On the block maxima method in
extreme value theory
Ana Ferreira
ISA, Univ Tecn Lisboa, Tapada da Ajuda 1349-017 Lisboa, Portugal
CEAUL, FCUL, Bloco C6 - Piso 4 Campo Grande, 749-016 Lisboa, Portugal
[email protected]
and
Laurens de Haan
Erasmus Univ Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
CEAUL, FCUL, Bloco C6 - Piso 4 Campo Grande, 749-016 Lisboa, Portugal
[email protected]
Abstract: In extreme value theory there are two fundamental approaches,
both widely used: the block maxima method and the peaks-over-threshold
(POT) method. Whereas much research has gone into the POT method,
the block maxima method has not been studied thoroughly. The present
paper aims at providing conditions under which the block maxima method
can be justified. In this paper we restrict attention to the independent and
identically distributed case and focus on the probability weighted moment
(PWM) estimators of Hosking, Wallis and Wood (1985).
MSC 2010 subject classifications: Primary 62G32, ; secondary 62G20,
62G30.
Keywords and phrases: block maxima, probability weighted moment
estimators, extreme value index, asymptotic normality, extreme quantile
estimation.
1. Introduction
The block maxima approach in extreme value theory (EVT), consists of dividing
the observation period into non-overlapping periods of equal size and restricts
attention to the maximum observation in each period. The new observations
thus created follow - under extreme value conditions - approximately an extreme
value distribution (Gγ for some real γ). Parametric statistical methods for the
extreme value distributions are then applied to those observations. Usually it
is taken for granted that the block maxima follow very well an extreme value
distribution. In this paper we take this misspecification into account. Since Gγ
is not the exact distribution for those observations, a bias may appear. The
procedure can be justified using (a strengthening of) the domain of attraction
conditions of EVT.
In the peaks-over-threshold approach in EVT one selects those of the initial
observations that exceed a certain high threshold. The probability distribution of
those selected observations - under extreme value conditions - is approximately a
generalized Pareto distribution (GPD). Parametric statistical methods for GPD
1
Ferreira and de Haan/On the block maxima method
2
are then applied to those observations. Again, a bias may appear since GPD is
not the exact distribution of those selected observations.
The block maxima method is the older one (see e.g. Gumbel, 1958). The POT
method has been developed by Pickands (1975) who provided the theoretical
framework and devised statistical tools.
In the case of the POT method, exact conditions under which the method is
justified are well-known (see e.g. de Haan and Ferreira 2006, Chapters 3-4, and
Drees 1998).
The POT method picks up all relevant high observations. The block maxima
method, on the one hand misses some of these high observations and, on the
other hand, retains some lower observations. Hence the POT seems to make
better use of the available information. However, there may be reasons for using
the block maxima method:
• The only available information may be block maxima (e.g. yearly maxima).
• The block maxima method may be preferable when the observations are
not exactly independent and identically distributed (i.i.d.). For example,
there may be a seasonal periodicity in case of yearly maxima or, there
may be short range dependence that plays a role within blocks but not
between blocks.
• The block maxima method may be easier to apply since the block periods
appear naturally in many situations. Hence the problem of choosing a high
threshold in the POT method (which is a difficult one) does not play a
role.
The present paper aims at formulating exact conditions under which the
block maxima method can be justified. Since some of the block maxima may
actually not be very high, one expects somewhat more strict conditions in this
case then in the POT case. However, as it turns out, the conditions are similar.
Throughout the paper we assume that the observations are i.i.d. When working with block maxima there are two major sets of estimators that are widely
used: the maximum likelihood estimators (e.g. Prescott and Walden, 1980)
