b] C (0.1]. Nn(I) is binomial

with parameters [nb]-[na], p n=l-F(Un) ( ] denoting integer part) and nPn--+ T

so that

P(Nn(I) = r) --+ e-T(b-a[T(b-a)]r/r! = P(N(I) = r).

Hence Nn(I) d-*N(I) and by independence if 11-..... k are disjoint

(Nn(I) .... Nn(I) d- (N(I1)...N(Y)

drwhich is sufficient ([2]) to show full weak convergence N nd N.

Now associate with each point of Nn (i/n such that Xi ) un) the

corresponding excess value X -u to give the "point process N of excesses".i n n

4

Technically perhaps N should be regarded as a "marked point process" (or as ann

atomic random measure) since the (Xi-un) are not necessarily integer valued but

it will be convenient (and legitimate) to call it a point process whose events

(at (i/n: Xi ) un)) have (not necessarily integer) multiplicities (X-un).

Since, as above, the positions of the excesses converge to a Poisson

Process, it seems evident that one should expect any limit N* for N* to be an

compound Poisson process having events at Poisson points with intensity T and

(independent) multiplicities with some common d.f. G. Such a result may be

shown (cf. [1]) but here we give sufficient conditions for such convergence.

Here and below we write N = CP(T,C) to denote a Compound Poisson point process

whose Poisson events have intensity T with (not necessarily integer-valued)

multiplicities having d.f. C.

Theorem 2.1. Let Xi i=1.2... be iid with d.f. F satisfying (1.4) for some g,

G, and let (un) be levels satisfying (2.1). Then a N* d N , CP(T,G) where Tn n nis as in (2.1), G as in (1.4) and an = I/g(u ).

Proof: If Gn(x) (= Fu (x/an)) denotes the conditional d.f. of an(Xl-un) givenn

X1 > un . then clearly

Sexp{-san(X1- U)+ = F(u n) + (u n) fSme-sxdGn(x).

But from (1.4), Gn(x) -- G(x) and hence f 0 e -sxdG (x) --+ (s) = ~e-sxdG(x) so

that if I = (a,b] C (0,1] contains mn points i/n (mn ~ n(b-a))

nn

n nnexp{-sa(Xl-U)+} = [I.-(T/n)(I-*(S))(l+o(1))]n

which is the Laplace Transform ge-sN (a'b] where NN is CP(rC). Since N (I) is

5

the sum I (Xi-un)+ of m iid terms, it follows that a N*(I) d-N *(I). Hencei/nEI

n

by independence

(anN*(i1) ... an(i)d. (Nn1)..Nni)

for any disjoint I I so that aN * d+ N* as required. 03l'''k .n n

Thus in the iid case the (normalized) point process a N of excesses overnTn

u has a Compound Poisson limit and may be regarded as approximately CP(T.G)

for large n. Note again that G is a GP distribution G for some fixed a..

This provides a strong basis for the POT model in the iid case.

3. Stationary sequences

A stationary sequence X1 ,X2 .... can exhibit both "long range" and "local"

dependence between the X Here the former, but not the latter, will be

restricted by a "mixing condition" which enables the collection of X.'s into1

groups which are approximately independent but with possible high dependence

within groups. Strong mixing will suffice to restrict long range dependence.

However one may "tailor" this to the problem at hand with a slightly weaker

restriction A(Vn) defined for any sequence of constants vn as follows. For

lj~k~n write 9jk(vn) = a((Xs-vn)+: j~s~k} and say that A(Vn) holds if for

le<n IP(AnB) - P(A)P(B)I a whenever A C 1.j(vn), B E J+e,n ) lj<n-t

and a --+0 for some e = o(n).n~en n

High local dependence is reflected in the presence of clustering of

exceedances of high levels. "Clusters" will be defined precisely below, but we

first note that for levels (u n) satisfying (2.1) the mean size of a cluster

typically converges to a parameter whose inverse 0 is sometimes called the

"extremal index" of the sequence (Xi). Specifically 0 is defined by the

property that if Mn = max(X, X2. .. Xn) and un satisfies (2.1) then

6

(3.1) P(Mn Un) --+ e ,

6 being independent of T. (For i.i.d. sequences 6 = 1). Such 6 exists under

general conditions (cf [4], Sec. 3.7)

