WEAK CONSISTENCY OF EXTREME VALUE ESTIMATORS IN C[0, 1] BY LAURENS DE HAAN AND TAO LIN Erasmus University Rotterdam and EURANDOM

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1 The Annals of Statistics 2003, Vol. 3, No. 6, Institute of Mathematical Statistics, 2003 WEAK CONSISTENCY OF EXTREME VALUE ESTIMATORS IN C[0, ] BY LAURENS DE HAAN AND TAO LIN Erasmus University Rotterdam EURANDOM We prove that when the distribution of a stochastic process in C[0, ] is in the domain of attraction of a max-stable process, then natural estimators for the extreme-value index which is now a continuous function) for the mean measure of the limiting Poisson process are consistent in the appropriate topologies. The ultimate goal, estimating probabilities of small failure) sets, will be considered later.. Introduction. Multivariate extreme value theory its statistical implications are by now well understood [Resnic 987), Smith 990) de Haan de Ronde 998) just to mention a few references]. Giné, Hahn Vatan 990) have characterized max-stable stochastic processes in C[0, ]. This seems to be the most sensible extension of extreme-value theory to infinite-dimensional spaces. De Haan Lin 200) have characterized the domain of attraction of max-stable processes in C[0, ]. The aim of the present paper is to initiate maing these results useful for statistical application by proving consistency of natural estimators for the main parameters of the max-stable process based on observations from a process which is in its domain of attraction. The result is stated in Theorem 2.. Infinite-dimensional extreme-value theory seems to be useful in a problem of coast protection [cf. de Haan Lin 200)]. Next we explain the framewor of our results. Consider a sequence of i.i.d. rom processes ξ,ξ 2,...in C[0, ]. Suppose the sequence of processes { } max i n ξ i t) b t n).) a t n) t [0,] converges in C[0, ] to a stochastic process η with nondegenerate marginals. Here a t n) > 0b t n) are nonrom normalizing constants chosen in such a way that the marginal limit distributions are stard extreme-value distributions of the form exp{ + γx) /γ } for some γ R, + γx>0 defined by continuity for γ = 0. We note the following structural property. The probability distribution of the limit process η is determined by the continuous function γ the extreme-value index) plus a measure ν on the Received October 2000; revised January AMS 2000 subject classifications. Primary 60G70; secondary 62G32, 62H. Key words phrases. Extreme values, convergence in C[0, ]. 996

2 HILL ESTIMATOR IN THE SPACE C 997 space C[0, ] which is homogeneous; that is, for any Borel set A a positive a,.2) νaa) = a νa). For each function f C[0, ],f >0, we have log P { + γt)ηt) ) /γ t) } <ft)for all t.3) = ν { g C + [0, ]; gt) ft)for some t } [Giné, Hahn Vatan 990), Proposition 3.2, part iv), where the measure ν is given in its polar representation]. We need the following result from de Haan Lin 200), Theorems PROPOSITION.. The sequence of stochastic processes.) converges in C[0, ] to a stochastic process η with nondegenerate marginals if only if the following hold: i).4) n n ζ i η i= in C[0, ] with ζ i t) := F t ξ i t)) ηt) := + γt)ηt))/γ t) for t [0.]. An equivalent statement is: there is a measure ν on C + [0, ] :={f C[0, ]; f 0,f 0} such that for each c>0 the restriction of the measure ν s defined by { } ν s ) := sp s ζ ) to S c := {f C + [0, ]; f c} converges wealy s ) to the restriction of ν to S c. The relation between η ν is as in.3). ii).5).6) U t sx) U t s) lim = xγt) s a t s) γt) a t sx) lim s a t s) = xγt) locally uniformly for x 0, ) uniformly for t [0, ], where a t s) := a t [s]) [ from.)].

