Maximum likelihood estimation for the Fréchet
distribution based on block maxima extracted
from a time series
AXEL BÜCHER1 and JOHAN SEGERS2

arXiv:1511.07613v2 [] 16 Sep 2016

1

Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstr. 150, 44780 Bochum, Germany.
E-mail: axel.buecher@ruhr-uni-bochum.de
2

Université catholique de Louvain, Institut de Statistique, Biostatistique et Sciences Actuarielles, Voie
du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium. E-mail: johan.segers@uclouvain.be
The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution
to a sample of block maxima extracted from an observed stretch of a time series. The method is usually
validated under two simplifying assumptions: the block maxima should be distributed exactly according
to an extreme-value distribution and the sample of block maxima should be independent. Both assumptions are only approximately true. The present paper validates that the simplifying assumptions can in
fact be safely made.
For general triangular arrays of block maxima attracted to the Fréchet distribution, consistency
and asymptotic normality is established for the maximum likelihood estimator of the parameters of
the limiting Fréchet distribution. The results are specialized to the common setting of block maxima
extracted from a strictly stationary time series. The case where the underlying random variables are
independent and identically distributed is further worked out in detail. The results are illustrated by
theoretical examples and Monte Carlo simulations.
Keywords: block maxima method, maximum likelihood estimation, asymptotic normality, heavy tails,
triangular arrays, stationary time series.

1. Introduction
For the analysis of extreme values, two fundamental approaches can be distinguished. First, the
peaks-over-threshold method consists of extracting those values from the observation period which
exceed a high threshold. To model such threshold excesses, asymptotic theory suggests the use of
the Generalized Pareto distribution (Pickands, 1975). Second, the block maxima method consists
of dividing the observation period into a sequence of non-overlapping intervals and restricting
attention to the largest observation in each time interval. Thanks to the extremal types theorem,
the probability distribution of such block maxima is approximately Generalized Extreme-Value
(GEV), popularized by Gumbel (1958). The block maxima method is particularly common in
environmental applications, since appropriate choices of the block size yield a simple but effective
way to deal with seasonal patterns.
For both methods, honest theoretical justifications must take into account two distinct features. First, the postulated models for either threshold excesses or block maxima arise from
asymptotic theory and are not necessarily accurate at sub-asymptotic thresholds or at finite
block lengths. Second, if the underlying data exhibit serial dependence, then the same will likely
be true for the extreme values extracted from those data.
How to deal with both issues is well-understood for the peaks-over-threshold method. The
model approximation can be justified under a second-order condition (see, e.g., de Haan and Ferreira,
1

2

A. Bücher and J. Segers

2006 for a vast variety of applications), while serial dependence is taken care of in Hsing (1991);
Drees (2000) or Rootzén (2009), among others. Excesses over large thresholds often occur in clusters, and such serial dependence usually has an impact on the asymptotic variances of estimators
based on these threshold excesses.
Surprisingly, perhaps, is that for the block maxima method, no comparable analysis has yet
been done. With the exception of some recent articles, which we will discuss in the next paragraph,
the commonly used assumption is that the block maxima constitute an independent random
sample from a GEV distribution. The heuristic justification for assuming independence over
time, even for block maxima extracted from time series data, is that for large block sizes, the
occurrence times of the consecutive block maxima are likely to be well separated.
A more accurate framework is that of a triangular array of block maxima extracted from
a sequence of random variables, the block size growing with the sample size. While Dombry
(2015) shows consistency of the maximum likelihood estimator (Prescott and Walden, 1980) for
the parameters of the GEV distribution, Ferreira and de Haan (2015) show both consistency and
asymptotic normality of the probability weighted moment estimators (Hosking, Wallis and Wood,
1985). In both papers, however, the random variables from which the block maxima are extracted
are supposed to be independent and identically distributed. In many situations, this assumption
is clearly violated. To the best of our knowledge, Bücher and Segers (2014) is the only reference
treating both the approximation error and the time series character, providing large-sample theory of nonparametric estimators of extreme-value copulas based on samples of componentwise
block maxima extracted out of multivariate stationary time series.
The aim of the paper is to show the consistency and asymptotic normality of the maximum
likelihood estimator for more general sampling schemes, including the common situation of extracting block maxima from an underlying stationary time series. For technical reasons explained
below, we restrict attention to the heavy-tailed case. The block maxima paradigm then suggests
to use the two-parametric Fréchet distribution as a model for a sample of block maxima extracted
from that time series.
The first (quite general) main result, Theorem 2.5, is that for triangular arrays of random variables whose empirical measures, upon rescaling, converge in an appropriate sense to a Fréchet
distribution, the maximum likelihood estimator for the Fréchet parameters based on those variables is consistent and asymptotically normal. The theorem can be applied to the common set-up
discussed above of block maxima extracted from an underlying time series, and the second main
result, Theorem 3.6, shows that, in this case, the asymptotic variance matrix is the inverse of the
Fisher information of the Fréchet family: asymptotically, it is as if the data were an independent
random sample from the Fréchet attractor. In this sense, our theorem confirms the soundness of
the common simplifying assumption that block maxima can be treated as if they were serially
independent. Interestingly enough, the result allows for time series of which the strong mixing
coefficients are not summable, allowing for some long range dependence scenarios.
Restricting attention to the heavy-tailed case is done because of the non-standard nature
of the three-parameter GEV distribution. The issue is that the support of a GEV distribution
depends on its parameters. Even for the maximum likelihood estimator based on an independent
random sample from a GEV distribution, asymptotic normality has not yet been established. The
article usually cited in this context is Smith (1985), although no formal result is stated therein.
Even the differentiability in quadratic mean of the three-parameter GEV is still to be proven;
Marohn (1994) only shows differentiability in quadratic mean for the one-parameter GEV family
(shape parameter only) at the Gumbel distribution. We feel that solving all issues simultaneously
(irregularity of the GEV model, finite block size approximation error and serial dependence) is
a far too ample program for one paper. For that reason, we focus on the analytically simpler
Fréchet family, while thoroughly treating the triangular nature of the array of block maxima

Maximum likelihood estimation for the Fréchet distribution

3

and the issue of serial dependence within the underlying time series. In a companion paper
(Bücher and Segers, 2016), we consider the maximum likelihood estimator in the general GEVmodel based on independent and identically distributed random variables sampled directly from
the GEV distribution. The main focus of that paper is devoted to resolving the considerable
technical issues arising from the dependence of the GEV support on its parameters.
We will build up the theory in three stages. First, we consider general triangular arrays of
observations that asymptotically follow a Fréchet distribution in Section 2. Second, we apply
the theory to the set-up of block maxima extracted from a strictly stationary time series in
Section 3. Third, we further specialize the results to the special case of block maxima formed
from independent and identically distributed random variables in Section 4. This section can
hence be regarded as a continuation of Dombry (2015) by reinforcing consistency to asymptotic
normality, albeit for the Fréchet domain of attraction only. We work out an example and present
finite-sample results from a simulation study in Section 5. The main proofs are deferred to
Appendix A, while some auxiliary results concerning the Fréchet distribution are mentioned in
Appendix B. The proofs of the less central results are postponed to a supplement.

2. Triangular arrays of block maxima
In this section, we summarize results concerning the maximum likelihood estimator for the parameters of the Fréchet distribution: given a sample of observations which are not all tied, the
Fréchet likelihood admits a unique maximum (Subsection 2.1). If the observations are based on a
triangular array which is approximately Fréchet distributed in the sense that certain functionals
admit a weak law of large numbers or a central limit theorem, the maximum likelihood estimator
is consistent or asymptotically normal, respectively (Subsections 2.2 and 2.3). Proofs are given
in Subsection A.1.

2.1. Existence and uniqueness
Let Pθ denote the two-parameter Fréchet distribution on (0, ∞) with parameter θ = (α, σ) ∈
(0, ∞)2 = Θ, defined through its cumulative distribution function
Gθ (x) = exp{−(x/σ)−α },

x > 0.

Its probability density function is equal to
α
pθ (x) = exp{−(x/σ)−α } (x/σ)−α−1 ,
σ
with log-likelihood function

x > 0,

ℓθ (x) = log(α/σ) − (x/σ)−α − (α + 1) log(x/σ),

x > 0,

and score functions ℓ̇θ = (ℓ̇θ,1 , ℓ̇θ,2 )T , with

ℓ̇θ,1 (x) = ∂α ℓθ (x) = α−1 + (x/σ)−α − 1 log(x/σ),

ℓ̇θ,2 (x) = ∂σ ℓθ (x) = 1 − (x/σ)−α α/σ.

(2.1)
(2.2)

Let x = (x1 , . . . , xk ) ∈ (0, ∞)k be a sample vector to which the Fréchet distribution is to be
fitted. Consider the log-likelihood function
L(θ | x) =

k
X
i=1

ℓθ (xi ).

(2.3)

4

A. Bücher and J. Segers

Further, define
1
Ψk (α | x) = +
α
σ(α | x) =

1
k

Pk

−α
log(xi )
i=1 xi
P
k
−α
1
i=1 xi
k
!−1/α

k
1 X −α
x
k i=1 i

−

k
1X
log(xi ),
k i=1

.

(2.4)

(2.5)

Lemma 2.1. (Existence and uniqueness) If the scalars x1 , . . . , xk ∈ (0, ∞) are not all equal
(k ≥ 2), then there exists a unique maximizer

θ̂(x) = α̂(x), σ̂(x) = arg max L(θ | x).
θ∈Θ

We have σ̂(x) = σ(α̂(x) | x) while α̂(x) is the unique zero of the strictly decreasing function
α 7→ Ψk (α | x):

(2.6)
Ψk α̂(x) | x = 0.

It is easily verified that the estimating equation for α is scale invariant: for any c ∈ (0, ∞),
we have Ψk (α | cx) = Ψk (α | x). As a consequence, the maximum likelihood estimator for the
shape parameter is scale invariant:
α̂(cx) = α̂(x).
Moreover, the estimator for σ is a scale parameter in the sense that
σ̂(cx) = σ(α̂(cx) | cx) = c σ(α̂(x) | x) = c σ̂(x).

Until now, the maximum likelihood estimator is defined only in case not all xi values are
identical. For definiteness, if x1 = . . . = xk , define α̂(x) = ∞ and σ̂(x) = min(x1 , . . . , xk ) = x1 .

2.2. Consistency
We derive a general condition under which the maximum likelihood estimator for the parameters
of the Fréchet distribution is consistent. The central result, Theorem 2.3 below, shows that, apart
from a not-all-tied condition, the only thing that is required for consistency is a weak law of large
numbers for the functions appearing in the estimating equation (2.6) for the shape parameter.
Suppose that for each positive integer n, we are given a random vector Xn = (Xn,1 , . . . , Xn,kn )
taking values in (0, ∞)kn , where kn ≥ 2 is a positive integer sequence such that kn → ∞
as n → ∞. One may think of Xn,i as being (approximately) Fréchet distributed with shape
parameter α0 > 0 and scale parameter σn > 0. This statement is made precise in Condition 2.2
below. On the event that the kn variables Xn,i are not all equal, Lemma 2.1 allows us to define
α̂n = α̂(Xn ),

(2.7)

the unique zero of the function 0 < α 7→ Ψkn (α | Xn ). Further, as in (2.5), put
σ̂n = σ(α̂n | Xn ) =

kn
1 X
X −α̂n
kn i=1 n,i

!−1/α̂n

.

(2.8)

Maximum likelihood estimation for the Fréchet distribution

5

For definiteness, put α̂n = ∞ and σ̂n = Xn,1 on the event {Xn,1 = . . . = Xn,kn }. Subsequently,
we will assume that this event is asymptotically negligible:
lim Pr(Xn,1 = . . . = Xn,kn ) = 0.

(2.9)

n→∞

We refer to (α̂n , σ̂n ) as the maximum likelihood estimator.
The fundamental condition guaranteeing consistency of the maximum likelihood estimator
concerns the asymptotic behavior of sample averages of f (Xn,i /σn ) for certain functions f . For
0 < α− < α+ < ∞, consider the function class
F1 (α− , α+ ) = {x 7→ log x} ∪ {x 7→ x−α : α− < α < α+ } ∪ {x 7→ x−α log x : α− < α < α+ },
(2.10)
all functions being from (0, ∞) into R. Let the arrow ‘ ’ denote weak convergence.
Condition 2.2. There exist 0 < α− < α0 < α+ < ∞ and a positive sequence (σn )n∈N such
that, for all f ∈ F1 (α− , α+ ),
kn
1 X
f (Xn,i /σn )
kn i=1

Z

∞

f (x) pα0 ,1 (x) dx,

0

n → ∞.

(2.11)

Theorem 2.3. (Consistency) Let Xn = (Xn,1 , . . . , Xn,kn ) be a sequence of random vectors in
(0, ∞)kn , where kn → ∞. Assume that Equation (2.9) and Condition 2.2 hold. On the complement of the event {Xn,1 = . . . = Xn,kn }, the random vector (α̂n , σ̂n ) is the unique maximizer of
the log-likelihood (α, σ) 7→ L(α, σ | Xn,1 , . . . , Xn,kn ). Moreover, the maximum likelihood estimator
is consistent in the sense that
(α̂n , σ̂n /σn )

(α0 , 1),

n → ∞.