Clusters of exceedances of high levels may be defined in various ways, the

most obvious being as runs of consecutive exceedances. However if A(vn) holds

for a given sequence {vn} of levels the following "block definition" is more

convenient (and often asymptotically equivalent to that using runs). Choose

integers k -- -, kn = o(n), satisfying

(3.2) kn(a ne + Rn/n) --+0.n

Write r n=[n/k ] and divide the integers (1.. .n) into consecutive blocks

J, = {(i-l)rn+l. (i-1)rn+2,....irn ldi~kn

J =kr~k +2 .... n}JR +1 = k knr n+ 1 ' k nr n "2. .n

n

and regard the exceedances (if any) in a block as forming a cluster. The

choice of kn (or equivalently rn) is flexible, subject to the growth

restriction (3.2).

Obviously this block definition can count a run of consecutive exceedances as

two or more clusters, if the run straddles more than one block so that the

block definition is less natural in some cases. However it can be more

appropriate than the runs definition for sequences with high local variability.

In any case we use the block definition for its convenience. Further if the

block Ji contains exceedances the first point. (i-l)rn+l, of Ji will be

regarded as the location of the cluster in Ji"

Define now a point process P of normalized cluster positions, i.e.nconsisting of the points {((i-1)rn+1)/n, 1 i kn: M(Ji) > Un} where M(E) is

7

written for max{Xj: jeE}. Associate with each point ((i-l)rn+l)/n of Pn the

corresponding maximum excess M(Ji)-un = max{(XJ-Un)+: j e Ji}. giving the

(again technically "marked") point process P* of peak excess values above then

level in the clusters. We show that a result like Theorem 2.1 holds in the

dependent case with P' replacing N*. In fact it may be shown (cf [5]) that forn n

i.i.d. sequences N and P are asymptotically equivalent in a strong sense asn n

also are N and P* so that P and P* generalize N and N* .n n n n n n

It was shown in Section 1 that in the iid case the exceedance point

process N has a Poisson limit with intensity T. This result is simplyn

generalized (cf. [1]) to show that under dependence Pn has a Poisson limit

(with intensity OT) whereas N itself has a Compound Poisson limit whose eventsn

occur a* the positions of clusters (i.e. points of P ), with multiplicities

given by (limiting) cluster sizes. However the (limiting) distribution for

multiplicities need not have a GP form and need not be totally determined by

the (tail of the) marginal d.f. F of the X1 . On the other hand it will be

shown below that P* has a CP(OT,G) distributional lintit where G is obtainedn

from the tail of F via (1.4). Thus G has GP form and the dependence influences

the limit only through the factor 0 in the intensity of the underlying Poisson

Process.

The Compound Poisson limit for P* will be obtained by considering then

asymptotic behavior of the maximum in a cluster and showing that the clusters

are essentially independent. The specific basic results needed are contained

in the following lemma,

Lemma 3.1. Let {Xn) be stationary with extremal index 0 > 0 and marginal d.f.

F satisfying (1.4). and let A(vn) hold with vn=un+xg(un), each x > 0. where un

satisfies (2.1). Let {kn} satisfy (3.2) and write an = Wu . Then (with

r = [n/k ]). as n--n n

8

(i) P(a (Mr -Un) > xI (x ) x 0n n

(ii) P(an(Mr -Un) xIM r >Un -d G(x)n n

k(iii) [Sexp{-sa n(Mr -un)+] n -- exp(-&r(1-#(s)))

n

where #(s) = O e-SXdG(x)

Proof: Since n[1-F(un+a nx)] ~ TF(un+xg(un) )/F (un) -- 70(x) and (Xn} has

extremal index 6. it follows that

P{M n Un+a-lx) --+ exp(-OiG(x)).

