Testing for a generalized Pareto process

0.42 MB·English

by  Stefan Aulbach

TESTING FOR A GENERALIZED PARETO PROCESS
STEFANAULBACHANDMICHAELFALK
2 Abstract. We investigate two models for the following setup: We consider
1 a stochastic process X ∈ C[0,1] whose distribution belongs to a parametric
0 family indexed by ϑ ∈ Θ ⊂ R. In case ϑ = 0, X is a generalized Pareto
2
process. Based on n independent copies X(1),...,X(n) of X, we establish
n
localasymptoticnormality(LAN)ofthepointprocessofexceedancesamong
a
J X(1),...,X(n) aboveanincreasingthresholdlineineachmodel.

1 The corresponding central sequences provide asymptotically optimal se-
3 quences of tests for testing H0 : ϑ = 0 against a sequence of alternatives

] Hn : ϑ = ϑn converging to zero as n increases. In one model, with an un-
T derlying exponential family, the central sequence is provided by the number
S ofexceedancesonly,whereasintheotheronetheexceedancesthemselvescon-
.
h tribute, too. However it turns out that, in both cases, the test statistics also
t dependonsomeadditionalandusuallyunknownmodelparameters.
a
m We,therefore,consideranomnibusteststatisticsequenceaswellandcom-
[ puteitsasymptoticrelativeefficiencywithrespecttotheoptimaltestsequence.

1
v
1
1 1. Introduction
6
6 In the recent three decades, the focus of univariate extreme value theory shifted
.
1 from the investigation of maxima (minima) in a sample to the investigation of
0
exceedances above a high threshold. This approach towards large observations
2
1 eased accessing the field of extreme value theory and became a crucial tool for
:
v various applied disciplines, such as building dykes.
i
X Since the publications of the articles by Balkema and de Haan (1974) and
r Pickands(1975)itisknownthatexceedancesaboveahighthresholdcanreasonably
a
be modeled only by (univariate) generalized Pareto distributions (GPD), resulting
in the peaks-over-threshold approach (POT).
Due to practical necessity, the focus of extreme value theory moved in recent
years to multivariate observations as well. Accordingly, the investigation of multi-
variate exceedances enforced the definition and investigation of multivariate GPD.
2010 Mathematics Subject Classification. Primary62F05,secondary60G70,62G32.
Key words and phrases. Functional extreme value theory, generalized Pareto process, point
process of exceedances, local asymptotic normality, central sequence, asymptotic optimal test
sequence,asymptoticrelativeefficiency.
1
2 STEFANAULBACHANDMICHAELFALK
This investigation is still lively continuing as even the definition of multivariate
GPD is under debate; see, for instance, Falk et al. (2010, Chapter 5).
As already mentioned by de Haan and Ferreira (2006, p.293): Infinite-dimen-
sional extreme value theory is not just a theoretical extension of multivariate ex-
tremevaluetheorytoamoreabstractcontext. Itservestosolveconcreteproblemsas
well. Such concrete problems are, e.g., observing dykes and tides along their whole
widthandnotonlyatafinitesetofobservationpoints. Thereis, consequently, the
need for a POT approach for functional data and for generalized Pareto processes
as well. Again, the data exceeding some kind of a high threshold are modeled by
a functional counterpart of a GPD; see Aulbach et al. (2012b). The current paper
dealswithoptimalteststhatcheckforparticularmodelswhetherthoseexceedances
do, in fact, arise from such a corresponding process.
1.1. Previous work. Following Buishand et al. (2008), a standard generalized
Pareto process, i.e., a generalized Pareto process with ultimately uniform tails in
the margins, is defined as follows. For convenience, we use bold font such as V for
stochastic processes and default font such as f for non stochastic functions. All
operations on functions such as f ≤0 are meant pointwise.
Definition 1.1. Let U be an on (0,1) uniformly distributed random variable (rv)
and let Z =(Z ) ∈C[0,1] be a stochastic process on the interval [0,1] having
t t∈[0,1]
continuous sample paths. We require that U and Z are independent and choose an
arbitrary constant M <0. Then
(cid:18) (cid:18) (cid:19)(cid:19)
U
(1) V :=(V ) := max − ,M .
t t∈[0,1] Z
t t∈[0,1]
defines a standard generalized Pareto process (GPP) if 0 ≤ Z ≤ m, E(Z ) = 1,
t t
t ∈ [0,1], hold for some constant m ≥ 1. A stochastic process Z ∈ C[0,1] with
these two properties will be called a generator.