2.3. Asymptotic distribution
We formulate a general condition under which the estimation error of the maximum likelihood
estimator for the Fréchet parameter vector converges weakly. The central result is Theorem 2.5
below.
For 0 < α− < α+ < ∞, recall the function class F1 (α− , α+ ) in (2.10) and define another one:
F2 (α− , α+ ) = F1 (α− , α+ ) ∪ {x 7→ x−α (log x)2 : α− < α < α+ }.

(2.12)

Furthermore, fix α0 > 0 and consider the following triple of real-valued functions on (0, ∞):
H = {f1 , f2 , f3 } = {x 7→ x−α0 log(x), x 7→ x−α0 , x 7→ log x}.

(2.13)

The following condition strengthens Condition 2.2.
Condition 2.4. There exist α0 ∈ (0, ∞) and a positive sequence (σn )n∈N such that the following
two statements hold:
(i) There exist 0 < α− < α0 < α+ < ∞ such that Equation (2.11) holds for all f ∈ F2 (α− , α+ ).
(ii) There exists a sequence 0 < vn → ∞ and a random vector Y = (Y1 , Y2 , Y3 )T such that,
denoting
!
Z ∞
kn
1 X
f (Xn,i /σn ) −
f (x) pα0 ,1 (x) dx ,
(2.14)
Gn f = vn
kn i=1
0

6

A. Bücher and J. Segers
we have, for fj as in (2.13),
(Gn f1 , Gn f2 , Gn f3 )T

n → ∞.

Y,

(2.15)

Let Γ be the Euler gamma function and let γ = −Γ′ (1) = 0.5772 . . . be the Euler–Mascheroni
constant. Recall Γ′′ (2) = (1 − γ)2 + π 2 /6 − 1. Define the matrix
 2

6
α0
α0 (1 − γ)
−α20
,
α0 ∈ (0, ∞).
(2.16)
M (α0 ) = 2
γ − 1 −(Γ′′ (2) + 1)/α0 1 − γ
π
Theorem 2.5. (Asymptotic distribution) Let Xn = (Xn,1 , . . . , Xn,kn ) be a sequence of random vectors in (0, ∞)kn , where kn → ∞. Assume that Equation (2.9) and Condition 2.4 hold.
As n → ∞, the maximum likelihood estimator (α̂n , σ̂n ) satisfies




Gn x−α0 log(x)
vn (α̂n − α0 )
 + op (1)
= M (α0 )  Gn x−α0
M (α0 )Y ,
(2.17)
vn (σ̂n /σn − 1)
Gn log(x)
where Y = (Y1 , Y2 , Y3 )T and M (α0 ) are given in Equations (2.15) and (2.16), respectively.

For block√maxima extracted from a strongly mixing stationary time series, Condition 2.4
with vn = kn , where kn denotes the number of blocks, will be derived from the Lindeberg
central limit theorem. In that case, the distribution of Y is trivariate Gaussian with some mean
vector µY (possibly different from 0, see Theorem 3.6 below for details) and covariance matrix


1 − 4γ + γ 2 + π 2 /3 α0 (γ − 2) π 2 /6 − γ
1 
α0 (γ − 2)
α20
−α0  .
(2.18)
ΣY = 2
α0
2
π /6 − γ
−α
π 2 /6
0

According to Lemma B.2 below, the right-hand side in (2.18) coincides with the covariance
T
matrix of the random vector X −α0 log(X), X −α0 , log(X) , where X is Fréchet distributed
with parameter (α0 , 1). From Lemma B.3, recall the inverse of the Fisher information matrix of
the Fréchet family at (α, σ) = (α0 , 1):


6
α20
(γ − 1)
−1
I(α
=
.
(2.19)
−2
0 ,1)
π 2 (γ − 1) α0 {(1 − γ)2 + π 2 /6}
Addendum 2.6. If Y is normally distributed with covariance matrix ΣY as in (2.18), then the
limit M (α0 )Y in Theorem 2.5 is also normally distributed and its covariance matrix is equal to
−1
the inverse of the Fisher information matrix of the Fréchet family, M (α0 ) ΣY M (α0 )T = I(α
.
0 ,1)

3. Block maxima extracted from a stationary time series
Let (ξt )t∈Z be a strictly stationary time series, that is, for any k ∈ N and τ, t1 , . . . , tk ∈ Z, the
distribution of (ξt1 +τ , . . . , ξtk +τ ) is the same as the distribution of (ξt1 , . . . , ξtk ). For positive
integer i and r, consider the block maximum
Mr,i = max(ξ(i−1)r+1 , . . . , ξir ).
Abbreviate Mr,1 = Mr . The classical block maxima method consists of choosing a sufficiently
large block size r and fitting an extreme-value distribution to the sample of block maxima

Maximum likelihood estimation for the Fréchet distribution

7

Mr,1 , . . . , Mr,k . The likelihood is constructed under the simplifying assumption that the block
maxima are independent. The present section shows consistency and asymptotic normality of
this method in an appropriate asymptotic framework.
For the block maxima distribution to approach its extreme-value limit, the block sizes must
increase to infinity. Moreover, consistency can only be achieved when the number of blocks grows
to infinity too. Hence, we consider a positive integer sequence rn , to be thought of as a sequence
of block sizes. The number of disjoint blocks of size rn that fit into a sample of size n is equal to
kn = ⌊n/rn ⌋, where ⌊x⌋ denotes the integer part of a real number x. Assume that both rn → ∞
and kn → ∞ as n → ∞.
The theory will be based on an application of Theorem 2.5 to the sample of left-truncated
block maxima Xn,i = Mrn ,i ∨ c (i = 1, . . . , kn ), for some positive constant c specified below. The
estimators α̂n and σ̂n are thus the ones in (2.7) and (2.8), respectively. The reason for the lefttruncation is that otherwise, some of the block maxima could be zero or negative. Asymptotically,
such left-truncation does not matter, since all maxima will simultaneously diverge to infinity in
probability (Condition 3.2 below).
In Section 4 below, we will specialize things further to the case where the random variables
ξt are independent. In particular, we will simplify the list of conditions given in this section.
The basic assumption is that the distribution of rescaled block maxima is asymptotically
Fréchet. The sequence of scaling constants should possess a minimal degree of regularity. The
assumption is satisfied in case the stationary distribution of the series is in the Fréchet domain
of attraction and the series possesses a positive extremal index; see Remark 3.7 below.
Condition 3.1. (Domain of attraction) The time series (ξt )t∈Z is strictly stationary and there
exists a sequence (σn )n∈N of positive numbers with σn → ∞ and a positive number α0 such that
Mn /σn

Fréchet(α0 , 1),

n → ∞.

(3.1)

Moreover, σmn /σn → 1 for any integer sequence (mn )n∈N such that mn /n → 1 as n → ∞.
The domain-of-attraction condition implies that, for every scalar c, we have Pr[Mn ≤ c] =
Pr[Mn /σn ≤ c/σn ] → 0 as n → ∞. In words, the block maxima become unboundedly large as
the sample size grows to infinity. Still, out of a sample of kn block maxima, the smallest of the
maxima might still be small, especially when the number of blocks is large, or, equivalently, the
block sizes are not large enough. The following condition prevents this from happening.
Condition 3.2. (All block maxima diverge) For every c ∈ (0, ∞), we have
lim Pr[min(Mrn ,1 , . . . , Mrn ,kn ) ≤ c] = 0.

n→∞

To control the serial dependence within the time series, we require that the Rosenblatt mixing
coefficients decay sufficiently fast: for positive integer ℓ, put

α(ℓ) = sup |Pr(A ∩ B) − Pr(A) Pr(B)| : A ∈ σ(ξt : t ≤ 0), B ∈ σ(ξt : t ≥ ℓ) ,

where σ( · ) denotes the σ-field generated by its argument.

Condition 3.3. (α-Mixing with rate) We have limℓ→∞ α(ℓ) = 0. Moreover, there exists ω > 0
such that
kn1+ω α(rn ) → 0,
n → ∞.
(3.2)

8

A. Bücher and J. Segers

Condition 3.3 can be interpreted as requiring the block sizes rn to be sufficiently large. For
instance, if α(ℓ) = O(ℓ−a ) for some a > 0, then (3.2) is satisfied as soon as rn is of larger
order than n(1+ε)/(1+a) for some 0 < ε < a; in that case, one may choose ω = ε. Note that the
exponent a is allowed to be smaller than one, in which case the sequence of mixing coefficients
is not summable.
In order to be able to integrate (3.1) to the limit, we require an asymptotic bound on certain
moments of the block maxima; more precisely, on negative power moments in the left tail and
on logarithmic moments in the right tail.
Condition 3.4. (Moments) There exists some ν > 2/ω with ω from Condition 3.3 such that


lim sup E gν,α0 (Mn ∨ 1)/σn < ∞,
(3.3)
n→∞

where gν,α0 (x) = {x−α0 1(x ≤ e) + log(x)1(x > e)}2+ν .

An elementary argument shows that if Condition 3.4 holds, then Mn ∨ 1 in the lim sup may
be replaced
R ∞ by Mn ∨ c, for arbitrary c > 0. Moreover, note that the limiting Fréchet distribution
satisfies 0 xβ pα0 ,1 (x) dx < ∞ if and only if β is less than α0 . In some scenarios, e.g., for the
iid case considered in Section 4 or for the moving maximum process considered in Section 5.1, it
can be shown that the following sufficient condition for Condition 3.4 is true:

β
lim sup E (Mn ∨ c)/σn
<∞
(3.4)
n→∞

for all c > 0 and all β ∈ (−∞, α0 ). In that case, Condition 3.4 is easily satisfied for any ν > 0.
By Condition 3.2 and Lemma A.5, the probability that all block maxima Mrn ,1 , . . . , Mrn ,kn
are larger than some positive constant c and that they are not all equal tends to unity. On
this event, we can study the maximum likelihood estimators (α̂n , σ̂n ) for the parameters of the
Fréchet distribution based on the sample of block maxima.
Fix c ∈ (0, ∞) and put
Xn,i = Mrn ,i ∨ c.

Let Gn be the empirical process associated to Xn,1 /σrn , . . . , Xn,kn /σrn as in (2.14) with vn =
√
kn . The empirical process is not necessarily centered, which is why we need a handle on its
expectation.
Condition 3.5. (Bias) There exists c ∈ (0, ∞) such that for every function f in H defined in
(2.13), the following limit exists:

Z ∞
p  

lim
f (x) pα0 ,1 (x) dx = B(f ).
(3.5)
kn E f (Mrn ∨ c)/σrn −
n→∞

0

Theorem 3.6. Suppose that Conditions 3.1 up to 3.5 are satisfied and fix c as in Condition 3.5.
Then, with probability tending to one, there exists a unique maximizer (α̂n , σ̂n ) of the Fréchet
log-likelihood (2.3) based on the block maxima Mrn ,1 , . . . , Mrn ,kn , and we have, as n → ∞,


 √

Gn x−α0 log(x)

 + op (1)
√ kn (α̂n − α0 )
= M (α0 )  Gn x−α0
N2 M (α0 ) B, Iα−1
.
,1
0
kn (σ̂n /σrn − 1)
Gn log(x)

Here, M (α0 ) and Iα−1
are defined in Equations (2.16) and (2.19), respectively, while B =
0 ,1
(B(f1 ), B(f2 ), B(f3 ))T , where B(f ) is the limit in (3.5) and where f1 , f2 , f3 are defined in (2.13).

Maximum likelihood estimation for the Fréchet distribution

9

The proof of Theorem 3.6 is given in Subsection A.2. The conditions imposed in Theorem 3.6
are rather high-level. In the setting of a sequence of independent and identically distributed
random variables, they can be brought down to analytical conditions on the tail of the stationary distribution function (Theorem 4.2). Moreover, all conditions will be worked out in a
moving maximum model in Section 5.1. Still, we admit that for more common time series models, such as linear time series with heavy-tailed innovations or solutions to stochastic recurrence
equations, checking the conditions in Theorem 3.6 may not be an easy matter. Especially the
bias Condition 3.5, which requires quite detailed knowledge on the distribution of the sample
maximum, may be hard to verify. Even in the i.i.d. case, where the distribution of the sample
maximum is known explicitly, checking Condition 3.5 occupies more than three pages in the
proof of Theorem 4.2 below.
Interestingly, the asymptotic covariance matrix
√ in Theorem 3.6 is unaffected by√serial dependence and the asymptotic standard deviation of kn (α̂n − α0 ) is always equal to ( 6/π) × α0 ≈
0.7797 × α0. The reason for this invariance is that even for time series, maxima over large disjoint
blocks are asymptotically independent because of the strong mixing condition.
Remark 3.7. (Domain-of-attraction condition for positive extremal index) Let F be
the cumulative distribution function of ξ1 . Assume that there exist 0 < an → ∞ and α0 ∈ (0, ∞)
such that
x ∈ (0, ∞).
lim F n (an x) = exp(−x−α0 ),
n→∞

Moreover, assume that the sequence (ξt )t∈Z has extremal index ϑ ∈ (0, 1] (Leadbetter, 1983): If
un → ∞ is such that F n (un ) converges, then
Pr(Mn ≤ un ) = F nϑ (un ) + o(1),
1/α0

Note that we assume that ϑ > 0. Putting σn = ϑ
for every x ∈ (0, ∞), we have

n → ∞.

an we obtain that Condition 3.1 is satisfied:


Pr(Mn /σn ≤ x) = F nϑ (σn x) + o(1) → exp −ϑ(ϑ1/α0 x)−α0 = exp(−x−α0 ),

n → ∞.