3 998 L. DE HAAN AND T. LIN Now we proceed as in the finite-dimensional case: we fit the appropriate limit distribution to the tail part of the distribution of the original process [cf..9)]. Next this limit distribution enables us to extend the original probability distribution beyond the range of the available data as follows: A failure region F defined, for example, by F ={ξt) ft)for some 0 t } with f a continuous function which is extreme with respect to the sample in the sense that ξ i t) < f t) for i =, 2,...,n 0 t, where ξ, ξ 2,..., ξ n are the observed realizations. Since this feature is essential to the problem, we want to eep it in our asymptotic approximation. This implies that f must depend on n, the sample size, that is, f = f n, in fact we assume that ) ncn.7) ft)= U t ht) n with h a fixed positive continuous function, c n a sequence of positive constants.8) ) U t x) := x) F t with F t x) := P {ξt) x} [cf. de Haan Sinha 999), relation.5)]. We shall now attempt to explain the intuitive reasoning that leads to a way to estimate PF):forn,= n), n)/n 0,.9) P {ξt) f n t) for some t [0, ]} { } = P F t ξt)) for some t [0, ] F t f n t)) {.7) = P F t ξt)) nc } n ht) for some t [0, ] { }} { log P n/ n n/){ F t ξt))} <c nht) for 0 t ) { { ) /γ t) log P + γt)ηt) <cn ht) for 0 t }} n.3) = n ν{ g C + [0, ]; gt) c n ht) for some 0 t }.2) = nc n ν { g C + [0, ]; gt) ht) for some 0 t }. The approximate equation ) follows from the convergence of.) see Proposition.).

4 HILL ESTIMATOR IN THE SPACE C 999 Since np {ξt) f n t) for some t [0, ]} is the mean number of realizations that fall in the failure set since we want this to go to zero this is the essential feature mentioned above), we need to assume c n n) = oc n ), n. Now to turn this into a useful statistics tool we need to estimate the measure ν. Moreover, we need to estimate the unnown function h which can be evaluated approximately as follows:.0) ht) = nc n { F t f n t))} nc n + γt) f nt) b t n/) a t n/) ) /γ t). Hence we also need to estimate the functions γt),a t n/) b t n/). The estimation of these four objects is the purpose of this paper. The actual estimation of PF)is the subject of future research. Finally we remar that all our results still hold if the time parameter runs through an arbitrary compact set in R d, not just [0, ]. 2. Result. Suppose that {ξ i,i } are i.i.d. rom elements of C[0, ] that F t x), the marginal distribution function of ξ i t), is a continuous increasing function of x for each t. Assume that 2.) { P inf t } ξ t) > 0 > 0 this can be achieved by applying a shift). Define, for x>, U t x) := Ft ) x the function a t ) by a t s) := a t [s]) for s>0 [cf..)]. From 2.) we can pose inf U t2) 0. 0 t We assume wea convergence of the maxima in C-space: ni= ξ i t) U t n) 2.2) ηt), a t n) where a n) is positive in C[0, ] η is a rom element of C[0, ] satisfying P ηt) x ) = exp { + γt)x ) /γ t) } for each t [0, ], γ C[0, ]; the extreme-value index of ξ t) is γt)for each t. Let ξ,n t) ξ 2,n t) ξ n,n t) be the order statistics of ξ i t), i =, 2,...,n.

5 2000 L. DE HAAN AND T. LIN We define the sample functions 2.3) M n j) t) = log ξn i,n t) log ξ n,n t) ) j, j =, 2. Now we define estimators for γt), a t n/) b t n/) as in Deers, Einmahl de Haan 989): 2.4) Hill estimator); 2.5) 2.6) moment estimator); 2.7) 2.8) ˆγ + n t) = M) n t) ˆγ n t) = { M) n ) 2 } 2 M n 2) ; ˆγ n t) =ˆγ n + t) +ˆγ n t) Û t n â t n ) = ξ n,n t), ) = ξ n,n t) ˆγ + n t) ˆγ n t)) location shift estimators). For fixed t these are well-nown one-dimensional estimators [cf., e.g., de Haan Rootzén 993)]. Next we denote, for i =, 2,...,n, ˆζ ) i t) := n { [ ξi t) Û t n/) +ˆγ n t) â t n/) ˆζ 2) i t) := with ˆF n,t x) := n ni= I {ξi t)>x}. Define the estimators 2.9) 2.0) ˆν ) n, ) = ˆν 2) n, ) = Fˆ n,t ξ i t)) n i= n i= { } ) I ˆζ i, n { } 2) I ˆζ i. n )]} / ˆγn t) ˆγ n + t) The latter estimator has been inspired by Huang 992), the former by de Haan Resnic 993).