The constant M is incorporated in the above definition to ensure that V >−∞
t
for each t ∈ [0,1]. Note that the finite-dimensional marginal distributions of V
provide multivariate GPD with ultimately uniform tails; see, e.g., Aulbach et al.
(2012a).
The process V is characterized by the fact that its functional distribution func-
tion (df) is given by
(cid:32) (cid:33)
P(V ≤f)=1−E sup (|f(t)|Z ) , f ∈E¯−[0,1], (cid:107)f(cid:107) ≤x ,
t ∞ 0
t∈[0,1]
for some x > 0. We set E¯−[0,1] := {f ∈ E[0,1] : f ≤ 0} where E[0,1] denotes
0
the set of all bounded functions f : [0,1] → R that have only a finite number of
TESTING FOR A GENERALIZED PARETO PROCESS 3
discontinuities. Then
(cid:32) (cid:33)
(cid:107)f(cid:107) :=E sup (|f(t)|Z ) , f ∈E[0,1],
D t
t∈[0,1]
defines a D-norm on E[0,1] with generator Z; see Aulbach et al. (2012c). This
representation of the df of V in terms of a D-norm is in complete analogy with the
multivariate case of a GPD. We refer to Falk et al. (2010, Section 5.1).
It was established by Aulbach et al. (2012c) that a stochastic process X ∈
C[0,1] is in the functional domain of attraction of a max-stable process (MSP)
ξ, denoted by X ∈ D(ξ), if and only if this is true for the univariate margins
together with convergence of the corresponding copula processes. A stochastic
process U = (U ) ∈ C[0,1] is called copula process if U is for each t ∈ [0,1]
t t∈[0,1] t
uniformly distributed on (0,1).
ForeachstandardGPPthereisacorrespondingstandard MSP,i.e.,astochastic
process η =(η ) ∈C[0,1] such that
t t∈[0,1]
(2) P(η ≤f)=exp(−(cid:107)f(cid:107) ), f ∈E¯−[0,1].
D
NotethatthisimpliesP(η ≤x)=exp(x),x≤0,t∈[0,1]. Ontheotherhand,the
t
df of each max-stable process η having standard negative exponential margins has
a representation as in Equation (2); we refer to Aulbach et al. (2012c) for details.
1.2. Overview of the current paper. We replace the rv U in Equation (1) by
a rv W ≥ 0 which is independent of Z, too. However, the distribution of W is
different from the uniform one and, thus, the process
(cid:18) (cid:18) (cid:19)(cid:19)
W
(3) X :=(X ) := max − ,M
t t∈[0,1] Z
t t∈[0,1]
is no longer a standard GPP.
This gives rise to the following problem: Based on the exceedances in a sample
ofnindependentcopiesX(1),...,X(n) ofX aboveahighthresholdline,howclose
can the df of W get to that of U with the difference still being detected? As we
consider exceedances above a high threshold, only the lower end of the df of W
matters. In other words, the problem suggests itself to define parametric models
{H : ϑ∈Θ} for the df H of W, such that we can derive optimal tests detecting
ϑ ϑ
the deviation of the distribution of the upper tail of X from that of V, i.e., the
deviationofϑfromzero. Thisisthecontentofthepresentpaper,whichisorganized
as follows.
In Section 2 we require that the df H of W has a density h near zero, which
ϑ ϑ
satisfies for some δ >0 the expansion
(4) h (u)=1+ϑuδ+o(uδ) as u↓0
ϑ
4 STEFANAULBACHANDMICHAELFALK
with some parameter ϑ∈Θ, where zero is an inner point of Θ⊂R. The standard
exponential df, for instance, satisfies this condition with δ = 1 and ϑ = −1. The
null-hypothesis ϑ=0 is meant to be the uniform distribution on (0,1).
In Section 3 we assume that the distribution of W belongs to an exponential
family given by the probability densities on the interval
(5) h (u)=C(ϑ)exp(ϑT(u)), 0≤u≤1, ϑ∈R.
ϑ
In both models we establish local asymptotic normality (LAN) of the point
process of exceedances among X(1),...,X(n) above an increasing threshold line.