and the probability weighted moment (PWM) estimators (Hosking, Wallis and
Wood, 1985). Recently, Dombry (2013) has proved consistency of the maximum
likelihood estimators. The present paper concentrates on the PWM estimators.
The asymptotic normality result for the PWM estimators is stated in Section
2, along with a similar result for the accompanying high quantile estimator. The
proofs (in Section 3) are based on a uniform expansion of the relevant quantile
process given in Proposition 3.1.
In future work we shall extend the results to the non - i.i.d. case and to the
maximum likelihood estimator. We shall also develop a theoretical comparison
between the peaks-over-threshold and the block maxima methods. A comparative simulation study has been carried out by S. Caires (2009).
Ferreira and de Haan/On the block maxima method
3
2. The estimators and their properties
Let X̃1 , X̃2 , . . . be i.i.d. random variables with distribution function F . Define
for m = 1, 2, . . . and i = 1, 2, . . . , k the block maxima
Xi =
max
(i−1)m<j≤im
X̃i .
(1)
Hence, the m × k observations are divided into k blocks of size m. Write n =
m × k, the total number of observations. We study the model for large k and m,
hence we shall assume that n → ∞; in order to obtain meaningful limit results,
we have to require that both m = mn → ∞ and k = kn → ∞, as n → ∞.
The main assumption is that F is in the domain of attraction of some extreme
value distribution
Gγ (x) = exp −(1 + γx)−1/γ , γ ∈ R, 1 + γx > 0.
Then Xi , when normalized, will follow approximately this extreme value distribution i.e., for appropriately chosen am > 0 and bm and all x
X i − bm
≤ x = lim F m (am x + bm ) = Gγ (x),
i = 1, 2, . . . , k.
lim P
m→∞
m→∞
am
(2)
This can be written as
lim
m→∞
1
1
= (1 + γx)1/γ ,
m − log F (am x + bm )
which is equivalent to the convergence of the inverse functions:
xγ − 1
V (mx) − bm
=
,
m→∞
am
γ
lim
x > 0,
←
with V = (−1/ log F ) . Hence bm can be chosen to be V (m). This is the first
order condition. For our analysis we also need a second order expansion as
follows.
Condition 2.1 (Second order condition). Suppose that for some positive function a and some positive or negative function A with limt→∞ A(t) = 0,
lim
t→∞
V (tx)−V (t)
a(t)
−
A(t)
xγ −1
γ
=
Z
x
sγ−1
1
Z
s
uρ−1 du ds = Hγ,ρ (x),
1
for all x > 0 (see e.g. de Haan and Ferreira, Corollary 2.3.4). Note that the
function |A| is regularly varying with index ρ ≤ 0.
The PWM estimators to estimate γ, as well as the location bm and scale am ,
are defined as follows. Let X1,k , . . . , Xk,k be the order statistics of X1 , . . . , Xk .
Let
k
1 X (i − 1) . . . (i − r)
Mr =
Xi,k , for r = 0, 1, 2; k > r.
(3)
k i=1 (k − 1) . . . (k − r)
Ferreira and de Haan/On the block maxima method
4
The PWM estimators are simple functionals of M0 , M1 and M2 . The estimator
γ̂k,m for γ is defined as the solution of the equation
3M2 − M0
3γ̂k,m − 1
.
=
γ̂
k,m
2M1 − M0
2
−1
(4)
The estimator âk,m of am is
âk,m =
γ̂k,m
γ̂
k,m
2
−
2M1 − M0
,
1 Γ (1 − γ̂k,m )
(5)
1 − Γ (1 − γ̂k,m )
,
γ̂k,m
(6)
and the estimator b̂k,m of bm is
b̂k,m = M0 + âk,m
R∞
where Γ(x) = 0 tx−1 e−t dt, x > 0 (Hosking, Wallis and Wood, 1985).
For γ̂k,m = 0 the estimators follow by continuity:
γ̂k,m = 0
if
log 3
3M2 − M0
=
,
2M1 − M0
log 2
2M1 − M0
and b̂k,m = M0 + âk,m Γ′ (1).
log 2
Note that −Γ′ (1) is Euler’s constant.
Clearly, the given estimators are quite different from the ones in the POT
case.
Pk i r
Remark 2.1. There are other variants for Mr , e.g. k1 i=1 k+1
Xi,k , but
they give theoretical problems.
âk,m =
2.1. Asymptotic normality
Next we state conditions for the asymptotic normality of the mentioned estimators.
Theorem 2.1. Assume that F is in the domain of attraction of an extreme value
distribution Gγ with γ < 1/2 and that Condition√2.1 holds. Let m = mn → ∞
and k = kn → ∞ as n → ∞, in such a way that kA(m) → λ ∈ R. Then
√
(r + 1)Mr − bm
− Dr (γ)
k
am
Z 1
sr−1 (− log s)−1−γ B(s) ds + λIr (γ, ρ) =: Qr , (7)
→d (r + 1)
0
as n → ∞, jointly for r = 0, 1, 2, where →d means convergence in distribution,
B is Brownian bridge,
Dr (ξ) =
(r + 1)ξ Γ(1 − ξ) − 1
,
ξ
ξ<1
Ferreira and de Haan/On the block maxima method
5
(Dr (0) = log(r + 1) − Γ′ (1) as defined by continuity), and
Ir (γ, ρ)
 1