4. Block maxima extracted from an iid sample
We specialize Theorem 3.6 to the case where the random variables ξ1 , ξ2 , . . . are independent and
identically distributed with common distribution function F . In this setting, fitting extremevalue distributions to block maxima is also considered in Dombry (2015) (consistency of the
maximum likelihood estimator in the GEV-family with γ > −1) and Ferreira and de Haan (2015)
(asymptotic normality of the probability weighted moment estimator in the GEV-family with
γ < 1/2). Assume that F is in the maximum domain of attraction of the Fréchet distribution
with shape parameter α0 ∈ (0, ∞): there exists a positive scalar sequence (an )n∈N such that, for
every x ∈ (0, ∞),
−α0
n → ∞.
(4.1)
F n (an x) → e−x ,

Because of serial independence, the conditions in Theorem 3.6 can be simplified considerably.
In addition, the mean vector of the asymptotic bivariate normal distribution of the maximum
likelihood estimator can be made explicit. Required is a second-order reinforcement of (4.1) in
conjunction with a growth restriction on the number of blocks.
Equation (4.1) is equivalent to regular variation of − log F at infinity with index −α0 (Gnedenko,
1943): we have F (x) < 1 for all x ∈ R and
− log F (ux)
= x−α0 ,
u→∞ − log F (u)
lim

x ∈ (0, ∞).

(4.2)

10

A. Bücher and J. Segers

The scaling constants in (4.1) may be chosen as any sequence (an )n∈N that satisfies
lim n {− log F (an )} = 1.

n→∞

(4.3)

Being constructed from the asymptotic inverse of a regularly varying function with non-zero
index, the sequence (an )n∈N is itself regularly varying at infinity with index 1/α0 .
The following condition reinforces (4.2) and thus (4.1) from regular variation to second-order
regular variation (Bingham, Goldie and Teugels, 1987, Section 3.6). With − log F replaced by
1 − F , it appears for instance in de Haan and Ferreira (2006, Theorem 3.2.5) in the context of
the asymptotic distribution of the Hill estimator. For τ ∈ R, define hτ : (0, ∞) → R by
 τ
Z x
 x − 1 , if τ 6= 0,
τ −1
τ
y
dy =
hτ (x) =
(4.4)
log(x), if τ = 0.
1
Condition 4.1. (Second-Order Condition) There exists α0 ∈ (0, ∞), ρ ∈ (−∞, 0], and a real
function A on (0, ∞) of constant, non-zero sign such that limu→∞ A(u) = 0 and such that, for
all x ∈ (0, ∞),


− log F (ux)
1
(4.5)
lim
− x−α0 = x−α0 hρ (x).
u→∞ A(u)
− log F (u)

The function A can be regarded as capturing the speed of convergence in (4.2). The form of
the limit function in (4.5) may seem unnecessarily specific, but actually, it is not, as explained
in Remark 4.3 below.
Let ψ = Γ′ /Γ denote the digamma function and recall the Euler–Mascheroni constant γ =
−Γ′ (1) = 0.5772 . . .. To express the asymptotic bias of the maximum likelihood estimators, we
will employ the functions b1 and b2 defined by

(1 + x) Γ(x){γ + ψ(1 + x)}, if x > 0,
(4.6)
b1 (x) = π 2
 ,
if x = 0,
6
and


 π2
+ (1 + x) Γ(x){Γ′′ (2) + γ + (γ − 1) ψ(1 + x)}, if x > 0,
−
b2 (x) =
6x
0,
if x = 0.

(4.7)

See Figure 1 for the graphs of these two functions. For (α0 , ρ) ∈ (0, ∞) × (−∞, 0], define the bias
function


6
b1 (|ρ| /α0 )
B(α0 , ρ) = − 2
.
(4.8)
b2 (|ρ| /α0 )/α20
π
The proof of the following theorem is given in Section A.3.

Theorem 4.2. Let ξ1 , ξ2 , . . . be independent random variables with common distribution function
F satisfiying Condition 4.1. Let the block sizes rn be such that rn → ∞ and kn = ⌊n/rn ⌋ → ∞
as n → ∞ and assume that
p
(4.9)
kn A(arn ) = λ ∈ R.
lim
n→∞

Then, with probability tending to one, there exists a unique maximizer (α̂n , σ̂n ) of the Fréchet
log-likelihood (2.3) based on the block maxima Mrn ,1 , . . . , Mrn ,kn , and we have


p  α̂n − α0 
−1
kn
N2 λ B(α0 , ρ), I(α
,
n → ∞,
(4.10)
0 ,1)
σ̂n /arn − 1

11

6

Maximum likelihood estimation for the Fréchet distribution

2

3

4

5

b1
b2

0

1

π2 6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure 1. Graphs of the functions b1 and b2 in (4.6) and (4.7).
−1
where I(α
denotes the inverse of the Fisher information of the Fréchet family as in (2.19) and
0 ,1)
with B(α0 , ρ) as in (4.8).

We conclude this section with a series of remarks on the second-order Condition 4.1 and its
link to the block-size condition in (4.9) and the mean vector of the limiting distribution in (4.10).
Remark 4.3. (Second-order regular variation) Let F satisfy (4.2). For x > 0 sufficiently
large such that F (x) > 0, define L(x) by
− log F (x) = x−α0 L(x).

(4.11)

In view of (4.2), the function L is slowly varying at infinity, that is,
lim

u→∞

L(ux)
= 1,
L(u)

x ∈ (0, ∞).

A second-order refinement of this would be that there exist A : (0, ∞) → (0, ∞) and h : (0, ∞) →
R, the latter not identically zero, such that limu→∞ A(u) = 0 and


L(ux)
1
− 1 = h(x),
x ∈ (0, ∞).
lim
u→∞ A(u)
L(u)
Writing g(u) = A(u) L(u), Theorem B.2.1 in de Haan and Ferreira (2006) (see also Bingham, Goldie and Teugels,
1987, Section 3.6) implies that there exists ρ ∈ R such that g and thus A = g/L are regularly
varying at infinity with index ρ. Since A vanishes at infinity, necessarily ρ ≤ 0. Furthermore,
there exists κ ∈ R\{0} such that h(x) = κ hρ (x) for x ∈ (0, ∞), with hρ as in (4.4). Incorporating
the constant κ into the function A, we can assume without loss of generality that κ = 1 and
we arrive at Condition 4.1. The function A then possibly takes values in (−∞, 0) rather than in
(0, ∞).
Remark 4.4. (Asymptotic mean squared error) According to (4.9) and (4.10), the distribution of the estimation error α̂n − α0 is approximately equal to


rn 6 2
6
α
.
N −A(arn ) 2 b1 (|ρ| /α0 ),
π
n π2 0

12

A. Bücher and J. Segers

The asymptotic mean squared error is therefore equal to
AMSE(α̂n ) = ABias2 (α̂n ) + AVar(α̂n ) = |A(arn )|2

rn 6 2
36
b1 (|ρ| /α0 )2 +
α .
4
π
n π2 0

The choice of the block size rn (or, equivalently, the number of blocks kn ), thus involves a bias–
variance trade-off; see Section 5. Alternatively, if ρ and A(arn ) could be estimated, then one
could construct bias-reduced estimators, just as in the case of the Hill estimator (see, e.g., Peng,
1998, among others) or probability weighted moment estimators (Cai, de Haan and Zhou, 2013).
Remark 4.5. (On the number of blocks) A version of (4.9) is used in Ferreira and de Haan
(2015) to prove asymptotic normality of probability weighted moment estimators. Equation (4.9)
also implies the following limit relation, which is imposed in Dombry (2015) and which we will
be needing later on as well:
kn log(kn )
= 0.
(4.12)
lim
n→∞
n
Indeed, in view of Remark 4.3 and regular variation of (an )n∈N , the sequence (|A(ar )|)r∈N
is regularly varying at infinity. Potter’s theorem (Bingham, Goldie and Teugels, 1987, Theorem 1.5.6) then implies that there exists β > 0 such that r−β = o(|A(ar )|) as r → ∞. But then
√
√
1/2+β
/nβ = o(1) as n → ∞. We obtain
kn (rn )−β = kn o(|A(arn )|) = o(1) by (4.9) and thus kn
1+1/(2β)
that kn
/n = o(1), which implies (4.12).
Remark 4.6. (No asymptotic bias) If λ = 0 in (4.9), then the limiting normal distribution in
(4.10) is centered and the maximum likelihood estimator is said to be asymptotically unbiased. If
the index, ρ, of regular variation of the auxiliary function |A| is strictly negative (see Remark 4.3),
then a sufficient condition for λ = 0 to occur is that kn = O(nβ ) for some β < |ρ|/(α0 /2 + |ρ|).

5. Examples and finite-sample results
5.1. Verification of conditions in a moving maximum model
For many stationary time series models, the distribution of the sample maximum is a difficult
object to work with. This is true even for linear time series models, since the maximum operator is
non-linear. In such cases, checking the conditions of Section 3 may be a hard or even impossible
task. An exception occurs for moving maximum models, where the sample maximum can be
linked directly to maxima of the innovation sequence.
Let (Zt )t∈Z be a sequence of independent and identically distributed random variables with
common distribution function F in the domain of attraction of the Fréchet distribution with
shape parameter α0 > 0, that is, such that (4.1) is satisfied for some sequence an → ∞. Let
p ∈ N, p ≥ 2, be fixed and let b1 , . . . , bp be nonnegative constants, b1 6= 0 6= bp , such that
P
p
i=1 bi = 1. We consider the moving maximum process ξt of order p, defined by
ξt = max{b1 Zt , b2 Zt−1 , . . . , bp Zt−p+1 },

t ∈ Z.

A simple calculation (see also the proof of Lemma 5.1 for the stationary distribution of ξt ) shows
that the extremal index of (ξt )t∈Z is equal to
Pp
0 −1 α0
b(p) ,
θ = { i=1 bα
i }

where b(p) = maxpi=1 bi . Let σn = b(p) an . The proof of the following lemma is given in Section D
in the supplementary material.

Maximum likelihood estimation for the Fréchet distribution

13

Lemma 5.1. The stationary time series (ξt )t∈Z satisfies Conditions 3.1, 3.3 and 3.4. If additionally (4.12) is met, then Condition 3.2 is satisfied as well. Finally, if F satisfies the Second-Order
Condition 4.1, if (4.9) is met and if kn = o(n2/3 ) as n → ∞, then Condition 3.5 is also satisfied,
with B(f ) denoting the same limit as in the iid case, that is, B(f ) = β with β as in (A.23).
As a consequence, Theorem 3.6 may be applied and the asymptotic bias of the maximum
likelihood estimator is the same as specified in Theorem 4.2 for the case of independent and
identically distributed random variables.