The results, which are stated in Theorem 2.2 and Theorem 3.2, provide in each
modelthecorrespondingcentralsequenceand,thus,optimaltestsfortestingϑ=0
againstasequenceofalternativesϑ convergingtozeroasthesamplesizeincreases.
n
It turns out that the particular values of the exceedances contribute to the central
sequence only in model (4), whereas in the exponential model (5) the number of
exceedances alone yields the central sequence. The fact that just the number of
realizations in shrinking sets provides the central sequence was characterized for
truncatedprocessesinquiteageneralframeworkinFalk(1998)andFalkandLiese
(1998).
Asthecentralsequencesand,thus,theasymptoticallyoptimaltestsalsodepend
on further parameters of the generator process Z, which might be unknown in
practice, we consider an omnibus test for testing ϑ = 0 as well. We compute its
asymptoticrelativeefficiency(ARE)withrespecttotheoptimaltestineachmodel.
While ARE is positive in model (4), it is zero in model (5).
2. Testing in δ-neighborhoods of a standard GPP
Thissectiondealswithoptimaltestsinthemodelintroducedin(4). Weassume
that the df of the rv W ≥0 in (3) belongs to a parametric family {H : ϑ∈Θ} of
ϑ
distributions, where Θ is an open subset of R containing 0. By H we denote the
0
uniform distribution on the interval (0,1). In addition to (4), it is required that
there is some u ∈(0,1) such that the density h (u) of H (u) exists for u∈[0,u ],
0 ϑ ϑ 0
ϑ∈Θ, and satisfies for some δ ∈(0,1] the expansion
(6) h (u)=1+ϑuδ+r (u), u∈[0,u ],
ϑ ϑ 0
where r (0)=0, ϑ∈Θ, and
ϑ
(cid:12) (cid:12)
(7) sup (cid:12)(cid:12)rϑ(u)(cid:12)(cid:12)=o(1)
(cid:12) ϑuδ (cid:12)
0<|ϑ|≤ε0
as u↓0 for some ε >0. Obviously, (7) is equivalent with r (u)=o(ϑuδ) as u↓0,
0 ϑ
uniformly for |ϑ| ≤ ε . Since we have h = 1 and r = 0, (6) and (7) imply in
0 0 0
TESTING FOR A GENERALIZED PARETO PROCESS 5
particular the representation
(cid:16) (cid:16) (cid:17)(cid:17)
h (u)=h (u) 1+O (H (u))δ ,
ϑ 0 0
i.e., the lower tail of H is in a δ-neighborhood of H ; see Falk et al. (2010, Sec-
ϑ 0
tion 2.2).
Moreover, we assume
(cid:18) (cid:19)
(8) A:=E inf Z >0.
t
t∈[0,1]
Asinf Z ≥0,thisconditionisequivalentwiththeassumptionthatinf Z
t∈[0,1] t t∈[0,1] t
is not the constant function zero. (8) is, for instance, satisfied if Z = 2U, where
U =(U ) ∈C[0,1]isacopulaprocesssuchthat−U ∈D(η),η ∈C[0,1]being
t t∈[0,1]
(cid:0) (cid:1)
astandardMSP.ThisisimpliedbythefactthatinthiscaseP inf U >0 =
t∈[0,1] t
1.
Note that (8) and H¨older’s inequality also give
(cid:18) (cid:19)
B :=E inf Z1+δ >0.
t
t∈[0,1]
2.1. Local asymptotic normality. In order to derive asymptotically optimal
tests in this model, we first establish local asymptotic normality (LAN) of the
point process of exceedances
(cid:88)
Nn,c(B):= εsup (cid:16)X(i)/c(cid:17)(B∩[0,1]), B ∈B,
t∈[0,1] t
i≤n
where X(i), i ≤ n, are independent copies of X in (3) and c < 0. B denotes the
σ-fieldofBorelsetsofRandε isthepointmeasurewithmassoneatx. Notethat
x
X
(9) sup t ≤u ⇐⇒ X ≥cu, u∈[0,1],
c
t∈[0,1]
i.e., the random point measure N actually represents those observations among
n,c
X(1),...,X(n) which exceed the constant threshold function c.