ρ (Dr (γ + ρ) − Dr (γ)) ,





′


=
 Dr (γ)γ (r+1)
Dr (γ)
′
=
−Γ
(1
−
γ)
+
log(r
+
1)Γ(1
−
γ)
−
γ
(r+1)γ ,







D′ (0) =

 1r 3
′′
′
2 log (r + 1) + Γ (1) − 3 log(r + 1)Γ (1) ,
R∞
Note that Γ′ (1 − γ) = 0 u−γ e−u (log u) du.
ρ 6= 0,
γ 6= 0, ρ = 0,
γ = 0, ρ = 0.
Remark 2.2. Explicit (complicated) expressions for the limiting covariance matrix can be found in Hosking, Wallis and Wood (1985), cf. vr,r , vr,r+1 and
vr,r+s (C.9)–(C.11) in their Appendix C. From there, var(Qr ) = (r + 1)2 vrr and
cov(Qr , Qs ) = (r + 1)(s + 1)vr,s .
√
Remark 2.3. The condition kA(m) → λ ∈ R means that the growth of kn ,
the number of blocks, is restricted with respect to the growth mn , the size of a
block, as n → ∞. In particular this condition implies that (log k)/m → 0, as
n → ∞.
The asymptotic normality of γ̂k,m , âk,m and b̂k,m follows from Theorem 2.1:
Theorem 2.2. Under the conditions of Theorem 2.1, as n → ∞,
√
k (γ̂k,m − γ) →d
−1 1
log 3
γ
log 2
γ
−
(Q
−
Q
)
−
(Q
−
Q
)
=: ∆,
2
0
1
0
Γ(1 − γ) 1 − 3−γ
1 − 2−γ
3γ − 1
2γ − 1
√
âk,m
k
− 1 →d
am
Γ′ (1 − γ)
−γ
γ
1
log 2
+
=: Λ
(Q1 − Q0 )+∆
+
(2γ − 1) Γ(1 − γ)
γ
1 − 2−γ
log 2
Γ(1 − γ)
√ b̂k,m − bm
→d
k
am
Q0 +
1 − Γ(1 − γ)
γΓ′ (1 − γ) − 1 + Γ(1 − γ)
∆+
Λ =: B;
γ2
γ
where for γ = 0 the formulas should read as (defined by continuity):
√
kγ̂k,m →d
log 3 log 2
−
2
2
−1 1
1
(Q2 − Q0 ) −
(Q1 − Q0 )
log 3
log 2
Ferreira and de Haan/On the block maxima method
√
k
6
1
log 2
âk,m
′
− 1 →d
(Q1 − Q0 ) + ∆
+ Γ (1)
am
log 2
2
√ b̂k,m − bm
k
→d Q0 − Γ′′ (1)∆ + Γ′ (1)Λ
am
Remark 2.4. The second order condition for the asymptotic normality in the
POT case (see e.g. de Haan and Ferreira, Theorem 3.6.1), is formulated in
terms of the function U = (1/(1 − F ))← , not the function V . For a comparison
between the two second order conditions see Drees, de Haan and Li (2003).
2.2. High quantile estimation
High quantile estimation is discussed in the next theorem. Let the total number
of observations be n, that is, n = mk as before. Let m = m(n) → ∞, k =
k(n) → ∞ as n → ∞. We want to estimate xn := F ← (1 − pn ) with pn small
and write xn = V (1/ (− log (1 − pn ))).
Our estimator for xn is
γ̂
x̂k,m := b̂k,m + âk,m
cnk,m − 1
γ̂k,m
with cn = (−m log(1 − pn ))−1 and we obtain the following result:
Theorem 2.3. Assume the conditions of Theorem 2.1 with the second order