5.2. Simulation results
We report on the results of a simulation study, highlighting some interesting features regarding
the finite-sample performance of the maximum likelihood estimator. Attention is restricted to
the estimation of the shape parameter, and particular emphasis is given to a comparison with
the common Hill estimator, which is based on the competing peaks-over-threshold method. Its
variance is of the order O(k −1 ), where k is the number of upper order statistics taken into
account for its calculation. The Hill estimator’s asymptotic variance is given by α20 , which is larger
than the asymptotic variance (6/π 2 ) × α20 of the block maxima maximum likelihood estimator.
Furthermore, numerical experiments (not shown) involving the probability weighted moment
estimator showed a variance that was higher, in all cases considered, than the one of the maximum
likelihood estimator.
We consider three time series models for (ξt )t∈Z : independent and identically distributed
random variables, the moving maximum process from Section 5.1, and the absolute values of a
GARCH(1,1) time series. In the first two models, three choices are considered for the distribution
function F of either the variables ξt in the first model and the innovations Zt in the second model:
absolute values of a Cauchy-distribution, the standard Pareto distribution and the Fréchet(1,1)
distribution itself. All three distribution functions are attracted to the Fréchet distribution with
α0 = 1. For the moving maximum process, we fix p = 4 and bj = j/10 for j ∈ {1, 2, 3, 4}. The
GARCH(1,1) model is based on standard normal innovations, that is, ξt = |Zt |, where Zt is the
stationary solution of the equations

Zt = εt σt ,
(5.1)
2
2
σt2 = λ0 + λ1 Zt−1
+ λ2 σt−1
,
with εt , t ∈ Z, independent standard normal random variables. The parameter vector (λ0 , λ1 , λ2 )
is set to either (0.5, 0.367, 0.367) or (0.5, 0.08, 0.91). By Mikosch and Stărică (2000), the stationary distribution associated to any of these two models is attracted to the Fréchet distribution
with shape parameter being (approximately) equal to α0 = 5.
We generate samples from all of the afore-mentioned models for a fixed sample size of n =
1 000. Based on N = 3 000 Monte Carlo repetitions, we obtain empirical estimates of the finite
sample bias, variance and mean squared error (MSE) of the competing estimators. The results
are summarized in Figure 2 for the iid and the moving maxima model, and in Figure 3 for
the GARCH-model. Additional details for the case of independent random sampling from the
absolute value of a Cauchy distribution are provided in the Supplement, Section F.
In general, (most of) the graphs nicely reproduce the bias-variance tradeoff, its characteristic
form however varying from model to model. Consider the iid scenario: since the Hill estimator
is essentially the maximum likelihood estimator in the Pareto family, it is to be expected that
it outperforms the block maxima estimator. On the other hand, by max-stability of the Fréchet
family, the block maxima estimator should outperform the Hill estimator for that family. These

14

A. Bücher and J. Segers

expectations are confirmed by the simulation results in the left column of Figure 2. For the
Cauchy distribution, it turns out that the block maxima maximum likelihood estimator shows a
better performance.
Now, consider the moving maxima time series scenarios (right column in Figure 2). Compared
to the iid case, we observe an increase in the mean squared error (note that the scale on the
axis of ordinates is row-wise identical). The block maxima method clearly outperforms the Hill
estimator, except for the Pareto model. The big increase in relative performance is perhaps
not too surprising, as the data points from a moving maximum process are already (weighted)
maxima, which principally favors the block maxima method with small block sizes.
Finally, consider the GARCH models in Figure 3. While, as in line with the theoretical findings,
the variance of the block maxima estimator is smaller than the one of the Hill estimator, the
squared bias turns out to be substantially higher for a large range of values for k. The MSEoptimal point is smaller for the Hill estimator.

Appendix A: Proofs
A.1. Proofs for Section 2
Proof of Lemma 2.1. The proof extends the development in Section 2 of Balakrishnan and Kateri
(2008). First, fix α > 0 and consider the function 0 < σ 7→ L(α, σ | x). By Equation (2.2), its
derivative is equal to
!
k
k
X
X
−α
α
.
xi
∂σ ℓθ (xi ) = (α/σ) k − σ
∂σ L(α, σ | x) =
i=1

i=1

We find that ∂σ L(α, σ | x) is positive, zero, or negative according to whether σ is smaller
than, equal to, or larger than σ(α | x), respectively. In particular, for fixed α, the expression
L(α, σ | x) is maximal at σ equal to σ(α | x). Hence we need to find the maximum of the function
0 < α 7→ L(α, σ(α | x) | x). By (2.1), its derivative is given by
k

X
d
∂α ℓα,σ (xi )
L(α, σ(α | x) | x) =
dα
i=1

+
σ=σ(α|x)

k
X

∂σ ℓα,σ (xi )

i=1

σ=σ(α|x)

×

d
σ(α | x).
dα

The second sum is equal to zero, by definition of σ(α | x). We obtain
d
L(α, σ(α | x) | x) = k Ψk (α | x),
dα
with Ψk as in (2.4). This is the same expression as Eq. (2.3) in Balakrishnan and Kateri (2008),
with their xi replaced by our x−1
i . Differentiating once more with respect to α, we obtain that
2
P
Pk
Pk
k
−α
−α
−α
2
log(xi )
−
i=1 xi
i=1 xi (log(xi ))
i=1 xi
k
d2
L(α, σ(α | x) | x) = − 2 − k
.
P
2
dα2
α
k
−α
x
i=1 i

(A.1)

By the Cauchy–Schwartz inequality, the numerator of the big fraction is nonnegative, whence
k
d2
L(α, σ(α | x) | x) ≤ − 2 < 0.
dα2
α

Maximum likelihood estimation for the Fréchet distribution

200

300

400

0.05
0.04
0.03
0.02

Squared Bias, Variance, MSE

0.01

500

100

200

300

400

Frechet(1) − i.i.d. model

Frechet(1) − moving maxima model

0.05

Effective Sample Size k

0.04
0.03
0.02

Squared Bias, Variance, MSE

Bias^2
Var
MSE
Block maxima MLE
Hill estimator

0.00

0.01

0.02

0.03

0.04

Bias^2
Var
MSE
Block maxima MLE
Hill estimator

500

0.01

0.05

Effective Sample Size k

0.00

200

300

400

500

300

400

Pareto(1) − i.i.d. model

Pareto(1) − moving maxima model

500

0.06

0.08

Bias^2
Var
MSE
Block maxima MLE
Hill estimator

0.00

0.00

0.02

0.04

Squared Bias, Variance, MSE

0.08

200

Effective Sample Size k

Bias^2
Var
MSE
Block maxima MLE
Hill estimator

0.06

100

Effective Sample Size k

0.04

100

0.02

Squared Bias, Variance, MSE

Bias^2
Var
MSE
Block maxima MLE
Hill estimator

0.00

0.01

0.02

0.03

0.04

Bias^2
Var
MSE
Block maxima MLE
Hill estimator

100

Squared Bias, Variance, MSE

15

Absolute Cauchy − moving maxima model

0.00

Squared Bias, Variance, MSE

0.05

Absolute Cauchy − i.i.d. model

100

200

300

Effective Sample Size k

400

500

100

200

300

400

500

Effective Sample Size k

Figure 2. Simulation results (Section 5.2). Effective sample size refers to the number of blocks (block maxima
MLE) or the number of upper order statistics (Hill estimator). Time series models: iid (left) and moving maximum
model (right). Innovations: absolute values of Cauchy (top), unit Fréchet (middle) and unit Pareto (bottom)
random variables. Block sizes r ∈ {2, 3, . . . , 24}, resulting in k ∈ {500, 333, . . . , 41} blocks.

16

A. Bücher and J. Segers
GARCH(1,1) with lambda = (0.5, 0.08, 0.91)

7
5
4
3
2

Squared Bias, Variance, MSE

6

Bias^2
Var
MSE
Block maxima MLE
Hill estimator

1

0.5

1.0

1.5

2.0

2.5

Bias^2
Var
MSE
Block maxima ML
Hill estimator

0

0.0

Squared Bias, Variance, MSE

3.0

GARCH(1,1) with lambda = (0.5, 0.367, 0.367)

20

40

60

80

100

120

20

Effective Sample Size k

40

60

80

100

120

140

160

Effective Sample Size k

Figure 3. Simulation results (Section 5.2). Effective sample size refers to the number of blocks (block maxima
MLE) or the number of upper order statistics (Hill estimator). Both panels refer to the GARCH(1,1) model in
(5.1), with (λ0 , λ1 , λ2 ) equal to (0.5, 0.367, 0.367) on the left and to (0.5, 0.08, 0.91) on the right.

Hence, α 7→ Ψk (α | x) is strictly decreasing. For α → 0, this function diverges to ∞, whereas for
P
α → ∞, it converges to log(min(x1 , . . . , xk )) − k −1 ki=1 log(xi ), which is less than zero given the
assumptions on x1 , . . . , xk . Hence, there exists a unique α̂(x) ∈ (0, ∞) such that this function is
zero. We conclude that the function θ 7→ L(θ | x) admits a unique maximum at θ̂(x).
Fix α0 ∈ (0, ∞). Let P denote the Fréchet distribution with parameter θ0 = (α0 , 1), with
support X = (0, ∞). The tentative limit of the functions α 7→ Ψk (α | x) is the function
R ∞ −α
Z ∞
x log(x) dP (x)
1
−
log(x) dP (x).
Ψ(α) = + 0 R ∞ −α
α
dP (x)
0
0 x
Let Γ be the gamma function and let ψ = Γ′ /Γ be the digamma function.

Lemma A.1. Fix α0 ∈ (0, ∞). We have
Ψ(α) =


1
ψ(1) − ψ(α/α0 ) ,
α0

α ∈ (0, ∞).

(A.2)

As a consequence, Ψ : (0, ∞) → R is a decreasing bijection with a unique zero at α = α0 .
Proof of Lemma A.1. By Lemma B.1,
Ψ(α) =

′

(−α−1
1
1
′
0 )Γ (1 + α/α0 )
+
− (−α−1
(α/α0 )−1 − ψ(1 + α/α0 ) + ψ(1) .
0 )Γ (1) =
α
Γ(1 + α/α0 )
α0

The digamma function satisfies the recurrence relation ψ(x + 1) = ψ(x) + x1 . Equation (A.2)
follows. The final statement follows from the fact that the digamma function ψ : (0, ∞) → R is
an increasing bijection.
Proof of Theorem 2.3. By Lemma 2.1, we only have to show the claimed convergence. Define
a random function Ψn on (0, ∞) by
Ψn (α) = Ψkn (α | Xn ) = Ψkn (α | Xn /σn ),

(A.3)

Maximum likelihood estimation for the Fréchet distribution

17

with Ψk (·|·) as in (2.4). Recall Ψ in (A.2). The hypotheses imply that, for each α ∈ (α− , α+ ),
Ψn (α)

Ψ(α),

n → ∞.

By Lemma A.1, the limit Ψ(α) is positive, zero, or negative according to whether α is less than,
equal to, or greater than α0 . Moreover, the function Ψn is decreasing and Ψn (α̂n ) = 0; see the
proof of Lemma 2.1.
Let δ > 0 be such that α− < α0 − δ < α0 + δ < α+ . Since Ψn (α0 − δ)
Ψ(α0 − δ) > 0 as
n → ∞, we find
Pr[α̂n ≤ α0 − δ] ≤ Pr[Ψn (α0 − δ) ≤ 0] → 0,

n → ∞.

Similarly, Pr[α̂n ≥ α0 + δ] → 0 as n → ∞. We can choose δ > 0 arbitrarily small, thereby
concluding that α̂n
α0 as n → ∞.
Second, Condition 2.2 also implies that, for each α ∈ (α− , α+ ) and as n → ∞,
1
σn

kn
1 X
X −α
kn i=1 n,i

!−1/α

!−1/α
kn
1 X
−α
(Xn,i /σn )
=
kn i=1
Z ∞
−1/α
−1/α
x−α pα0 ,1 (x) dx
= Γ(1 + α/α0 )
,
0

where we used Lemma B.1 for the last identity. Both the left-hand and right-hand sides are
continuous, nonincreasing functions of α. Since α̂n
α0 as n → ∞ and since the right-hand
side evaluates to unity at α = α0 , a standard argument then yields
1
σ̂n
=
σn
σn

kn
1 X
X −α̂n
kn i=1 n,i

!−1/α̂n

1,

n → ∞.

The proof of Theorem 2.5 is decomposed into a sequence of lemmas. Recall Ψn and Ψ in (A.3)
and (A.2), respectively, and define Ψ̇n (α) = (d/dα)Ψn (α) and Ψ̇(α) = (d/dα)Ψ(α). By (A.1),
Ψ̇n (α) = −

Pn [x−α (log x)2 ] Pn x−α − (Pn x−α log x)2
1
−
,
α2
(Pn x−α )2

(A.4)

n
and where
where Pn denotes the empirical distribution of the points (Xn,i /σn )ki=1

Pn f =

kn
1 X
f (Xn,i /σn ).
kn i=1

The asymptotic distribution of vn (α̂n − α0 ) can be derived from the asymptotic behavior of Ψ̇n
and vn Ψn , which is the subject of the next two lemmas, respectively.
Lemma A.2. (Slope) Let Xn = (Xn,1 , . . . , Xn,kn ) be a sequence of random vectors in (0, ∞)kn ,
where kn → ∞. Suppose that Equation (2.9) and Condition 2.4(i) are satisfied. If α̃n is a random
sequence in (0, ∞) such that α̃n
α0 as n → ∞, then
Ψ̇n (α̃n )

Ψ̇(α0 ) = −

π2
,
6α20

n → ∞.

18

A. Bücher and J. Segers

Proof. For α ∈ (0, ∞) and m ∈ {0, 1, 2}, define

fm,α (x) = x−α (log x)m ,

x ∈ (0, ∞),

with (log x)0 = 1 for all x ∈ (0, ∞). Suppose that we could show that, for m ∈ {0, 1, 2} and some
ε > 0,
Z ∞
0,
n → ∞.
(A.5)
Pn fm,α −
fm,α (x) pα0 (x) dx
sup
α:|α−α0 |≤ε

0

Then from weak convergence of α̃n to α0 , Slutsky’s lemma (van der Vaart, 1998, Lemma 2.8)
and Lemma B.1 below, it would follow that
Ψ̇n (α̃n )

−

−1 ′
′′
2
1
α−2
0 Γ (2) Γ(2) − (α0 Γ (2))
−
,
2
2
α0
(Γ(2))

n → ∞.