6 HILL ESTIMATOR IN THE SPACE C 200 THEOREM 2.. As, /n 0, we have 2.) 2.2) 2.3) 2.4) 2.5) γ n + t) γ + t) P 0 with γ + t) = γt) 0, 0 t γn t) γ t) P 0 with γ t) = γt) 0, 0 t ˆγ n t) γt) P 0, 0 t Û t n/) U t n/) a t n/) P 0, â t n/) a t n/) P 0, 0 t 0 t 2.6) ˆν ) n d S c ν Sc, 2.7) ˆν 2) n d S c ν Sc in the space of finite measures on C + [0, ], with c>0 S c := {f C + [0, ]; f c}. REMARK 2.2. Relations 2.6) 2.7) mean that ˆν i) n E) νe), i =, 2, in probability, where E S c is a Borel ν-continuous set cf. proof of Lemma 3.). 3. Proof. We first prove some auxiliary results. LEMMA 3.. Define the rom measures ν n, ) = n { } I n ζ i F t ξ i t)). i= with ζ i t) := As, /n 0 c>0, d 3.) ν n, Sc ν Sc in the space of finite measures on C + [0, ]. PROOF. According to Daley Vere-Jones [988), Theorem 9..VI ], for 3.) we only need to prove, for any Borel ν-continuous sets E,E 2,..., E m S c, ν n, E ), ν n, E 2 ),..., ν n, E m )) νe ), νe 2 ),..., νe m )) in

7 2002 L. DE HAAN AND T. LIN probability. Since the limit is not rom, this is equivalent to the following: for any Borel ν-continuous set E S c, 3.2) ν n, E) νe) in probability. Using characteristic functions we now that this is equivalent to ) n 3.3) P n ζ i E νe), which is the same as 3.4) ν n/ Sc ν Sc. This has been proved in Proposition.i). Next we show the convergence of the tail empirical distribution functions. LEMMA 3.2. For each t, let ζ,n t) ζ 2,n t) ζ n,n t) be the order statistics of ζ i t), i =, 2,...,n, define G n,t x) = n I {ζi t)>n/)x}. i= Then, for any positive c, G n,tx) 3.5) 0 t,x c x P 0 0 t,x c n ζ n x,nt) 3.6) x P 0. Also, pose µ λ are continuous functions defined on [0, ] with µ<, λ<, µ + λ<. Then ζ n i,n t)/ζ n,n t)) µt) P 3.7) 0 0 t µt) µt) ζ n i,n t)/ζ n,n t)) µt) 0 t µt) 3.8) ζ n i,nt)/ζ n,n t)) λt) λt) 2 µt) λt) P 0. µt) λt)) µt)) λt))