parameter ρ negative, or zero with γ negative. Moreover assume
lim cn = ∞
n→∞
and
lim
n→∞
log cn
√ = 0.
k
Then, as n → ∞,
√
γ−
k (x̂k,m − xn ) d
→ ∆ + (γ− )2 B − γ− Λ − λ
am qγ (cn )
γ− + ρ
where γ− := min(0, γ) and
qγ (t) :=
Z
t
sγ−1 log s ds.
1
Remark 2.5. This is the result for a high quantile of the original distribution
F . One may also want to estimate a high quantile of the distribution of the block
maximum. In that case we need to estimate xn := V (m/ (− log (1 − pn ))). The
result is as above with cn replaced by mcn .
Ferreira and de Haan/On the block maxima method
7
3. Proofs
Throughout this section Z represents a unit Fréchet random variable, i.e. one
k
with distribution function F (x) = e−1/x , x > 0, and {Zi,k }i=1 are the order statistics from the associated i.i.d. sample of size k, Z1 , . . . , Zk . Similarly,
k
{Xi,k }i=1 represents the order statistics of the block maxima X1 , . . . , Xk from
(1) and, X⌈u⌉,k := Xr,k for r − 1 < u ≤ r, r = 1, . . . , k. Recall the function V
from Section 2. The following representation will be useful,
X =d V (mZ).
(8)
We start by formulating a number of auxiliary results.
1. As k → ∞,
Lemma 3.1.
(log k)Z1,k →P 1.
2. (Csörgő and Horváth 1993, p. 381) Let 0 < ν < 1/2. With {Bk }k≥1 an
appropriate sequence of Brownian bridges,
Bk (s) s(− log s) √
sup
= oP (1),
ν k (− log s)Z⌈ks⌉,k − 1 −
s(− log s) 1/(k+1)≤s≤k/(k+1) (s(1 − s))
as k → ∞ (⌈u⌉ represents the smallest integer larger or equal to u).
3. Similarly, with 0 < ν < 1/2 for an appropriate sequence {Bk }k≥1 of
Brownian bridges and ξ ∈ R,
sup
1/(k+1)≤s≤k/(k+1)
√
ks(− log s)1+ξ
(s(1 − s))
−ν
ξ
Z⌈ks⌉,k
−1
ξ
(− log s)−ξ − 1
−
ξ
!
as k → ∞.
− Bk (s) = oP (1),
Lemma 3.2. Under Condition 2.1, there are functions A(t) ∼ A0 (t) and a(t) =
a0 (t) (1 + o (A0 (t))), as t → ∞, such that for all ε, δ > 0 there exists t0 = t0 (ε, δ)
such that for t, tx > t0 ,
V (tx)−V (t) xγ −1
− γ
a(t)
(9)
− Hγ,ρ (x) ≤ ε max xγ+ρ+δ , xγ+ρ−δ .
A(t)
Moreover,
and
a(tx)
ρ
a(t) − xγ
x
−
1
γ
≤ ε max xγ+ρ+δ , xγ+ρ−δ
−x
A(t)
ρ A(tx)
ρ
ρ+δ
ρ−δ
A(t) − x ≤ ε max(x , x ).
(10)
Ferreira and de Haan/On the block maxima method
8
Remark 3.1. This is an easily obtained variant of Theorem B.3.10 of de Haan
and Ferreira (2006).
Note that,
Hγ,ρ (x) =