Since Γ(2) = 1, Γ′ (2) = 1 − γ and Γ′′ (2) = (1 − γ)2 + π 2 /6 − 1, the conclusion would follow.
It remains to show (A.5). We consider the three cases m ∈ {0, 1, 2} separately. Let ε > 0 be
small enough such that α− < α0 − ε < α0 + ε < α+ .
R∞
First, let m = 0. The maps α 7→ (Pn f0,α )1/α and α 7→ ( 0 f0,α pα0 ,1 )1/α are monotone by
R
Lyapounov’s inequality [i.e., kf kr ≤ kf ks for 0 < r < s, where kf kr = ( X |f |r dµ)1/r denotes
the Lr -norm of some real-valued function f on a measurable space (X , µ)], and the second one
is also continuous by Lemma B.1. Pointwise convergence of monotone functions to a monotone,
continuous limit implies locally uniform convergence (Resnick, 1987, Section 0.1). This property
easily extends to weak convergence, provided the limit is nonrandom. We obtain
Z ∞
1/α
(Pn f0,α )1/α −
sup
f0,α (x) pα0 (x) dx
0,
n → ∞.
α:|α−α0 |≤ε

0

Uniform continuity of the map (y, α) 7→ y α on compact subsets of (0, ∞)2 then yields (A.5) for
m = 0.
R∞
Second, let m = 1. The maps α 7→ Pn f1,α and α 7→ 0 f1,α pα0 ,1 are continuous and nonincreasing (their derivatives are nonpositive). Pointwise weak convergence at each α ∈ (α− , α+ )
then yields (A.5) for m = 1.
Finally, let m = 2. With probability tending to one, not all variables Xn,i are equal to σn ,
and thus Pn (log x)2 > 0. On the latter event, we have
(
1/α )α
Pn x−α (log x)2
−α
2
2
Pn x (log x) = Pn (log x)
.
Pn (log x)2
By Lyapounov’s inequality, the expression in curly braces is nondecreasing in α. For each α ∈
(α− , α+ ), it converges weakly to {Γ′′ (1+α/α0 )/Γ′′ (1)}1/α , which is nondecreasing and continuous
in α; see Lemma B.1. It follows that

1/α
1/α  ′′
Pn x−α (log x)2
Γ (1 + α/α0 )
sup
0,
n → ∞.
−
Pn (log x)2
Γ′′ (1)
α:|α−α0 |≤ε
Equation (A.5) for m = 2 follows.
Lemma A.3. Assume Condition 2.4. Then, as n → ∞,
vn Ψn (α0 ) = Gn x−α0 log(x) +

1−γ
Gn x−α0 − Gn log(x) + op (1).
α0

The expression on the right converges weakly to Y ≡ Y1 +

1−γ
α0 Y2

− Y3 .

(A.6)

Maximum likelihood estimation for the Fréchet distribution

19

Proof. Recall that
Ψn (α0 ) = Ψkn (α0 | Xn /σn ) =

Pn x−α0 log(x)
1
+
− Pn log(x).
α0
Pn x−α0

Define φ : R × (0, ∞) × R → R by
φ(y1 , y2 , y3 ) =

1
y1
+
− y3 .
α0
y2

The previous two displays allow us to write

Ψn (α0 ) = φ Pn x−α0 log(x), Pn x−α0 , Pn log(x) .

Recall Lemma B.1 and put



−1 ′
−1
−1
′
y0 = −α−1
0 Γ (2), Γ(2), −α0 Γ (1) = α0 (γ − 1), 1, α0 γ .

−1
−1
As already noted in the proof of Lemma A.1, we have φ(y0 ) = α−1
0 + α0 (γ − 1) − α0 γ = 0.
As a consequence,


vn Ψn (α0 ) = vn φ Pn x−α0 log(x), Pn x−α0 , Pn log(x) − φ(y0 ) .

In view of Condition 2.4 and the delta method, as n → ∞,

vn Ψn (α0 ) = φ̇1 (y0 ) Gn x−α0 log(x) + φ̇2 (y0 ) Gn x−α0 + φ̇3 (y0 ) Gn log(x) + op (1),
where φ̇j denotes the first-order partial derivative of φ with respect to yj for j ∈ {1, 2, 3}.
Elementary calculations yield
φ̇1 (y0 ) = 1,

φ̇2 (y0 ) = α−1
0 (1 − γ),

φ̇3 (y0 ) = −1.

The conclusion follows by Slutsky’s lemma.
Proposition A.4. (Asymptotic expansion for the shape parameter) Assume that the
conditions of Theorem 2.5 hold. Then, with Y as defined in Lemma A.3,
 6α2
vn α̂n − α0 = 20 vn Ψn (α0 ) + op (1)
π

6α20
Y,
π2

n → ∞.

(A.7)

Proof. Recall that, with probability tending to one, α̂n is the unique zero of the random function
α 7→ Ψn (α). Recall that Ψ̇n in (A.4) is the derivative of Ψn . With probability tending to one, we
have, by virtue of the mean-value theorem,

0 = Ψn (α̂n ) = Ψn (α̂n ) − Ψn (α0 ) + Ψn (α0 ) = (α̂n − α0 ) Ψ̇n (α̃n ) + Ψn (α0 );
here α̃n is a convex combination of α̂n and α0 . Since Ψ̇n (α) ≤ −1/α2 < 0 (argument as in the
proof of Lemma 2.1), we can write

vn α̂n − α0 = −

1
vn Ψn (α0 ).
Ψ̇n (α̃n )

By weak consistency of α̂n , we have α̃n
α0 as n → ∞. Lemma A.2 then gives Ψ̇n (α̃n )
−π 2 /(6α20 ) as n → ∞. Apply Lemma A.3 and Slutsky’s lemma to conclude.

20

A. Bücher and J. Segers

Proof of Theorem 2.5 and Addendum 2.6. Combining Equations (A.7) and (A.6) yields


6α2
1−γ
vn (α̂n − α0 ) = 20 Gn x−α0 log(x) +
Gn x−α0 − Gn log(x) + op (1)
π
α0
as n → ∞. This yields the first row in (2.17).
By definition of σ̂n , we have (σ̂n /σn )−α̂n = Pn x−α̂n . Consider the decomposition



vn (σ̂n /σn )−α̂n − 1 = vn Pn x−α̂n − Pn x−α0 + vn Pn x−α0 − 1 .

(A.8)

By the mean value theorem, there exists a convex combination, α̃n , of α̂n and α0 such that
Pn x−α̂n − Pn x−α0 = −(α̂n − α0 ) Pn x−α̃n log(x).
By the argument for the case m = 1 in the proof of Lemma A.2, we have
Pn x−α̃n log(x)

−

1−γ
1 ′
Γ (2) = −
,
α0
α0

n → ∞.

By Proposition A.4 and Lemma A.3, it follows that, as n → ∞,

1−γ
+ op (1)
vn Pn x−α̂n − Pn x−α0 = vn (α̂n − α0 )
α0
6α0 (1 − γ)
=
vn Ψn (α0 ) + op (1)
π2


1−γ
6α0 (1 − γ)
−α0
−α0
log(x) +
Gn x
− Gn log(x) + op (1).
=
Gn x
π2
α0
This expression in combination with (A.8) yields, as n → ∞,
vn (σ̂n /σn )−α̂n − 1
=

6α0 (1 − γ)
π2





1−γ
Gn x−α0 log(x) +
Gn x−α0 − Gn log(x) + Gn x−α0 + op (1). (A.9)
α0

Write Zn = (σ̂n /σn )−α̂n , which converges weakly to 1 as n → ∞. By the mean value theorem,
vn (σ̂n /σn − 1) = vn (Zn−1/α̂n − 1) = vn (Zn − 1) (−1/α̂n ) Z̃n−1/α̂n −1 ,
where Z̃n is a random convex combination of Zn and 1. But then Z̃n
by consistency of α̂n and Slutsky’s lemma,
vn (σ̂n /σn − 1) = (−1/α0 ) vn ((σ̂n /σn )−α̂n − 1) + op (1),

1 as n → ∞, whence,
n → ∞.

Combinining this with (A.9), we find
6(1 − γ)
vn (σ̂n /σn − 1) = −
π2



−α0

Gn x


1−γ
−α0
log(x) +
− Gn log(x)
Gn x
α0

−α0
− α−1
+ op (1)
0 Gn x

as n → ∞. This is the second row in (2.17).
The proof of Addendum 2.6 follows from a tedious but straightforward calculation.

Maximum likelihood estimation for the Fréchet distribution

21

A.2. Proofs for Section 3
Lemma A.5. (Block maxima rarely show ties) Under Conditions 3.1 and 3.3, for every
c ∈ (0, ∞), we have Pr[Mrn ,1 ∨ c = Mrn ,3 ∨ c] → 0 as n → ∞.
Proof of Lemma A.5. By the domain-of-attraction condition combined with the strong mixing property, the sequence of random vectors ((Mrn ,1 ∨ c)/σrn , (Mrn ,3 ∨ c)/σrn ) converges weakly
to the product of two independent Fréchet(α0 , 1) random variables. Apply the Portmanteau
lemma – the set {(x, y) ∈ R2 : x = y} is closed and has zero probability in the limit.
Lemma A.6. (Moments of block maxima converge) Under Conditions 3.1 and 3.4, we
have, for every c ∈ (0, ∞),
Z ∞

lim E[f (Mn ∨ c)/σn ] =
f (x) pα0 ,1 (x) dx
n→∞

0

for every measurable function f : (0, ∞) → R which is continuous almost everywhere and for
which there exist 0 < η < ν such that |f (x)| ≤ gη,α0 (x) = {x−α0 1(x ≤ e) + log(x)1(x > e)}2+η .

Proof of Lemma A.6. An elementary argument shows that we may replace Mn ∨1 by Mn ∨c in
(3.3). Since c/σn → 0 as n → ∞, the sequence (Mn ∨c)/σn converges weakly to the Fréchet(α0 , 1)
distribution in view of Condition 3.1. The result follows from Example 2.21 in van der Vaart
(1998).
In order to separate maxima over consecutive blocks by a time lag of at least ℓ, we clip off the
final ℓ − 1 variables within each block:
[ℓ]

Mr,i = max{ξt : (i − 1)r + 1 ≤ t ≤ ir − ℓ + 1}.

(A.10)

[ℓ]

Clearly, Mr,i ≥ Mr,i . The probability that the maximum over a block of size r is attained by any
of the final ℓ − 1 variables should be small; see Lemma A.8 below.
Lemma A.7. (Short blocks are small) Assume Condition 3.1. If ℓn = o(rn ) and if α(ℓn ) =
o(ℓn /rn ) as n → ∞, then for all ε > 0,
Pr[Mℓn ≥ εσrn ] = O(ℓn /rn ),

n → ∞.

(A.11)

Proof of Lemma A.7. Let Fr be the cumulative distribution function of Mr . By Bücher and Segers
(2014, Lemma 7.1), for every u > 0,
Pr[Frn (Mℓn ) ≥ u] = O(ℓn /rn ),

n → ∞.

Fix ε > 0. By assumption,
lim Frn (εσrn ) = exp(−ε−α0 ).

n→∞

For sufficiently large n, we have
Pr[Mℓn ≥ εσn ] ≤ Pr[Frn (Mℓn ) ≥ Frn (εσn )] ≤ Pr[Frn (Mℓn ) ≥ exp(−ε−α0 )/2].
Set u = exp(−ε−α0 )/2 in (A.12) to arrive at (A.11).

(A.12)

22

A. Bücher and J. Segers

Lemma A.8. (Clipping doesn’t hurt) Assume Condition 3.1. If ℓn = o(rn ) and if α(ℓn ) =
o(ℓn /rn ) as n → ∞, then
Pr[Mrn > Mrn −ℓn ] → 0,

n → ∞.

(A.13)

Proof of Lemma A.8. Recall Lemma A.7. For every ε > 0 we have, by stationarity,
Pr[Mrn > Mrn −ℓn ] ≤ Pr[Mrn −ℓn ≤ εσrn ] + Pr[Mℓn > εσrn ].
Since σrn −ℓn /σrn → 1 as a consequence of Condition 3.1 and the fact that ℓn = o(rn ) as n → ∞,
the first term converges to exp(−ε−α0 ) as n → ∞, whereas the second one converges to 0 by
Lemma A.7. Since ε > 0 was arbitrary, Equation (A.13) follows.
Proof of Theorem
3.6. We apply Theorem 2.5 and Addendum 2.6 to the array Xn,i = Mrn ,i ∨
√
c and vn = kn , where c ∈ (0, ∞) is arbitrary and i ∈ {1, . . . , kn }. By Condition 3.2, we have
limn→∞ Pr[∀i = 1, . . . , kn : Xn,i = Mrn ,i ] = 1.
The not-all-tied property (2.9) has been established in Lemma A.5.
We need to check Condition 2.4, and in particular that the distribution of the random vector
Y in (2.15) is N3 (B, ΣY ) with B as in the statement of Theorem 3.6 and ΣY as in (2.18).
Essentially, the proof employs the Bernstein big-block-small-block method in combination with
the Lindeberg central limit
p theorem.
√
Let ℓn = max{sn , ⌊rn α(sn )⌋}, where sn = ⌊ rn ⌋. Clearly,
ℓn → ∞,

ℓn = o(rn ) and α(ℓn ) = o(ℓn /rn ), as n → ∞.