8 HILL ESTIMATOR IN THE SPACE C 2003 PROOF. Fix c>0. From Lemma 3. use of Sorohod construction we can pose ν n, Sc ν Sc a.s., where S c ={f C + [0, ], f c}. Note that this is convergence of finite rom measures. For any finite measures ν,µ, define the metric [cf. Daley Vere-Jones 988), A2.5.)] dν,µ) := inf { ε>0 : for all closed sets F C + [0, ], νf) µf ε ) + ε µf ) νf ε ) + ε }, where F ε := {f C + [0, ], f g ε for some g F }. Now for any positive ε eventually Next define the closed set d ν n, Sc,ν Sc ) ε a.s. E x,t ={f C + [0, ]; ft) x}. Note that in our situation the set E ε x,t is the same as E x ε,t.alsoνe x,t ) = /x [Giné, Hahn Vatan 990), pages 50 5]. It follows that for x>c, 0 t, G n,t x) = ν n, f C + [0, ]; ft)>x ) ν n, E x,t ) νe x ε,t ) + ε = x ε + ε G n,t x) ν n, E x+ε,t ) νe x+2ε,t ) ε = x + 2ε ε. This proves 3.5). Statement 3.6) follows because the uniform convergence of the function G n,t x) to /x is equivalent to the uniform convergence of its inverse /n)ξ n x,n t) to the same function. For 3.7), observe that ζ n i,n t)/ζ n,n t)) µt) µt) n = n = ζ n,n t) ζ n,n t) ) µt) ζ n i,n t)/n)) µt) ζ n,n t)/n)) µt) µt) ) µt) x /n)ζ n +,n t) /n)ζ n +,n t) ) s µt) ds dg n,t x)

9 2004 L. DE HAAN AND T. LIN n = ) µt) Gn,t x) ) x µt) dx ζ n,n t) /n)ζ n +,n t) ) µt) Gn,t x) ) x µt) dx ζ n,n t) ) n µt) + ζ n,n t) n = /n)ζ n +,n t) Gn,t x) ) x µt) dx. From 3.5) 3.6) the second part converge to 0. So we only need to prove Gn,t x) ) x µt) 3.9) dx 0 t µt) P 0. Let Y i := t ζ i t), i =, 2,...,n. These are i.i.d. r.v. s. From Proposition. we have n n n Y n i = n ζ i t) = i= t t n ζ it) ηt) =: Y t in distribution, where we now { ) x } PY x) = exp c for some c>. Let Y n i,n,i=, 2,...,n, be the order statistics of Y i,i =, 2,...,n, F n x) := n I Y i,n >x n ). We have y G n,t x) F n x). Hence for any y>0byone-dimensional results, Gn,t x) ) x µt) dx Fn x) ) x µt) dx y Moreover, Gn,t x) ) x µt) dx P y y c y x xµt) dx = cyµt) µt). x µt) 2 dx = yµt) µt) uniformly in t. By letting y we get 3.9). Hence we have proved 3.7). The proof of 3.8) is similar.

10 HILL ESTIMATOR IN THE SPACE C 2005 LEMMA 3.3. Suppose a t s) > 0,at s), g t s) are locally bounded in t [0, ], 0 s<,γt) C[0, ] 3.0) g t sx) g t s) a t s) xγt), γt) a t sx) 3.) a t s) xγt) locally uniformly for x>0, 0 t. Then for any positive ε, there exists s 0 such that for s>s 0, sx > s 0 we have a t sx) a t s) 3.2) xγt) <εxγt) exp{ε log x } or, alternatively, ε)x γt) exp{ ε log x } < a tsx) 3.3) a t s) < + ε)xγt) exp{ε log x } 3.4) g t sx) g t s) a t s) xγt) γt) <ε + x γt) exp{ε log x } ). PROOF. The proof is not much more complicated than, has the same structure as, the proof of the same result with t fixed. We refer to de Haan Pereira [999), Appendix], which contains a simplification, applicable in this case, of a result of Drees [998), Lemma 2.]. LEMMA 3.4. have, for s, With the same conditions as in Lemma 3.3 g t s) > 0, we a t s) 3.5) g t s) γ + t) uniformly in t. For any positive ε, there exists an s 0 > 0 such that if s,sx > s 0, we have 3.6) log g t sx) log g t s) a t s)/g t s) xγ t) γ t) <ε + x γ t) exp{ε log x } ). PROOF. 3.7) For 3.5), we need to prove, for any t n t 0 s n, { g tn s n ) a tn s n ) γt0 ), for γt 0 )>0,, for γt 0 ) 0.