1
ρ
1
γ
xγ+ρ −1
γ+ρ
−
xγ log x −
ρ
1 x −1
−
ρ
ρ
1
2
2 (log x) ,
xγ −1
γ
γ
x −1
γ
log x
,
,
ρ 6= 0 6= γ
,
ρ = 0 6= γ
ρ 6= 0 = γ
ρ=0=γ.
Proposition 3.1. Assume the conditions of Theorem 2.1. Let 0 < ε < 1/2 and
k
{Xi,k }i=1 be the order statistics of the block maxima X1 , X2 , . . . , Xk . Then,
√
k
√
X⌈ks⌉,k − bm
1
Bk (s)
(− log s)−γ − 1
=
+ kA(m)Hγ,ρ
−
am
γ
s(− log s)1+γ
− log s
+ s−1/2−ε (1 − s)−1/2−γ−ρ−ε + B(s)s−1 (− log s)−1−γ−ρ−ε oP (1),
as n → ∞, where the oP (1) term is uniform for 1/(k + 1) ≤ s ≤ k/(k + 1).
Remark 3.2. This proposition should also be useful when analysing other estimators for the block maxima approach, like the maximum likelihood estimators.
Proof of Proposition 3.1. By representation (8),
X⌈ks⌉,k − bm
(− log s)−γ − 1
−
am
γ