(A.14)

Consider the truncated and rescaled block maxima
Zr,i = (Mrn ,i ∨ c)/σr ,

[ℓ ]

[ℓ ]

Zr,in = (Mrnn,i ∨ c)/σr ,

[ℓ ]

with Mr,in as in (A.10). Consider the following empirical and population probability measures:
Pn f =
n]
P[ℓ
n f =

kn
1 X
f (Zrn ,i ),
kn i=1
kn
1 X
[ℓ ]
f (Zrnn,i ),
kn i=1

Pn f = E[f (Zrn ,i )],
[ℓ ]

Pn[ℓn ] f = E[f (Zrnn,i )].

Abbreviate the tentative limit distribution by P = Fréchet(α0 , 1). We will also need the following
empirical processes:
p
Gn = kn (Pn − P )
(uncentered),
p
G̃n = kn (Pn − Pn )
(centered),
p
n]
(centered).
G̃[ℓ
= kn (Pn[ℓn ] − Pn[ℓn ] )
n

Finally, the bias arising from the finite block size is quantified by the operator
p
Bn = kn (Pn − P ).

Proof of Condition 2.4(i). Choose η ∈ (2/ω, ν) and 0 < α− < α0 < α+ . Additional constraints
on α+ will be imposed below, while the values of η and α− do not matter. Recall the function
class F2 (α− , α+ ) in (2.12). For every f ∈ F2 (α− , α+ ), we just need to show that
Pn f = P f + op (1),

n → ∞.

Maximum likelihood estimation for the Fréchet distribution

23

The domain-of-attraction property (Condition 3.1) and the asymptotic moment bound (Condition 3.4) yield
E[Pn f ] = Pn f → P f,
n → ∞,
by uniform integrability, see Lemma A.6 (note that |f | is bounded by a multiple of g0,α0 if α+ is
chosen suitably small: α+ < 2α0 must be satisfied). Further,
1
Pn f − Pn f = √ G̃n f.
kn
Below, see (A.16), we will show that
n]
G̃n f = G̃[ℓ
n f + op (1) = Op (1) + op (1) = Op (1),

n → ∞.

(A.15)

It follows that, as required,
Pn f = (Pn f − Pn f ) + Pn f = op (1) + P f + o(1) = P f + op (1),

n → ∞.

Proof of Condition 2.4(ii). We can decompose the empirical process Gn in a stochastic term
and a bias term:
p
p
Gn = kn (Pn − Pn ) + kn (Pn − P ) = G̃n + Bn .

For f ∈ H = {f1 , f2 , f3 }, the bias term Bn f converges to B(f ) thanks to Condition 3.5. It
remains to treat the stochastic term G̃n f , for all f ∈ F2 (α− , α+ ) [in view of the proof of item
(i); see (A.15) above]. We will show that the finite-dimensional distributions of G̃n converge to
those of a P -Brownian bridge, G, i.e., a zero-mean, Gaussian stochastic process with covariance
function given by


f, g ∈ F2 (α− , α+ ).
cov(Gf, Gg) = P (f − P f )(g − P g) = covP f (X), g(X) ,
Decompose the stochastic term in two parts:

n]
G̃n = G̃[ℓ
n + ∆n .

(A.16)

We will show that ∆n converges to zero in probability and that the finite-dimensional distribun]
tions of G̃[ℓ
converge to those of G.
n
n]
First, we treat the main term, G̃[ℓ
n . By the Cramér–Wold device, it suffices to show that
[ℓn ]
G̃n g
Gg as n → ∞, where g is an arbitrary linear combination of functions f ∈ F2 (α− , α+ ).
Define


[ℓ ]
φni (t) = exp − itkn−1/2 {g(Zrnn,i ) − Pn[ℓn ] g} ,

with i the imaginary unit. Note that the characteristic function of G̃n[ℓn] g can be written as
Qkn
t 7→ E[ i=1
φni (t)]. Successively applying Lemma 3.9 in Dehling and Philipp (2002), we obtain
that



# k
"k
kn
n
n

 Y
Y
Y
kn
φnj (t)  ,
E[φni (t)] ≤ 2πkn max α σ{φni (t)}, σ
φni (t) −
E
i=1


i=1

i=1

j=i+1

where α(A1 , A2 ) denotes the alpha-mixing coefficient between the sigma-fields A1 and A2 . Since
[ℓn ]
over different blocks are based on observations that are at least ℓn observations
the maxima Zr,i
apart, the expression on the right-hand side of the last display is of the order O(kn α(ℓn )), which

24

A. Bücher and J. Segers

converges to 0 as a consequence of Equation (3.2). We can conclude that the weak limit of G̃n[ℓn ] g
is the same as the one of
)
(
kn
p
1 X
[ℓn ]
[ℓn ]
[ℓn ]
g(Z̄rn ,i ) − Pn g ,
H̃n g = kn
kn i=1
where Z̄r[ℓnn,i] are independent over i ∈ N and have the same distribution as Zr[ℓnn,i] . By the classical
central limit theorem for row wise independent triangular arrays, the weak limit of H̃[ℓ]
n g is Gg:
first, its variance
[ℓn ] 2
[ℓn ] 2
n]
Var(H̃[ℓ
n g) = Pn g − (Pn g)
converges to Var(Gg) by Lemma A.6. Note that the square of any linear combination g of
functions f ∈ F2 (α− , α+ ) can be bounded by a multiple of gη,α0 , after possibly decreasing the
value of α+ > α0 . Second, the Lyapunov Condition is satisfied: for all δ > 0,
1
1+δ/2

kn

kn
X


[ℓ ]
E |g(Z̄rnn,i ) − Pn[ℓn ] g|2+δ
i=1

converges to 0 as n → ∞ again as a consequence of Lemma A.6, as |g|2+δ can also be bounded
by a multiple of gη,α0 if δ ∈ (0, η) and α+ > α0 are chosen sufficiently small.
[ℓ ]
Now, consider the remainder term ∆n in (A.16). Since G̃n f and G̃n n f are centered, so is
∆n f , and
!
kn
X
1
[ℓn ]
2
E[(∆n f ) ] = var(∆n f ) =
∆rn ,i f ,
var
kn
i=1
[ℓ ]

[ℓ ]

where ∆r,in f = f (Zr,i ) − f (Zr,in ). By stationarity and the Cauchy–Schwartz inequality,
kn −1




2 X
[ℓ ]
[ℓ ]
[ℓ ]
(kn − h) cov ∆rnn,1 f, ∆rnn,1+h f
E[(∆n f )2 ] = var ∆rnn,1 f +
kn
h=1



[ℓ ]



≤ 3 var ∆rnn,1 f + 2

kX
n −1
h=2



[ℓ ]
[ℓ ]
cov ∆rnn,1 f, ∆rnn,1+h f .

(A.17)

Please note that we left the term h = 1 out of the sum; whence the factor three in front of the
variance term.
Since ℓn = o(rn ) as n → ∞ by Condition 3.3, we have σrn −ℓn +1 /σrn → 1 as n → ∞ by
Condition 3.1. The asymptotic moment bound in Condition 3.4 then ensures that we may choose
δ ∈ (2/ω, ν) and α+ > α0 such that, for every f ∈ F2 (α− , α+ ), we have, by Lemma A.6,


2+δ
[ℓn ]
lim sup E ∆rn ,1 f
< ∞.
(A.18)
n→∞

[ℓ ]

On the event that Mrn ,1 = Mrn −ℓn +1 , we have ∆rnn,1 f = 0. The mixing rate in (A.14) together
with Lemma A.8 then imply
[ℓ ]
∆rnn,1 f = op (1),
n → ∞.
Lyapounov’s inequality and the asymptotic moment bound (A.18) then ensure that


2+δ
[ℓn ]
lim E ∆rn ,1 f
= 0,
f ∈ F2 (α− , α+ ).
n→∞

(A.19)

Maximum likelihood estimation for the Fréchet distribution

25

Recall Lemma 3.11 in Dehling and Philipp (2002): for random variables ξ and η and for
numbers p, q ∈ [1, ∞] such that 1/p + 1/q < 1,
|cov(ξ, η)| ≤ 10 kξkp kηkq {α(σ(ξ), σ(η))}1−1/p−1/q ,

(A.20)

where α(A1 , A2 ) denotes the strong mixing coefficient between two σ-fields A1 and A2 . Use
inequality (A.20) with p = q = 2 + δ to bound the covariance terms in (A.17):
[ℓ ]

[ℓ ]

E[(∆n f )2 ] ≤ 3 k∆rnn,1 f k22 + 20 kn k∆rnn,1 f k22+δ {α(rn )}δ/(2+δ) .
In view of (A.19) and Condition 3.3, the right-hand side converges to zero since ω < 2/δ.

A.3. Proof of Theorem 4.2
Proof of Theorem 4.2. We apply Theorem 3.6. To this end, we verify its conditions.
Proof of Condition 3.1. The second-order regular variation condition (4.5) implies the firstorder one in (4.2), which is in turn equivalent to weak convergence of partial maxima as in (4.1).
Condition 3.1 follows with scaling sequence σn = an . The latter sequence is regularly varying
(Resnick, 1987, Proposition 1.11) with index 1/α0 , which implies that limn→∞ amn /an = 1
whenever limn→∞ mn /n = 1.
Proof of Condition 3.2. For any real c we have, since log F (c) < 0 and since log(kn ) = o(rn )
by (4.12),
Pr[min(Mrn ,1 , . . . , Mrn ,kn ) ≤ c] ≤ kn F rn (c) = exp{log(kn ) + rn log F (c)} → 0,

n → ∞.

Proof of Condition 3.3. Trivial, since α(ℓ) = 0 for integer ℓ ≥ 1.
Proof of Condition 3.4. This follows from Lemma C.1 in the supplementary material (which
in turn is a variant of Proposition 2.1(i) in Resnick, 1987), where we prove that the sufficient
Condition (3.4) is satisfied.
Proof of Condition 3.5. Recall Remark 4.3 and therein the functions L and g(u) = A(u)L(u).
We begin by collecting some non-asymptotic bounds on the function L. Fix δ ∈ (0, α0 ). Potter’s
theorem (Bingham, Goldie and Teugels, 1987, Theorem 1.5.6) implies that there exists some
constant x′ (δ) > 0 such that, for all u ≥ x′ (δ) and x ≥ x′ (δ)/u,
L(u)
≤ (1 + δ) max(x−δ , xδ ).
L(ux)

(A.21)

As a consequence of Theorem B.2.18 in de Haan and Ferreira (2006), accredited to Drees
(1998), there exists some further constant x′′ (δ) > 0 such that, for all u ≥ x′′ (δ) and x ≥ x′′ (δ)/u,
L(ux) − L(u)
≤ c(δ) max(xρ−δ , xρ+δ ),
g(u)

(A.22)

for some constant c(δ) > 0. Define x(δ) = max{x′ (δ), x′′ (δ), 1}.
We are going to show Condition 3.5 for c = x(δ) and σrn = arn . For i = 1, . . . , kn , define
Xn,i = Mrn ,i ∨ x(δ). Let Pn denote the common distribution of the rescaled, truncated
block
√
maxima Xn,i /arn and let P denote the Fréchet(α0 , 1) distribution. Write Bn = kn (Pn − P )
and define the three-by-one vector β by

|ρ|
|ρ| 
2 − γ − Γ(2 + α
) − Γ′ (2 + α
)
0
0

λ 
|ρ|


β=
(A.23)
) − α0
α0 Γ(2 + α

0
|ρ| α0 
|ρ|
1 − Γ(1 + α0 )

26

A. Bücher and J. Segers

if ρ < 0 and by



γ − (1 − γ)2 − π 2 /6
λ 

α0 (1 − γ)
β= 2
α0
γ

if ρ = 0. We will show that

T
lim Bn x−α0 log x, Bn x−α0 , Bn log x = β.