(cid:16) (cid:17) (cid:16) (cid:17)
Denote those observations among sup X(i)/c with sup X(i)/c ≤
t∈[0,1] t t∈[0,1] t
1, i≤n, by Y ,...,Y in the order of their outcome. Then we have
1 τ(n)
(cid:88)
N (B)= ε (B), B ∈B.
n,c Yk
k≤τ(n)
By Theorem 1.4.1 in Reiss (1993) we may assume without loss of generality that
Y ,Y ,... are independent copies of a rv Y with df
1 2
P (X ≥cu)
P (Y ≤u)= ϑ , 0≤u≤1,
ϑ P (X ≥c)
ϑ
underparameterϑ, andthattheyareindependentofthetotalnumberτ(n), which
is binomial B(n,P (X ≥c))-distributed.
ϑ
6 STEFANAULBACHANDMICHAELFALK
In the next lemma we provide the density f of sup (X /c) and, thus, the
ϑ,c t∈[0,1] t
densityofY underϑ,whichisf /P (X ≥c). ByP∗Z wedenotethedistribution
ϑ,c ϑ
of a rv Z.
Lemma 2.1. Suppose that the distribution of the rv W in (3) belongs to the family
{H : ϑ∈Θ}. Thenthereissome c <0suchthatthedensity f (u), with respect
ϑ 0 ϑ,c
to the Lebesgue measure, of the rv sup (X /c) exists for ϑ ∈ Θ, c ∈ [c ,0),
0≤t≤1 t 0
u∈[0,1], and it is given by
(cid:90) m (cid:18) (cid:19)
f (u)=|c| zh (|c|zu) P ∗ inf Z (dz).
ϑ,c ϑ t
0 t∈[0,1]
Furthermore there exists ε >0 such that
0
(cid:16) (cid:17)
f (u)=|c|A+ϑ|c|1+δBuδ+o ϑ|c|1+δ
ϑ,c
uniformly for |ϑ|≤ε and u∈[0,1] as c↑0; note that f (u)=|c|A.
0 0,c
Proof. LetmbegivenasinDefinition1.1andu ,ε ,δ begivenasinEquation(6).
0 0
Then we obtain for c > max{M,−u /m}, ϑ ∈ Θ, c ∈ [c ,0) and u ∈ [0,1] by
0 0 0
conditioning on inf Z =z and Fubini’s theorem
t∈[0,1] t
(cid:18) (cid:18) (cid:19) (cid:19)
W
P (X ≥cu)=P max − ,M ≥cu, t∈[0,1]
ϑ ϑ Z
t
=P (W ≤|c|uZ , t∈[0,1])
ϑ t
(cid:18) (cid:19)
=P W ≤|c|u inf Z
ϑ t
t∈[0,1]
(cid:90) m (cid:18) (cid:19)
= P (W ≤|c|zu) P ∗ inf Z (dz)
ϑ t
0 t∈[0,1]
(cid:90) m (cid:18) (cid:19)
= H (|c|zu) P ∗ inf Z (dz)
ϑ t
0 t∈[0,1]
(cid:90) u (cid:90) m (cid:18) (cid:19)
= |c| zh (|c|zx) P ∗ inf Z (dz)dx.