m
− bm
V
V
mZ
−
b
−
log
s
m
⌈ks⌉,k

−
=d 
am
am
 
m
V − log
−γ
s − bm
(−
log
s)
−
1

+
−
am
γ
=
I (random part) + II (bias part).
We start with part I,


m
m


V (− log s)Z⌈ks⌉,k − log
−
V
s
− log s
(− log s)−γ
I =
m


a − log
s

 m

 a − log
s
(− log s)γ
×

 a(m)
=
I.1 × I.2.
Ferreira and de Haan/On the block maxima method
9
According to (10) of Lemma 3.2, for each ε, δ > 0 there exists t0 such that the
factor I.2 is bounded (above and below) by
(− log s)−ρ − 1
−ρ+δ
−ρ−δ
1 + A(m)
± ε max (− log s)
, (− log s)
ρ
provided m ≥ t0 and s ≥ e−m/t0 . According to (9) of Lemma 3.2, for factor I.1
we have the bounds
γ
−1
−γ (− log s)Z⌈ks⌉,k
(− log s)
γ
m
+A
(− log s)−γ × Hγ,ρ (− log s)Z⌈ks⌉,k
− log s
γ+ρ+δ
γ+ρ−δ , (− log s)Z⌈ks⌉,k
± ε max (− log s)Z⌈ks⌉,k
= I.1a + I.1b
provided s√≥ e−m/t0 and m/ log k ≥ t0 (the latter inequality eventually holds
true since kA(m) is bounded). Note that m/ log k ≥ t0 implies mZ1,k ≥ 2t0
which implies (Lemma 3.1) mZ⌈ks⌉,k ≥ 2t0 for all s.
For term I.1a we use Lemma 3.1.3:
γ
Z⌈ks⌉,k − 1 (− log s)−γ − 1
−
γ
γ
is bounded (above and below) by
ν
1
Bk (s)
ε (s(1 − s))
√
±√
,
1+γ
s(−
log
s)
k
k s(− log s)1+γ
for some ε > 0, 0 < ν < 1/2 and all s ∈ [1/(k + 1), k/(k + 1)].
m
Next we turn to term I.1b. By Lemma 3.2, (− log s)−γ A − log
s is bounded
(above and below) by
A(m) (− log s)−γ−ρ ± ε max (− log s)−γ−ρ+δ , (− log s)−γ−ρ−δ
provided s > e−m/t0 and m/ log k > t0 . Furthermore for ρ 6= 0 6= γ and s ∈
[1/(k + 1), k/(k + 1)], by Lemma 3.1.3,
Hγ,ρ (− log s)Z⌈ks⌉,k
(
)
γ+ρ
γ
(− log s)Z⌈ks⌉,k
−1
(− log s)Z⌈ks⌉,k − 1
1
=
−
ρ
γ+ρ
γ
(
#
" γ+ρ
−γ−ρ
Z
−
1
(−
log
s)
−
1
1
⌈ks⌉,k
(− log s)γ+ρ
−
=
ρ
γ+ρ
γ+ρ
" γ
#)
Z⌈ks⌉,k − 1 (− log s)−γ − 1
γ
−(− log s)
−
γ
γ
Ferreira and de Haan/On the block maxima method
10
is bounded by
ν
Bk (s)
(s(1 − s))
1
1
ε
√
(− log s)γ+ρ √
±
ρ
k s(− log s)1+γ+ρ
k s(− log s)1+γ+ρ
ν Bk (s)
ε (s(1 − s))
1
√
∓
− (− log s)γ √
k s(− log s)1+γ
k s(− log s)1+γ
ν
2ε (s(1 − s))
= ± √
,
ρ k s(− log s)
and similarly
for cases other than ρ 6= 0 6= γ. The remaining
part of I.1b, namely
γ+ρ−δ γ+ρ+δ
, is similar.
±ε max (− log s)Z⌈ks⌉,k
, (− log s)Z⌈ks⌉,k
Part II, by the inequalities of Lemma 3.2, is bounded by
1
± ε max (− log s)−γ−ρ+δ , (− log s)−γ−ρ−δ
A(m) Hγ,ρ
− log s
√
1
to the result.
hence it contributes kA(m)Hγ,ρ − log
s
Collecting all the terms, one finds the result.
Proof of Theorem 2.1. Let, for r = 0, 1, 2,
(r)
Jk (s) =
(⌈ks⌉ − 1) . . . (⌈ks⌉ − r)
,
(k − 1) . . . (k − r)
s ∈ [0, 1].
(r)
Note that Jk (s) → sr , as k → ∞, uniformly in s ∈ [0, 1], and,