11 2006 L. DE HAAN AND T. LIN i) For γt 0 )>0, from Lemma 3.3 for any ε 0,γt 0 )), wecanfinds 0 > 0 such that for s n >s 0 we have g tn s n ) g tn s 0 ) s 0/s n ) γt n) ) γtn a tn s n ) γt n ) <ε s0 ) ε) +, s n which implies 3.8) g tn s n ) g tn s 0 ) lim = n a tn s n ) γt 0 ). From Lemma 3.3 we get a t s), s, uniformly for t {t : γt)> 2 γt 0)}, which implies 3.9) g tn s 0 ) lim n a tn s n ) = 0. From 3.8) 3.9) we get the first part of 3.7). ii) For γt 0 ) 0, first define ψ t s) := g t sx) g t s) dx 3.20) x 2 = s g t x) dx s x 2 g ts) for s>0 for t E c := {0 t,γt) c} with any c<. Then we get dg t s) + ψ t s)) = g t x) dx ds s x 2 g ts) = ψ ts). s s This implies, for any positive s 0, s ψ t x) 3.2) g t s) = dx ψ t s) + s 0 s 0 x s 0 Next we will prove ψ t s) 3.22) lim s a t s) γt) Note that g t x) x 2 dx for s s 0. uniformly with t E c. ψ t s) a t s) = g t sx) g t s) dx a t s) x 2. From Lemma 3.3 we now for any ε 0, c) there exists a positive s 0 such that g t s) g t sx) a t s) x 2 <x 2 x γt) γt) + ε + x γt)+ε)) for s>s 0,x [, ).

12 HILL ESTIMATOR IN THE SPACE C 2007 Since the right-h side is integrable on x [, ), weget g t sx) g t s) dx lim s a t s) x 2 = lim x γt) dx s γt) x 2 = γt). This leads to 3.22). Now bac to the proof of 3.7) for γt 0 ) 0. From 3.22) we get g tn s n ) a tn s n ) g t n s n ) ψ tn s n ) γt n ). Since g t s) > 0, from 3.22), 3.2) Lemma 3.3, we get, for any ε>0, lim inf n g tn s n ) a tn s n ) lim inf n lim inf n lim inf n = lim inf n γt n ) γt n ) γt n ) ψ tn s n u) s 0 /s n ψ tn s n ) du u ) s 0 /s n 2ε)u γt n) +2ε du s 0 /s n 2ε)u ε du 2ε s 0 /s n ) ε γt n ) ε γt 0 ) γt 0 ) γt 0 ) = 2ε γt 0 ))ε γt 0 ). Letting ε 0 we get the second part of 3.7). The proof of the first part of the lemma is complete. For 3.6), according to 3.5) Lemma 3.3 we only need to prove log g t sx) log g t s) a t s)/g t s) xγ t) γ t) locally uniformly in t x. Thatis,foranyt n t 0,x n x 0 > 0,s n, log g tn s n x n ) log g tn s n ) lim n a tn s n )/g tn s n ) For γt 0 )>0, from 3.0) 3.5), we get log g tn s n x n ) log g tn s n ) lim n a tn s n )/g tn s n ) 0 γ. t 0 ) = xγ t 0 ) = lim n log[g tn s n x n ) g tn s n ))/a tn s n )]a tn s n )/g tn s n )) + ) a tn s n )/g tn s n )