ϑ t
0 0 0≤t≤1
This representation implies that sup (X /c) has the density
t∈[0,1] t
(cid:90) m (cid:18) (cid:19)
f (u)=|c| zh (|c|zu) P ∗ inf Z (dz)
ϑ,c ϑ t
0 t∈[0,1]
(cid:90) m (cid:18) (cid:19)
=|c| z(cid:0)1+ϑ(|c|zu)δ+r (|c|zu)(cid:1) P ∗ inf Z (dz)
ϑ t
0 t∈[0,1]
(cid:18) (cid:19) (cid:32)(cid:18) (cid:19)1+δ(cid:33) (cid:16) (cid:17)
=|c|E inf Z +ϑ|c|1+δuδE inf Z +o ϑ|c|1+δ
t t
t∈[0,1] t∈[0,1]
(cid:16) (cid:17)
=|c|A+ϑ|c|1+δuδB+o ϑ|c|1+δ
TESTING FOR A GENERALIZED PARETO PROCESS 7
(cid:16) (cid:17)
as c ↑ 0, uniformly for |ϑ| ≤ ε and u ∈ [0,1]. (Note that r (|c|zu) = o ϑ|c|δ
0 ϑ
uniformly for |ϑ| ≤ ε , u ∈ [0,1] and z ∈ [0,m] as c ↑ 0.) As H (u) = u, we have
0 0
h (u)=1, u∈[0,1], which completes the proof. (cid:3)
0
If c ≤c<0, we have P (Y ≤u)=u, i.e., the distribution L (Y) of Y under
0 0 ϑ,c
ϑ is then dominated by L (Y). Thus the distribution L (N ) of N under ϑ
0,c ϑ n,c n,c
is, in this case, dominated by L (N ) – see, e.g., Theorem 3.1.2 in Reiss (1993) –
0 n,c
and we obtain from Reiss (1993, Example 3.1.2) the L (N )-density of L (N )
0 n,c ϑ n,c
 
µ([0,1])
dLϑ(Nn,c)(µ)= (cid:89) fϑ,c(yi)P0(X ≥c)×
dL (N ) f (y )P (X ≥c)
0 n,c 0,c i ϑ
i=1
(cid:18)P (X ≥c)(cid:19)µ([0,1])(cid:18)1−P (X ≥c)(cid:19)n−µ([0,1])
× ϑ ϑ
P (X ≥c) 1−P (X ≥c)
0 0
whereµ=(cid:80)µ([0,1])ε ,0≤y ,...,y ≤1andµ([0,1])≤n. Theloglikelihood
i=1 yi 1 µ([0,1])
ratio is, consequently,
(cid:26) (cid:27)
dL (N )
L (ϑ|0):=log ϑ n,c (N )
n,c dL (N ) n,c
0 n,c
(cid:18) (cid:19) (cid:18) (cid:19)
(10) = (cid:88) log fϑ,c(Yi)P0(X ≥c) +τ(n)log Pϑ(X ≥c)
f (Y )P (X ≥c) P (X ≥c)
0,c i ϑ 0
i≤τ(n)
(cid:18) (cid:19)
1−P (X ≥c)
+(n−τ(n))log ϑ .
1−P (X ≥c)
0
Weletinthesequelc=c dependonthesamplesizenwithc ↑0and,equally,
n n
ϑ=ϑ with ϑ →0 as n→∞. Precisely, we put with arbitrary ξ ∈R
n n
ξ
(11) ϑ :=ϑ (ξ):= .
n n (cid:16) (cid:17)1/2
n|c |1+2δ
n
The following result finally proves the desired LAN property of N ; it is a crucial
n,c
tool for deriving asymptotically optimal tests in the subsequent subsection.
Theorem 2.2. Suppose that c ↑ 0, n|c |1+2δ → ∞ as n → ∞. Then we obtain
n n
for ϑ as in (11) the expansion
n
ξB ξ2B2
L (ϑ |0)= (Z +Z )− +o (1)
n,cn n (1+δ)A1/2 n1 n2 2A(2δ+1) P0
(cid:18) ξ2B2 ξ2B2 (cid:19)
→ N − ,
D0 2A(2δ+1) A(2δ+1)
with
τ(n)−n|c |A
Z := n → N(0,1),
n1 (n|c |A)1/2 D0
n
8 STEFANAULBACHANDMICHAELFALK
and
1+δ (cid:88) (cid:18) 1 (cid:19) (cid:18) δ2 (cid:19)
Z := Yδ− → N 0,
n2 τ(n)1/2 k 1+δ D0 2δ+1
k≤τ(n)
being independent.
Proof. First we compile several facts that will be used in the proof. From Lemma
2.1 we obtain
(Fact 1) P (X ≥c )=P(V ≥c )=|c |A
0 n n n
and, thus, a suitable version of the central limit theorem implies
τ(n)−n|c |A
(Fact 2) n → N(0,1).
(n|c |A)1/2 D0
n
Moreover, we conclude from Lemma 2.1 for |ϑ|≤ε
0
ϑ (cid:16) (cid:17)
(Fact 3) P (X ≥c )−P (X ≥c )=|c |1+δ B+o ϑ|c |1+δ
ϑ n 0 n n 1+δ n
and, thus,
(cid:18) (cid:19)
P (X ≥c )−P (X ≥c ) 1 ξB 1
(Fact 4) ϑn n 0 n = +o .