Z 1
Z 1
k
1
1 X (i − 1) . . . (i − r)
(r)
Jk (s)ds =
sr ds.
=
=
k i=1 (k − 1) . . . (k − r)
r+1
0
0
Ferreira and de Haan/On the block maxima method
11
Then,
√
(r + 1)Mr − bm
(r + 1)γ Γ(1 − γ) − 1
k
−
am
γ
!
R1
Z 1
(r)
√
(r + 1) 0 X⌈ks⌉,k Jk (s) ds − bm
(− log s)−γ − 1 r
k
=
− (r + 1)
s ds
am
γ
0
Z 1
√
X⌈ks⌉,k − bm
(− log s)−γ − 1
(r)
Jk (s) ds
−
k(r + 1)
=
a
γ
m
0
Z 1
√
(− log s)−γ − 1 r
(r)
s − Jk (s) ds
− k(r + 1)
γ
0
Z 1/(k+1) √
X⌈ks⌉,k − bm
(− log s)−γ − 1
(r)
=
k(r + 1)
Jk (s) ds
−
am
γ
0
Z k/(k+1) √
X⌈ks⌉,k − bm
(− log s)−γ − 1
(r)
Jk (s) ds
−
+ k(r + 1)
a
γ
m
1/(k+1)
Z 1
√
X⌈ks⌉,k − bm
(− log s)−γ − 1
(r)
+ k(r + 1)
Jk (s) ds
−
am
γ
k/(k+1)
Z 1
√
(− log s)−γ − 1 r
(r)
− k(r + 1)
s − Jk (s) ds
γ
0
= I1 + I2 + I3 + I4.
√
(r)
For I4: since sr − Jk (s) = O(1/k) uniformly in s, I4= O(1/ k).
For I1, note that
Z 1/(k+1) √
X⌈ks⌉,k − bm r
s ds = oP (1).
(11)
k
am
0
This follows since, the left-hand side of (11) equals, in distribution,
√
k
V (mZ1,k ) − V (m)
(k + 1)r+1
am
which, by Lemma 3.1.1, Lemma 3.2 and the fact that m/ log k → ∞, is bounded
(below and above) by
( γ
)
√
Z1,k − 1
k
γ+ρ+δ
γ+ρ−δ
.
, Z1,k
+ A(m)Hγ,ρ (Z1,k ) ± εA(m) max Z1,k
(k + 1)r+1
γ
√
ξ
This is easily seen to converge to zero in probability, since Z1,k
/ k =
√
√
{(log k) Z1,k }ξ log−ξ k/ k →P 0 for all real ξ and kA(m) → λ. Hence, I1=
oP (1).
Next we show that
Z 1
√ X⌈ks⌉,k − bm (r)
Jk (s) ds = oP (1).
(12)
k
am
k/(k+1)
Ferreira and de Haan/On the block maxima method
12
(r)
The left-hand side equals, in distribution (since Jk (s) ≡ 1 for s ∈ (k(k +
1)−1 , 1))
√ V (mZk,k ) − V (m)
k
1−
k
.
k+1
am
Lemma 3.1 yields
γ
n
o
Zk,k
−1
V (mZk,k ) − V (m)
γ+ρ+δ
=
+ A(m) Hγ,ρ (Zk,k ) ± εZk,k
am
γ
γ
which is (since Zk,k
/k γ converges to a positive random variable) of the order
√
OP (k γ ). Hence the integral is of order (k + 1)−1 kk γ which tends to zero since
γ < 1/2.
Finally, I2 has the same asymptotic behaviour as
(r + 1)
Z
k/(k+1)
1/(k+1)
√
k
X⌈ks⌉,k − bm
(− log s)−γ − 1
−
am
γ
which, by Proposition 3.1 tends to
Z
Z 1
sr−1 (− log s)−1−γ B(s) ds + λ(r + 1)
(r + 1)
0
1
Hγ,ρ
0
sr ds
1
− log s
sr ds.
For the evaluation of the latter integral note that for ξ < 1,
Z ∞
Z 1
v −ξ e−v dv = (r + 1)ξ−1 Γ(1 − ξ).
sr (− log s)−ξ ds = (r + 1)ξ−1
(r + 1)
0
0
Moreover, note that
Z 1
(r + 1)ξ Γ(1 − ξ) − 1
(− log s)−ξ − 1
ds =
,
sr
(r + 1)
ξ
ξ
0
ξ<1
(Dr (0) = log(r + 1) − Γ′ (1) as defined by continuity), and
Z 1
1
Hγ,ρ
(r + 1)
sr ds =
− log s
0
 1
(Dr (γ + ρ) − Dr (γ)) ,