But it follows in a standard way from the mixing condition A(Vn) (cf. [1. Lemmak k

2.3]) that P{Mn VnI - p nM vnI --+ 0 so that p n(M Un +anlx --* exp(-eOT(x))n n

from which (i) readily follows. The left hand side of (ii) is

1 - >u+ax/PM >U -=1 21 ?T +n n n n

by (i). so that (ii) follows. Finally (iii) follows by the first calculation

of Theorem 2.1 with Mr replacing Xi. using (ii). 0

n

The main result now follows in a similar way to Theorem 2.1 on using

approximate independence between the clusters.

Theorem 3.2. Let Xn) be stationary with extremal index 6 > 0, and marginal

d.f F satisfying (1.4). and let A(v n) hold with vn = u + xg(u ) each x 0.--n naif (32 n = nnad

where un satisfies (2.1). Let kn satisfy (3.2), rn = [nn and

an = (g(u))-l Then the point process Pn of peak values above un satisfies

a p* d-+ P* where P* is CP(OT,G) and G is as in (1.4).n n

9

Proof: Let I be a subinterval of (0,1] and J, = {(i-l)rn+l .... irn) as defined

above, 1 I k . Then it follows from the mixing conditions along the samen

lines as Lemma 2.2 of [3] (or Lemma 2.2 of [1]) that

9exp{-sa P*(I)} - 17 exp(-sanPn(Ji)} --+ 0(i:JiCIn

as n -- so that

knm(I)(l+o(1))

9exp(-saP*(I)) = [8exp-sa (M -U nn

--- exp(-OTm(I)(1-#(s))}

by Lemma 3.1, where #(s) = e-SXdG(x). Hence anPn() !- P(I) where P is

CP(OT,G) on (0,1]. Now if Il .... Ik are disjoint subintervals of (0,1] it

follows also as in Lemma 2.2 of [3] (or Lemma 2.2 of [1]) that

k k9exp{-a I s.P(I) - lexp-a n P (I )} --)0n J=l in j=l

from which it follows that

(anP *(i1)... anP *k) d. ((i).p*(ik

and hence that a p* i+ po.nn

Finally we reiterate that this result is one of many which can be obtained

involving different aspects of cluster structure. For example the Compound

Poisson limit for N was cited above, the multiplicities corresponding ton

cluster sizes. More complicated functions - such as the sum of powers of

excess values in a cluster - may also be considered and will lead to Compound

Poisson limits. Such cases may be useful in applications where damage from

high levels (e.g. high pollution episodes) may be modeled as a specific

10

function of the excess values. However the case of (excess) peak values in a

cluster is especially important (e.g. in describing severe floods, where damage

can well be a function of flood level). Theorem 3.2 establishing the POT

approximation is particularly useful since the multiplicity distribution then

depends only on the marginal d.f. F and moreover is known to have GP form.

References

[1] Hsing, T., HIUsler, J., Leadbetter, M.R., "On the exceedance point process for a

stationary sequence", Prob. Theor. Rel. Fields, 78, 97-112, 1988.

[2] Kallenberg, 0., Random Measures, Academic Press, N.Y. 1976

[3] Leadbetter, M.R. and Hsing, T., "Limit theorems for strongly mixing stationaryrandom measures" To appear in J. Stoch. Proc. and Applns., 1990.

[4] Leadbetter, M.R.. Lindgren, G., Rootz~n, H., "Extremes and related propertiesof random sequences and processes", Springer Statistics Series,1983.

[5] Leadbetter, M.R. and Nandagopalan. S., "On exceedance point processes undermild oscillation restrictions" Extreme Value Theory, J. HUsler andR.-D. Reiss (eds.) Proc. Oberwolfach 1987, Springer Lecture Notes inStatistics, No. 51, 1988.

[6] Pickands, J., "Statistical inference using extreme order statistics" Ann.Statist. 3, 1975, p. 119-131.

[7] Smith, R.L., "Threshold methods for sample extremes" in Statistical Extremesand Applications, J. Tiago de Oliveira Ed., NATO ASI Series, Reidel,Dordrecht, 1985, p. 623-638.

[8] Smith, R.L. "Estimating tails of probability distributions" Ann. Statist. 15,1987, p. 1174-1207.

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