13 2008 L. DE HAAN AND T. LIN = logγ t 0)x γt 0) 0 )/γ t 0 ) + ) = log x 0 ; γt 0 ) for γt 0 ) 0sincea tn s n )/g tn s n ) 0, log g tn s n x n ) log g tn s n ) lim n a tn s n )/g tn s n ) log[g tn s n x n ) g tn s n ))/a t s n )]a tn s n )/g tn s n )) + ) = lim n a tn s n )/g tn s n ) g tn s n x n ) g tn s n ) = lim n a tn s n ) This completes the proof of this lemma. 0. γt 0 ) = xγt 0) PROOF OF THEOREM 2.. We first prove 2.4). Note {U t ζ i t)), i =, 2,...}={ξ i t), i =, 2,...}.Then Û t n/) U t n/) a t n/) = ξ n,nt) U t n/) a t n/) From.5) Lemma 3.2 we get 2.4). Next we consider 2.), 2.2) 2.3). For any ε>0, d = U tζ n,n t)) U t n/). a t n/) M n ) t) a t ζ n,n t))/u t ζ n,n t)) = log ξ n i,n t) log ξ n,n t) a t ζ n,n t))/u t ζ n,n t)) d = From Lemma 3.4, M n ) t) a t ζ n,n t))/u t ζ n,n t)) <ε = ε + ζn i,n t) ζ n,n t) 2 + γ t) + ε ) log U t ζ n i,n t)) log U t ζ n,n t)). a t ζ n,n t))/u t ζ n,n t)) ) γ t)+ε ) ζ n i,n t)/ζ n,n t)) γ t) γ t) ζ n i,n t)/ζ n,n t)) γ t)+ε ) γ. t) + ε UsingLemma3.2weget M ) n t) a t ζ n,n t))/u t ζ n,n t)) 3.23) γ t) P 0. 0 t

14 HILL ESTIMATOR IN THE SPACE C 2009 Similarly we get M n 2) t) 0 t a t ζ n,n t))/u t ζ n,n t))) 3.24) 2 2 γ t)) 2γ t)) P 0. From Lemma 3.4 we get lim a t s) s 0 t U t s) γ + t) = 0. Hence a t ζ n,n t)) 0 t U t ζ n,n t)) γ ) t) P 0. From 3.23) 3.25) we get 2.). From 3.23) 3.25) we get M n ) t)) 2 0 t M n 2) 2γ t) 3.26) t) γ t) P 0, which implies 2.2). Hence 2.3) is obtained. Now we prove 2.5). Note â t n/) a t n/) = a tζ n,n t)) M n ) t) ˆγ t) ) ; a t n/) a t ζ n,n t))/u t ζ n,n t)) from 3.2), 3.6), 3.23) 2.2) we get 2.5). Finally we shall prove 2.6) 2.7). A similar argument as in the proof of Lemma 3. shows we only need to prove that, for any ν-continuous set E S c with any a>0, 3.27) ˆν ) n, E) P νe) ˆν 2) 3.28) n, E) P νe). For 3.28), we only need to prove that, for any ε>0, lim P { 2) d ˆν n n, ) } 3.29) S c,ν n, Sc ε = with d, a metric of finite measures, defined in the proof of Lemma 3.2. Note 2) ˆζ i = n /) n = j= I {ξj t)>ξ i t)} /) n j= I {ζj t)>ζ i t)} 3.30) ) = G n,t n ζ it).

15 200 L. DE HAAN AND T. LIN From 3.5) we get, for any 0 <c<b, 3.3) x) x G P 0. n,t 0 t,c x b Since G n,t x) 0 is a monotone function of x we get 3.32) x) x G P 0. n,t 0 t,0 x b Note νs b ) = C b with a positive constant C := νs ). Wecanfindb ε such that νs b 2ε ) ε. Suppose given the conditions 3.33) x) x 0 t,c x b G ε n,t 3.34) d ν n, Sc,ν Sc ) ε we have the following: i) For any closed set E S [c,b] := {f C + [0, ], 0 t ft) [c,b]} we have similarly These imply Hence 3.35) ˆν 2) n, E) = ν n, 2) d ˆν n, ) S [c,b],ν S[c,b] G n,t E) ) [from 3.30)] ν n, E ε ) [from 3.34)] ν n, E) ˆν 2) n, Eε ). d ν n, S[c,b], ˆν 2) n, S [c,b] ) ε. d ν n, S[c,b], ˆν 2) n, S [c,b] ) + d νn, S[c,b],ν S[c,b] ) 2ε. ii) For any closed set E S b, note from 3.33) we get { } S b ) = f G n,t G ft) C+ [0, ], ft) b S b ε n,t 0 t from 3.34) we get ν n, S b ε ) ν S b ε ) ε) + ε νs b 2ε + ε) 2ε;