P (X ≥c ) (n|c |)1/2(1+δ)A (n|c |)1/2
0 n n n
Hence, Taylor expansion log(1+ε)=ε−ε2/2+o(ε2) as ε→0 implies
(cid:18) (cid:19) (cid:18) (cid:19)
P (X ≥c ) 1−P (X ≥c )
τ(n)log ϑn n +(n−τ(n))log ϑn n
P (X ≥c ) 1−P (X ≥c )
0 n 0 n
(cid:40)P (X ≥c )−P (X ≥c ) 1(cid:18)P (X ≥c )−P (X ≥c )(cid:19)2
=τ(n) ϑn n 0 n − ϑn n 0 n
P (X ≥c ) 2 P (X ≥c )
0 n 0 n
+O(cid:32)(cid:12)(cid:12)(cid:12)Pϑn(X ≥cn)−P0(X ≥cn)(cid:12)(cid:12)(cid:12)3(cid:33)(cid:41)
(cid:12) P (X ≥c ) (cid:12)
0 n
(cid:26)
P (X ≥c )−P (X ≥c )
+(n−τ(n)) 0 n ϑn n
1−P (X ≥c )
0 n
(cid:16) (cid:17)(cid:27)
+O |P (X ≥c )−P (X ≥c )|2
ϑn n 0 n
P (X ≥c )−P (X ≥c ) P (X ≥c )−P (X ≥c )
=τ(n) ϑn n 0 n +(n−τ(n)) 0 n ϑn n
P (X ≥c ) 1−P (X ≥c )
0 n 0 n
τ(n)(cid:18)P (X ≥c )−P (X ≥c )(cid:19)2
− ϑn n 0 n +o (1)
2 P (X ≥c ) P0
0 n
as
τ(n)(cid:12)(cid:12)(cid:12)(cid:12)Pϑn(X ≥Pc(nX)−≥Pc0()X ≥cn)(cid:12)(cid:12)(cid:12)(cid:12)3 ∼E0(τ(n))O(cid:18)(n|c1|)3/2(cid:19)
0 n n
(cid:18) (cid:19)
1
=nP (X ≥c )O → 0,
0 n (n|c |)3/2 n→∞
n
TESTING FOR A GENERALIZED PARETO PROCESS 9
where∼denotesasymptoticequivalence. Theprecedingconvergencetozerofollows
from the condition n|c |1+2δ → ∞ and the equivalence
n n→∞
(n−τ(n))|P (X ≥c )−P (X ≥c )|2 ∼O(|c |)→ 0.
ϑn n 0 n n n→∞
From the law of large numbers we obtain
(cid:18)P (X ≥c )−P (X ≥c )(cid:19)2
τ(n) ϑn n 0 n
P (X ≥c )
0 n
(cid:18) 1 ξB (cid:18) 1 (cid:19)(cid:19)2
∼E(τ(n)) +o
(n|c |)1/2(1+δ)A (n|c |)1/2
n n
(cid:18) 1 ξB (cid:18) 1 (cid:19)(cid:19)2
=n|c |A +o
n (n|c |)1/2(1+δ)A (n|c |)1/2
n n
ξ2B2
→ .
n→∞ (1+δ)2A
Moreover,
P (X ≥c )−P (X ≥c ) P (X ≥c )−P (X ≥c )
τ(n) ϑn n 0 n +(n−τ(n)) 0 n ϑn n
P (X ≥c ) 1−P (X ≥c )
0 n 0 n
P (X ≥c )−P (X ≥c )
= ϑn n 0 n (τ(n)−nP (X >c ))
P (X ≥c )(1−P (X ≥c )) 0 n
0 n 0 n
(cid:18) (cid:18) (cid:19)(cid:19)
1 ξB 1
=(n|c |A)1/2 +o ×
n (n|c |)1/2(1+δ)A(1+o(1)) (n|c |)1/2
n n
τ(n)−nP (X ≥c )
× 0 n
(n|c |A)1/2
n
ξB
= Z +o (1).
(1+δ)A1/2 n1 P0
Altogether we have shown so far that
(cid:18) (cid:19) (cid:18) (cid:19)
P (X ≥c ) 1−P (X ≥c )
τ(n)log ϑn n +(n−τ(n))log ϑn n
P (X ≥c ) 1−P (X ≥c )
0 n 0 n
ξB ξ2B2
= Z − +o (1).