 ρ



 D′ (γ) =

 r γ
Dr (γ)
(r+1)
′
=
−Γ
(1
−
γ)
+
log(r
+
1)Γ(1
−
γ)
−
γ
(r+1)γ ,







D′ (0) =

 1r 3
′′
′
2 log (r + 1) + Γ (1) − 3 log(r + 1)Γ (1) ,
ρ 6= 0,
γ 6= 0, ρ = 0,
γ = 0, ρ = 0.
Ferreira and de Haan/On the block maxima method
13
Proof of Theorem 2.2. From Theorem 2.1 we obtain,
√
2γ − 1
2M1 − M0
−
Γ(1 − γ) →d Q1 − Q0
k
am
γ
√
3γ − 1
3M2 − M0
−
Γ(1 − γ) →d Q2 − Q0
k
am
γ
hence, by Cramér’s delta method,
γ̂k,m
√
√
− 1 3γ − 1
3
3M2 − M0
3γ − 1
k
= k
− γ
−
2M1 − M0
2 −1
2γ̂k,m − 1 2γ − 1
γ
1
3
−
1
γ
γ
d
→
(Q2 − Q0 ) − γ
(Q1 − Q0 ) .
Γ(1 − γ) 2γ − 1 3γ − 1
2 −1
It follows that γ̂k,m →P γ and hence
γ̂k,m
√ rγ̂k,m −γ − 1
√
r
−1
k
k
−
1
=
rγ − 1
1 − r−γ
has the same limit distribution as
√
log r
,
k (γ̂k,m − γ)
1 − r−γ
r = 2, 3.
It follows that
γ̂k,m
√
− 1 3γ − 1
3
k
−
2γ̂k,m − 1 2γ − 1
γ̂k,m
γ̂k,m
√
3γ − 1 √
3
−1
2
−1
k
=
−1 − k
−1
2γ − 1
3γ − 1
2γ − 1
has the same limit distribution as
3γ − 1 √
k (γ̂k,m − γ)
2γ − 1
log 2
log 3
−
1 − 3−γ
1 − 2−γ
and, consequently,
√
k (γ̂k,m − γ)
−1 log 3
log 2
γ
γ
1
−
Q
)
−
−
Q
)
.
−
(Q
(Q
→d
2
0
1
0
Γ(1 − γ) 1 − 3−γ
1 − 2−γ
3γ − 1
2γ − 1
For the asymptotic distribution of âk,m we write,
√
âk,m
γ̂k,m
− 1 = γ̂k,m
am
(2
− 1) Γ (1 − γ̂k,m )
γ
√
√
2M1 − M0
2 −1
2γ − 1
2γ̂k,m − 1
k
−
Γ(1 − γ) + k
Γ(1 − γ) −
Γ (1 − γ̂k,m )
,
am
γ
γ
γ̂k,m
k
Ferreira and de Haan/On the block maxima method
14
and the statement follows e.g. by Cramér’s delta method.
For the asymptotic distribution of b̂k,m we write,
!
√
b̂k,m − bm
M 0 − bm
Γ(1 − γ) − 1
k
= k
−
am
am
γ
√
âk,m
Γ(1 − γ̂k,m ) − 1 Γ(1 − γ) − 1
âk,m
Γ(1 − γ) − 1 √
−
k
k
+
−
−1
am
γ̂k,m
γ
γ
am
√
and the statement follows e.g. by Cramér’s delta method.
Proof of Theorem 2.3. The proof follows the line of the comparable result for
the peaks-over-threshold method (see e.g. de Haan and Ferreira 2006, Chapter
4.3)
√
√
!
γ̂
k (x̂k,m − xn )
k
cnk,m − 1
1
b̂k,m + âk,m
=
−V
am qγ (cn )
am qγ (cn )
γ̂k,m
− log(1 − pn )
!
√
√
γ̂
k b̂k,m − bm
k
cnk,m − 1 cγn − 1
âk,m
=
+
−
qγ (cn )
am
am qγ (cn )
γ̂
γ


√
m
k  V −m log(1−pn ) − V (m) cγn − 1 
âk,m
cγn − 1 √
k
−
−
−1 .
+
qγ (cn )
am
γ
γqγ (cn )
am
Similarly as on pages 135–137 of de Haan and Ferreira (2006) this converges in
distribution to
γ−
.
∆ + (γ− )2 B − γ− Λ − λ
γ− + ρ
Acknowledgements. Research partially supported by FCT Project PTDC
/MAT /112770 /2009 and FCT- PEst-OE/MAT/UI0006/2011.
We thank Holger Drees for a useful suggestion.
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15
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