16 lim inf n P { d HILL ESTIMATOR IN THE SPACE C 20 then ) ˆν 2) n, E) ˆν2) n, S b) = ν n, S b ) ν n, S b ε ) 2ε. G n,t Note νe) νs b ) ε.weget 2) d ˆν n, ) 3.36) S b,ν Sb 2ε. Hence 2) d ˆν n, ) S c,ν Sc d 2) ˆν n, ) S [c,b],ν S[c,b] + d 2) ˆν n, ) S b,ν Sb 4ε. Finally we have 2) ˆν n, ) } S c,ν Sc 5ε { lim inf P d ν ) n, Sc,ν Sc ε, n 0 t,c x b For the latter equality we use Lemma ). This completes the proof of 3.29). For 3.27) note ) ˆζ i t) = n with H n,t x) := [ + ˆγ n t) Ut ζ i t)) Û t n/) â t n/) { [ Ut n/)x) Û t n/) +ˆγ n t) â t n/) { = +ˆγ n t) [ at n/) â t n/) Ut n/)x) U t n/) a t n/) + U tn/) Û t n/) â t n/) 0 t,c x b } x) x G ε =. n,t ))] / ˆγn t) ) ˆγ n + = H n,t t) n ζ it) )]} / ˆγn t) ˆγ n + t) )) )]} / ˆγn t) ˆγ n +. t) From 2.3) 2.5) Proposition.ii) we get, for any 0 <c<b, 3.37) H n,t x) x P 0. Since H n,t x) 0 is a monotone function of x we get 3.38) 0 t,0 x b H n,t x) x P 0. The rest of the proof is similar to the proof of 3.28). The theorem has been proved.

17 202 L. DE HAAN AND T. LIN Acnowledgment. The remars of a referee have helped us improve the presentation remove some errors. REFERENCES BILLINGSLEY, P. 968). Convergence of Probability Measures. Wiley, New Yor. DALEY, D. J. VERE-JONES, D. 988). An Introduction to the Theory of Point Processes. Springer, Berlin. DEKKERS, A.L.M.,EINMAHL, J.H.J.DE HAAN, L. 989). A moment estimator for the index of an extreme-value distribution. Ann. Statist DREES, H. 998). On smooth statistical tail functionals. Sc. J. Statist GINÉ, E., HAHN, M. VATAN, P. 990). Max-infinitely divisible max-stable sample continuous processes. Probab. Theory Related Fields DE HAAN, L.LIN, T. 200). On convergence towards an extreme value distribution in C[0, ]. Ann. Probab DE HAAN, L.PEREIRA, T. T. 999). Estimating the index of a stable distribution. Statist. Probab. Lett DE HAAN, L.RESNICK, S. I. 993). Estimating the limit distribution of multivariate extremes. Comm. Statist. Stochastic Models DE HAAN, L. DE RONDE, J. 998). Sea wind: Multivariate extremes at wor. Extremes DE HAAN, L. ROOTZÉN, H. 993). On the estimation of high quantiles. J. Statist. Plann. Inference DE HAAN, L. SINHA, A. K. 999). Estimating the probability of a rare event. Ann. Statist HUANG, X. 992). Statistics of bivariate extremes. Dissertation. Tinbergen Research Series 22, Erasmus Univ. Rotterdam. RESNICK, S. I. 987). Extreme Values, Regular Variation, Point Processes. Springer, New Yor. SMITH, R. L. 990). Extreme value theory. In Hboo of Applicable Mathematics W. Ledermann, ed.) 7. Wiley, New Yor. ECONOMETRIC INSTITUTE ERASMUS UNIVERSITY ROTTERDAM P.O. BOX DR ROTTERDAM THE NETHERLANDS ldehaan@few.eur.nl EURANDOM P.O. BOX MB EINDHOVEN THE NETHERLANDS lin@eurom.tue.nl