(A.24)

n→∞

Elementary calculations yield that M (α0 ) β = λ B(α0 , ρ) as required in (4.8).
Equation (A.24) can be shown coordinatewise. We begin by some generalities. For any f ∈ H
as in (2.13), we can write, for arbitrary x, x0 ∈ (0, ∞),

R
f (x0 ) − x0 f ′ (y) dy, if 0 < x ≤ x0 ,
x
f (x) =
f (x0 ) + R x f ′ (y) dy, if x0 < x < ∞.
x0
By Fubini’s theorem, with Gn and G denoting the cdf-s of Pn and P , respectively,
Z
Z
Pf =
f (x) dP (x) +
f (x) dP (x)
(0,x0 ]

= f (x0 ) −
= f (x0 ) −
= f (x0 ) −

(x0 ,∞)

Z

x∈(0,x0 ]
x0 Z

Z

y=0
Z x0

Z

x0

f ′ (y) dy dP (x) +

y=x

dP (x) f ′ (y) dy +

x∈(0,y]
′

G(y) f (y) dy +

0

Z

Z

x∈(x0 ,∞)
∞ Z

y=x0

Z

∞

x0

Z

x

f ′ (y) dy dP (x)

y=x0

dP (x) f ′ (y) dy

x∈(y,∞)

{1 − G(y)} f ′ (y) dy,

and the same formula holds with P and G replaced by Pn and Gn , respectively. We find that
Z ∞p
p
kn {Gn (y) − G(y)} f ′ (y) dy.
Bn f = kn (Pn − P )f = −
0

Note that

G(y) = exp(−y −α0 ) 1(0,∞) (y),

Gn (y) = F rn (arn y) 1[x(δ)/arn ,∞) (y),

From the definition of L in (4.11), we can write, for y ≥ x(δ)/arn ,


L(arn y)
.
Gn (y) = exp −y −α0 rn {− log F (arn )}
L(arn )
For the sake of brevity, we will only carry out the subsequent parts of the proof in the case where
F is ultimately continuous, so that rn {− log F (arn )} = 1 for all sufficiently large n. In that case,
Bn f = Jn1 (f ) + Jn2 (f ) where
p Z
kn

x(δ)/arn

exp(−y −α0 )f ′ (y) dy,
0




p Z ∞
L(arn y)
exp −y −α0
Jn2 (f ) = − kn
− exp(−y −α0 ) f ′ (y) dy,
L(arn )
x(δ)/arn
Jn1 (f ) =

Maximum likelihood estimation for the Fréchet distribution

27

Let us first show that Jn1 (f ) converges to 0 for any f ∈ H. For that purpose, note that any
f ∈ H satisfies |f ′ (x)| ≤ Kx−α0 −ε−1 for any ε < 1 and for some constant K = K(ε) > 0. As a
consequence, by (4.9), for sufficiently large n,
Z x(δ)/arn
K
max |Jn1 (f )| ≤ {λ + o(1)}
exp(−y −α0 )y −α0 −ε−1 dy.
f ∈H
A(arn ) 0
Since A(x) is bounded from below by a multiple of xρ−ε for sufficiently large x (by Remark 4.3
and Potter’s theorem), the expression on the right-hand side of the last display can be easily
seen to converge to 0 for n → ∞.
For the treatment of Jn2 , note that
Z ∞

J(f, ρ) ≡
hρ (y) exp −y −α0 y −α0 f ′ (y) dy
0R ∞
−α0

) y −2α0 −1 (1 − α0 log y) dy , f (y) = y −α0 log y
R0 hρ (y) exp (−y
∞
=
, f (y) = y −α0
hρ (y) exp (−y −α0 ) (−α0 y −2α0 −1 ) dy
0

R ∞
−α0 −1
−α0
)y
dy
, f (y) = log y
0 hρ (y) exp (−y

−α0
−α0

(α−1
log y
E[hρ (Y )Y
0 − log Y )] , f (y) = y
−α
−α
= −E[hρ (Y )Y 0 ]
, f (y) = y 0

 −1
α0 E[hρ (Y )]
, f (y) = log y,

where Y denotes a Fréchet(α0 , 1) random variable. By Lemma B.1 this implies
o
1 n
|ρ|
|ρ|
) + Γ′ (2 + α
) − 1 − Γ′ (2)
Γ(2 + α
J(x−α0 log x, ρ) =
0
0
ρα0
o
1 n
|ρ|
|ρ|
′
2 − γ − Γ(2 + α
)
−
Γ
(2
+
)
,
=
α0
0
|ρ| α0
o
o
1n
1 n
|ρ|
|ρ|
Γ(2) − Γ(2 + α
Γ(2
+
J(x−α0 , ρ) =
)
=
)
−
1
,
α
0
0
ρ
|ρ|
o
o
1 n
1 n
|ρ|
|ρ|
Γ(1 + α
1
−
Γ(1
+
J(log x, ρ) =
)
−
1
=
)
α0
0
ρα0
|ρ| α0
for ρ < 0 and

1 
1
{Γ′ (2) + Γ′′ (2)} = 2 γ − (1 − γ)2 − π 2 /6 ,
α20
α0
1 ′
1−γ
J(x−α0 , 0) =
Γ (2) =
,
α0
α0
γ
1
J(log x, 0) = − 2 Γ′ (1) = 2 .
α0
α0
T
Hence, β = λ J(x−α0 log x, ρ), J(x−α0 , ρ), J(log x, ρ) and it is therefore sufficient to show that,
for any f ∈ H,
J(x−α0 log x, 0) = −

Jn2 (f ) → λ J(f, ρ)
as n → ∞. By the mean value theorem, we can write Jn2 (f ) as
Z ∞
p

L(arn y) − L(arn )
exp −y −α0 ξn (y) y −α0 f ′ (y) dy
Jn2 (f ) = kn A(arn )
)
)L(a
A(a
rn
rn
x(δ)/arn

(A.25)

28

A. Bücher and J. Segers

for some ξn (y) between L(arn y)/L(arn ) and 1. For n → ∞, the factor in front of this integral
converges to λ by assumption (4.9), while the integrand in this integral converges to

hρ (y) exp −y −α0 y −α0 f ′ (y),

pointwise in y ∈ (0, ∞), by Condition 4.1. Hence, the convergence in (A.25) follows from dominated convergence if we show that

fn (y) = 1 y >

x(δ)
ar n

 L(a y) − L(a )

rn
rn
exp −y −α0 ξn (y) y −α0 f ′ (y)
A(arn )L(arn )

can be bounded by an integrable function on (0, ∞). We split the proof into two cases.
First, for any 1 ≥ y ≥ x(δ)/arn ,
L(arn y) − L(arn )
≤ c(δ)y ρ−δ
A(arn )L(arn )
from (A.22) and


L(arn y)
≥ (1 + δ)−1 y δ
ξn (y) ≥ min 1,
L(arn )
from (A.21). Moreover, for any f ∈ H, the function f ′ (y) is bounded by a multiple of y −α0 −δ−1
for y ≤ 1. Therefore, for any y ∈ (0, 1),
fn (y) ≤ c′ (δ) exp{−(1 + δ)−1 y −α0 +δ }y −2α0 −2δ−1+ρ
and the function on the right is integrable on (0, 1) since δ < α0 .
Second, for y ∈ [1, ∞), we have
L(arn y) − L(arn )
≤ c(δ)y ρ+δ
A(arn )L(arn )
from (A.22) and


L(arn y)
≥ (1 + δ)−1 y −δ
ξn (y) ≥ min 1,
L(arn )
from (A.21). Moreover, f ′ (y) is bounded by a multiple of y −1 for any y ≥ 1 and any f ∈ H.
Therefore,
fn (y) ≤ c′′ (δ) y −α0 −1+ρ+δ
which is easily integrable on [1, ∞).

Appendix B: Auxiliary results
R∞
Let Γ(x) = 0 tx−1 e−t dt be the gamma function and let Γ′ and Γ′′ be its first and second
derivative, respectively. All proofs for this section are given in Section E in the supplementary
material.

Maximum likelihood estimation for the Fréchet distribution

29

Lemma B.1. (Moments) Let P denote the Fréchet distribution with parameter vector (α0 , 1),
for some α0 ∈ (0, ∞). For all α ∈ (−α0 , ∞),
Z ∞
x−α dP (x) = Γ(1 + α/α0 ),
0
Z ∞
1
x−α log(x) dP (x) = − Γ′ (1 + α/α0 ),
α0
Z ∞0
1
x−α (log(x))2 dP (x) = 2 Γ′′ (1 + α/α0 ).
α0
0
Lemma B.2. (Covariance matrix) Let X be a random variable whose distribution is Fréchet
with parameter vector (α0 , 1). The covariance matrix of the random vector Y = (Y1 , Y2 , Y3 )T =
T
X −α0 log(X), X −α0 , log(X) is equal to


1 − 4γ + γ 2 + π 2 /3 α0 (γ − 2) π 2 /6 − γ
1
α0 (γ − 2)
α20
−α0  .
cov(Y ) = 2 
α0
π 2 /6 − γ
−α0
π 2 /6
Lemma B.3. (Fisher information) Let Pθ denote the Fréchet distribution with parameter
θ = (α, σ) ∈ (0, ∞)2 . The Fisher information Iθ = Pθ (ℓ̇θ ℓ̇Tθ ) is given by
 


{(1 − γ)2 + π 2 /6}/α2 (1 − γ)/σ
ι11 ι12
=
.
Iθ =
(1 − γ)/σ
α2 /σ 2
ι21 ι22
Its inverse is given by
Iθ−1 =

6
π2



α2
(γ − 1)σ

(γ − 1)σ
(σ/α)2 {(1 − γ)2 + π 2 /6}



.

Acknowledgments
The authors would like to thank two anonymous referees and an Associate Editor for their
constructive comments on an earlier version of this manuscript, and in particular for suggesting a
sharpening of Conditions 2.2 and 2.4 and for pointing out the connection between Equations (4.9)
and (4.12).
The research by A. Bücher has been supported by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Project A7) of the German Research
Foundation, which is gratefully acknowledged. Parts of this paper were written when A. Bücher
was a visiting professor at TU Dortmund University.
J. Segers gratefully acknowledges funding by contract “Projet d’Actions de Recherche Concertées” No. 12/17-045 of the “Communauté française de Belgique” and by IAP research network
Grant P7/06 of the Belgian government (Belgian Science Policy).

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Maximum likelihood estimation for the Fréchet distribution

31

Supplementary Material on
“Maximum likelihood estimation for the Fréchet distribution
based on block maxima extracted from a time series”
AXEL BÜCHER and JOHAN SEGERS
Ruhr-Universität Bochum and Université catholique de Louvain
This supplementary material contains a lemma on moment convergence of block maxima used in
the proof of Theorem 4.2 (in Section C), the proof of Lemma 5.1 (in Section D) and the proofs of
auxiliary lemmas from Section B (in Section E) from the main paper. Furthermore, we present
additional Monte Carlo simulation results to quantify the finite-sample bias and variance of the
maximum likelihood estimator (in Section F).

Appendix C: Moment convergence of block maxima
The following Lemma is a variant of Proposition 2.1(i) in Resnick (1987). It is needed in the
proof of Theorem 4.2.
Lemma C.1. Let ξ1 , ξ2 , . . . be independent random variables with common distribution function
F satisfying (4.2). Let Mn = max(ξ1 , . . . , ξn ). For every β ∈ (−∞, α0 ) and any constant c > 0,
we have
β 

< ∞.
lim sup E (Mn ∨ c)/an
n→∞

Proof of Lemma C.1. Since the case β = 0 is trivial, there are two cases to be considered:
β ∈ (−∞, 0) and β ∈ (0, α0 ). Write Zn = (Mn ∨ c)/an and note that
Pr[Zn < y] = Pr[Mn ∨ c < an y] = F n (an y) 1(c/an ,∞) (y).
Case β ∈ (−∞, 0). We have
Z ∞
Z
E[Znβ ] =
Pr[Znβ > x] dx =
0

∞

Pr[Zn < x1/β ] dx =

0

Z

0

=

Z

∞

Pr[Zn < y] |β| y β−1 dy

∞

c/an

F n (an y) |β| y β−1 dy.