(1+δ)A1/2 n1 2(1+δ)2A P0
Next we show that
(cid:88) log(cid:18)fϑn,cn(Yk) P0(X ≥cn) (cid:19)= ξB Z − ξ2B2δ2 +o (1).
f (Y ) P (X ≥c ) A1/2(1+δ) n2 2A(2δ+1)(1+δ)2 P0
k≤τ(n) 0,cn k ϑn n
We have by Lemma 2.1
f (Y ) ξ B
ϑn,cn k =1+ Yδ+r (Y ,ϑ ,c ),
f (Y ) (n|c |)1/2 A k 0 k n n
0,cn k n
(cid:16) (cid:17)
where r (Y ,ϑ ,c )=o (n|c |)−1/2 uniformly for k and n with
0 k n n n
E (r (Y ,ϑ ,c ))
0 0 1 n n
10 STEFANAULBACHANDMICHAELFALK
(cid:90) 1 f (t)
= r (t,ϑ ,c ) 0,cn dt
0 n n P (X ≥c )
0 0 n
(cid:90) 1(cid:18)f (t) ξ B (cid:19) f (t)
= ϑn,cn −1− tδ 0,cn dt
f (t) (n|c |)1/2 A P (X ≥c )
0 0,cn n 0 n
P (X ≥c )−P (X ≥c ) ξ B (cid:90) 1
= ϑn n 0 n − tδdt
P (X ≥c ) (n|c |)1/2 A
0 n n 0
P (X ≥c )−P (X ≥c ) ξ B
= ϑn n 0 n −
P (X ≥c ) (n|c |)1/2(1+δ)A
0 n n
and
Var (r (Y ,ϑ ,c ))≤E (cid:0)r2(Y ,ϑ ,c )(cid:1)=o(1/(n|c |)).
0 0 1 n n 0 0 1 n n n
UsingagaintheTaylorexpansionlog(1+ε)=ε−ε2/2+O(ε3)asε→0,wededuce
(cid:18) (cid:19)
(cid:88) log fϑn,cn(Yk) P0(X ≥cn)
f (Y ) P (X ≥c )
k≤τ(n) 0,cn k ϑn n
(cid:18) (cid:19)
(cid:88) ξ B
= log 1+ Yδ+r (Y ,ϑ ,c )
(n|c |)1/2 A k 0 k n n
n
k≤τ(n)
(cid:18) (cid:19)
P (X ≥c )−P (X ≥c )
−τ(n)log 1+ ϑn n 0 n
P (X ≥c )
0 n
(cid:88) (cid:18) ξ B ξ2 B2 (cid:19)
= Yδ+r (Y ,ϑ ,c )− Y2δ
(n|c |)1/2 A k 0 k n n 2n|c |A2 k
n n
k≤τ(n)
(cid:32)P (X ≥c )−P (X ≥c ) 1(cid:18)P (X ≥c )−P (X ≥c )(cid:19)2(cid:33)
−τ(n) ϑn n 0 n − ϑn n 0 n
P (X ≥c ) 2 P (X ≥c )
0 n 0 n
+o (1)
P0
(cid:18) (cid:18) (cid:19) (cid:19)
(cid:88) ξ B 1
= Yδ− +r (Y ,ϑ ,c )−E (r (Y ,ϑ ,c ))
(n|c |)1/2 A k 1+δ 0 k n n 0 0 1 n n
n
k≤τ(n)
ξ2 B2 (cid:88) τ(n) ξ2B2
− Y2δ+ +o (1)
2n|c |A2 k n|c |2(1+δ)2A2 P0
n n
k≤τ(n)
τ(n)1/2 1 ξB (cid:88) (cid:18) 1 (cid:19)
= Yδ−
(n|c |)1/2τ(n)1/2 A k 1+δ
n
k≤τ(n)
τ(n) ξ2B2 1 (cid:88) τ(n) ξ2B2
− Y2δ+ +o (1)
2n|c | A2 τ(n) k n|c |2(1+δ)2A2 P0
n n
k≤τ(n)
ξB 1 (cid:88) (cid:18) 1 (cid:19) ξ2B2δ2
= Yδ− − +o (1)
A1/2τ(n)1/2 k 1+δ 2A(2δ+1)(1+δ)2 P0
k≤τ(n)
(cid:18) ξ2B2δ2 ξ2B2δ2 (cid:19)
→ N − ,
D0 2A(2δ+1)(1+δ)2 A(2δ+1)(1+δ)2