We split the integration domain in two pieces. For y ∈ (1, ∞), the integrand is bounded by
|β| y β−1 , which integrates to unity. Hence we only need to consider the integral over y ∈ (c/an , 1].
We have


− log F (an y)
F n (an y) = exp{n log F (an y)} = exp −n{− log F (an )}
.
− log F (an )
Fix δ ∈ (0, α0 ). By (4.3), we have n{− log F (an )} ≥ 1 − δ for all n larger than some n(δ). By
Potter’s theorem (Bingham, Goldie and Teugels, 1987, Theorem 1.5.6), there exists x(δ) > 0
such that, for all n such that an ≥ x(δ) and for all y ∈ (x(δ)/an , 1],
− log F (an )
≤ (1 + δ) y α0 −δ .
− log F (an y)

32

A. Bücher and J. Segers

Without loss of generality, assume x(δ) > c. For all y ∈ (c/an , x(δ)/an ], we have
− log F (an )
− log F (an )
≤
≤ (1 + δ) (x(δ)/an )α0 −δ ≤ (1 + δ) x(δ)α0 −δ cδ−α0 y α0 −δ .
− log F (an y)
− log F (x(δ))
Combining the previous two displays, we see that there exists a constant c(δ) > 0 such that
− log F (an y)
≥ c(δ) y −α0 +δ
− log F (an )
for all y ∈ (c/an , 1]. We conclude that, for all sufficiently large n and all y ∈ (c/an , 1],

F n (an y) ≤ exp −c(δ) y −α0 +δ ,

where c(δ) is a positive constant, possibly different from the one in the previous equation. For
such n, we have
Z 1
Z 1

n
β−1
F (an y) |β| y
dy ≤
exp −c(δ) y −α0 +δ |β| y β−1 dy < ∞.
c/an

0

Case β ∈ (0, α0 ). Let δ > 0 be sufficiently small such that β + δ < α. Let x(δ) > 0 be as
in Potter’s theorem. Let n(δ) be sufficiently large such that an ≥ x(δ) ∨ c for all n ≥ n(δ). Put
K = supn≥1 n{1 − F (an )}, which is finite by (4.3) and the fact that − log F (x) ∼ 1 − F (x) for
x → ∞. For n ≥ n(δ), we have
Z ∞
Z ∞
E[Znβ ] =
Pr[Zn > x1/β ] dx =
Pr[Mn ∨ c > an x1/β ] dx
0
0
Z ∞
≤1+
Pr[Mn > an x1/β ] dx
1
Z ∞
≤1+
n{1 − F (an x1/β )} dx
1

≤1+K

Z

1

∞

1 − F (an x1/β )
dx.
1 − F (an )

By Potter’s theorem, the integral on the last line is bounded by
Z ∞
(1 + δ)
(x1/β )−α0 +δ dx.
c

The latter integral is finite, since (−α0 + δ)/β < −1.

Appendix D: Proofs for Section 5
Proof of Lemma 5.1. We only give a sketch proof for the case p = 2, the general case being
similar, but notationally more involved. Set b1 = b and b2 = 1 − b, so that b(2) = b ∨ (1 − b).
Clearly,
−1
−1
Pr(Mn ≤ x) = Pr{Z0 ≤ x(1 − b)−1 , Z1 ≤ xb−1
}
(2) , . . . , Zn−1 ≤ xb(2) , Zn ≤ xb

= F (x(1 − b)−1 ) · F (xb−1 ) · F n−1 (xb−1
(2) ).

Maximum likelihood estimation for the Fréchet distribution

33

As a consequence, with b(1) = b ∧ (1 − b),
Hn (x) = Pr(Mn ≤ xb(2) an )
b

(2)
= F (an x 1−b
) · F (an x

b(2)
b )

b

) · F n (an x).
= F (an x b(2)
(1)

· F n−1 (an x)

Since, by assumption, F n (xan ) → exp(−x−α0 ), Condition 3.1 is satisfied.
Condition 3.3 is trivial, since the process is p-dependent.
The proof of Condition 3.4 can be be carried out along the lines of the proof of Lemma C.1.
For β < 0, simply use that
Pr{(Mn ∨ c)/σn ≤ x} = Hn (x) 1(x ≥ c/σn ) ≤ F n (xan ) 1(x ≥ c/σn ),
while, for β > 0,
Pr(Mn > σn x1/β ) ≤ 2n · Pr(Z1 > σn x1/β b(2) ) = 2n{1 − F (an x1/β )},
for any x > 1.
Since log kn = o(rn ), Condition 3.2 follows from
Pr[min(Mrn ,1 , . . . , Mrn ,kn ) ≤ c] ≤ kn Pr(Mrn ≤ c)

−1
= exp{log kn + (rn − 1) log F (cb−1
) · F (cb−1 ).
(2) )} · F (c(1 − b)

Finally, consider Condition 3.5. As in the proof of Theorem 4.2, write
Z ∞√
p



kn E f (Mrn ∨ c)/σrn − P f = −
kn {H̃n (y) − G(y)}f ′ (y) dy,
0

where G(y) = exp(−y −α0 ) and where

H̃n (y) = P{(Mrn ∨ c)/σrn ≤ y} = An (y)Gn (y)
with

b

Gn (y) = F rn (yarn )1(y ≥ c/σrn ).

(2)
An (y) = F (arn y b(1)
),

Write
Z

0

∞

√

c/σrn √
kn {H̃n (y) − G(y)}f (y) dy = −
k n G(y)f ′ (y) dy
0
Z ∞ √
+
kn An (y){Gn (y) − G(y)}f ′ (y) dy

Z

′

c/σrn

+

Z

∞

c/σrn

√
k n {An (y) − 1}G(y)f ′ (y) dy. (D.1)

The first integral converges to 0 as shown in the proof of Theorem 4.2, treatment of Jn1 (f ). The
integrand of the second integral converges pointwise to the same limit as in the iid case. The
integrand can further be bounded by an integrable function as shown in the treatment of Jn2 in
the proof of Theorem 4.2, after splitting the integration domain at 1. Hence, the limit of that
integral is the same as in the iid case by dominated convergence.

34

A. Bücher and J. Segers
Consider the last integral in the latter display. Decompose
√
p
− log An (y)
kn 1 − An (y)
kn |An (y) − 1| =
·
,
·
rn − log An (y) − log F (arn )

where we used the fact that rn {− log F (arn )} = 1. The second factor is bounded by 1, since
log(x) ≤ x − 1 for all x > 0. Consider the third factor. With L(x) = − log{F (x)}xα0 , we have
L(arn yb(2) /b(1) )
− log An (y)
= (yb(2) /b(1) )−α0
.
− log F (arn )
L(arn )
The fraction on the right-hand side is bounded by a multiple of y δ ∨ y −δ by Potter’s theorem, for
some 0 < δ < α0 . Further note that, up to a factor, f ′ (y) ≤ y −α0 −δ−1 for y ≤ 1 and f ′ (y) ≤ y −1
for y > 1. We obtain that the integrand of the third integral on the right-hand side of (D.1) is
bounded by a multiple of
√
kn /rn · exp(−y −α0 )y −2α0 −2δ−1
for y ≤ 1 and by a multiple of

√
k n /rn · y −α0 −1+δ

for √
y > 1. Both functions are integrable on its respective domains. Since kn = o(n2/3 ) is equivalent
to k n = o(rn ), the third integral converges to 0. Hence, Condition 3.5 is satisfied.

Appendix E: Proofs for Section B
Proof of Lemma B.1. If Y is a unit exponential random variable, then the law of Y −1/α0 is
α/α0
equal to P . The integrals stated in the lemma are equal to E[Y α/α0 ], −α−1
log(Y )], and
0 E[Y
−2
α/α0
2
α0 E[Y
(log Y ) ], respectively. First,
Z ∞
Z ∞
y α/α0 exp(−y) dy = Γ(1 + α/α0 ).
x−α dP (x) =
0

0

Second,
Z

∞

Z

∞

0

1
α0

Z

∞

1
α20

Z

∞

x−α log(x) dP (x) = −

0

log(y) y α/α0 exp(−y) dy = −

1 ′
Γ (1 + α/α0 ).
α0

Third,
x−α (log x)2 dP (x) =

0

(log y)2 y α/α0 exp(−y) dy =

0

1 ′′
Γ (1 + α/α0 ).
α20

Proof of Lemma B.2. Recall a few special values of the first two derivatives of the Gamma
function:
Γ′ (1) = −γ,

Γ′ (2) = 1 − γ,

Γ′ (3) = 3 − 2γ,

Γ′′ (1) = γ 2 + π 2 /6,
Γ′′ (2) = (1 − γ)2 + π 2 /6 − 1,

Γ′′ (3) = 2((3/2 − γ)2 + π 2 /6 − 5/4).

Applying the formulas in Lemma B.1 with α ∈ {0, α0 , 2α0 }, we find
 ′′
2
2
Γ (3) − (Γ′ (2))2 = α−2
var(Y1 ) = α−2
0 (1 − 4γ + γ + π /3),
0
var(Y2 ) = Γ(3) − (Γ(2))2 = 1,


2
Γ′′ (1) − (Γ′ (1))2 = α−2
var(Y3 ) = α−2
0 π /6,
0

Maximum likelihood estimation for the Fréchet distribution

35

as well as

cov(Y1 , Y2 ) = α0−1 (−Γ′ (3)) − (−Γ′ (2))Γ(2) = α−1
0 (γ − 2),

−2
′′
′
′
2
cov(Y1 , Y3 ) = α0 Γ (2) − (−Γ (2))(−Γ (1)) = α−2
0 (π /6 − γ),

cov(Y2 , Y3 ) = α0−1 (−Γ′ (2)) − Γ(2)(−Γ′ (1)) = −α−1
0 .

Proof of Lemma B.3. If X ∼ P(α,σ) , then Z = X/σ ∼ P(α,1) . Therefore, by (2.1) and
Lemma B.1,


ι11 = E {α−1 + (Z −α − 1) log Z}2

1 
= 2 1 − 2{Γ′ (2) − Γ′ (1)} + {Γ′′ (3) − 2Γ′′ (2) + Γ′′ (1)}
α
1
= 2 {(1 − γ)2 + π 2 /6}.
α
Similarly, by (2.1) and (2.2),

α 
E (1 − Z −α ){α−1 + (Z −α − 1) log Z}
σ
α  −1
=
α {Γ(1) − Γ(2)} + α−1 {Γ′ (1) − 2Γ′ (2) + Γ′ (3)}
σ
1−γ
=
.
σ

ι12 =

Finally,
ι22 =

α2
α2
α2
−α 2
E[(1
−
Z
)
]
=
{Γ(1)
−
2Γ(2)
+
Γ(3)}
=
.
σ2
σ2
σ2

Appendix F: Finite-sample bias and variance
We work out the second-order Condition 4.1 and the expressions for the asymptotic bias and
variance of the maximum likelihood estimator of the Fréchet shape parameter for the case of
block maxima extracted from an independent random sample from the absolute value of a Cauchy
distribution. Furthermore, we compare these expressions to those obtained in finite samples from
Monte Carlo simulations.
If the random variable ξ is Cauchy-distributed, then |ξ| has distribution function
F (x) = P{|ξ| ≤ x} =

2
arctan(x) 1(x > 0),
π

x ∈ R.

Based on the asymptotic expansion
2 2
1

2
− log arctan(x) = log
+
+ 2 2 +O 3 ,
π
πx π x
x

x → ∞,

one can show that − log F is regularly varying at infinity with index −α0 = −1 and that the
limit relation


− log F (ux)
1
− x−1 = x−1 hρ (x)
lim
u→∞ A(u)
− log F (u)

is satisfied for

ρ = −1

and

A(u) = −

1
.
1 + πu

36

A. Bücher and J. Segers

In addition, the normalizing sequence (an )n∈N can be chosen as an = 2n
π .
By Theorem 4.2, these facts imply that the theoretical bias and variance of α̂n are given by
Bias = −A(arn )

12
6
b1 (|ρ|) = 2
,
π2
π (1 + 2rn )

Variance =

1 π2
.
kn 6

In particular, the mean squared error is of the order O(1/rn2 ) + O(1/kn ), which can be minimized
by balancing the block size rn and the number of blocks kn , that is, by choosing rn = O(n1/3 )
2
12
and kn = O(n2/3 ) so that rn2 ≈ kn . More precisely, the equations n = kr and ( π2 (1+2r)
)2 = k1 π6
2
imply that 864
π 6 n = r(1 + 2r) , which for n = 1 000 implies that r ≈ 6 and k ≈ 174. These values
are quite close to the optimal finite-sample values of r = 4 and k = 250 to be observed in the
upper-left panel of Figure 2.
In Figure 4, we depict results of a Monte-Carlo simulation study on the finite-sample approximation of the theoretical bias, multiplied by r, and of the theoretical variance, multiplied by k.
Three scenarios have been considered:
• fixed number of blocks k = 200 and block sizes r = 4, . . . , 50;
• fixed block size r = 25 and number of blocks k = 40, . . . , 400;
• block sizes r = 8, 9, . . . , 32 and number of blocks k = r2 .
We find that the variance approximation improves with increasing r or k. For the bias approximation to improve, both r and k must increase.

Maximum likelihood estimation for the Fréchet distribution

37

Fixed block size r=25

Theoretical Bias
Empirical Bias
Theoretical Variance
Empirical Variance

0.2

10

20

30

40

50

0.60
0.55

Theoretical Bias
Empirical Bias
Theoretical Variance
Empirical Variance

0.50

0.4

0.65

0.6

0.70

0.8

0.75

1.0

0.80

Fixed number of blocks k=200

50

100

Block size r

150

200

250

300

350

400

Number of blocks k

0.8

Number of blocks k=r^2

0.4

0.5

0.6

0.7

Theoretical Bias
Empirical Bias
Theoretical Variance
Empirical Variance

200

400

600

800

1000

Number of blocks k

Figure 4. Simulation results in the iid Cauchy-model (see Section F). Theoretical bias multiplied with r and
theoretical variance multiplied with k, together with finite-sample approximations based on N = 5 000 simulation
runs. In the upper left picture, the number of blocks is fixed at k = 200; in the upper right picture, the size of the
blocks is fixed at r = 25; in the lower picture, finally, the number of blocks k and the block size r satisfy r 2 = k,
as suggested by (approximately) minimizing the mean squared error.

