ON THE MAXIMUM LIKELIHOOD ESTIMATOR FOR THE
GENERALIZED EXTREME-VALUE DISTRIBUTION

arXiv:1601.05702v3 [] 15 Mar 2017

AXEL BÜCHER AND JOHAN SEGERS
Abstract. The vanilla method in univariate extreme-value theory consists of fitting
the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of
block maxima. Despite claims to the contrary, the asymptotic normality of the maximum likelihood estimator has never been established. In this paper, a formal proof
is given using a general result on the maximum likelihood estimator for parametric
families that are differentiable in quadratic mean but whose supports depend on the
parameter. An interesting side result concerns the (lack of) differentiability in quadratic mean of the GEV family.
Key words. Differentiability in quadratic mean; M-estimator; maximum likelihood;
empirical process; Fisher information; Generalized Extreme-Value distribution; Lipschitz condition; support.

1. Introduction
Asymptotic normality of maximum likelihood estimators in regular parametric models
is a classic subject, but when the support of the distribution depends on the parameter,
the mathematics are not routine. The family of univariate, three-parameter Generalized
Extreme-Value distributions is a case in point. Its cumulative distribution function Gθ
at a parameter vector θ = (γ, µ, σ) ∈ R × R × (0, ∞) is given by
Gγ,µ,σ (x) = Gγ,0,1 ((x − µ)/σ),

x ∈ R,

(1.1)

where

exp(−e−z )





exp −(1 + γz)−1/γ
Gγ,0,1 (z) =

0



1

if
if
if
if

γ
γ
γ
γ

= 0,
6= 0 and 1 + γz > 0,
> 0 and z ≤ −1/γ,
< 0 and z ≥ −1/γ.

The support of Gθ is an interval, Sθ = {x ∈ R : σ + γ(x − µ) > 0}, whose endpoints
depend on θ. The above parameterization, due to von Mises (1936) and Jenkinson
(1955), generalizes and unifies the Fréchet/Gumbel/Weibull trichotomy in Fisher and
Tippett (1928) across all signs of the shape parameter γ.
Fitting a generalized extreme-value distribution to a sample of annual maxima is the
earliest statistical method in extreme-value theory. Various inference procedures have
been explored, including quantile or probability matching (Gumbel, 1958), the probability weighted moment method (Hosking et al., 1985), and the maximum likelihood
Date: March 16, 2017.
1

2

AXEL BÜCHER AND JOHAN SEGERS

method (Prescott and Walden, 1980; Hosking, 1985). The present paper revisits the
maximum likelihood estimator and its asymptotic distribution.
In general, deriving large-sample asymptotics of the maximum likelihood estimator for
a distribution family with varying support is a difficult problem. The classical regularity
conditions of Cramér (1946) are not fulfilled, and the same is true for the weaker Lipschitz
conditions stemming from empirical process theory (van der Vaart, 1998, Theorem 5.39).
Up to our knowledge, a general theory does not exist.
Smith (1985) was the first to consider maximum likelihood estimation in a large class
of non-regular parametric families on the real line. More precisely, he studied densities
which can be written as
f (x; µ, φ) = (x − µ)α−1 g(x − µ; φ),

x ∈ (µ, ∞),

where µ is a location parameter, φ is a parameter vector, α = α(φ) is a smooth function
of φ, and g is a known function. The formulation is general enough to include location
versions of the Weibull, Gamma, Beta and log-Gamma distribution. Depending on
the value of α, Smith (1985) shows that the rate of convergence and the asymptotic
distribution of the maximum likelihood estimator for µ is n−1/2 and normal or faster
than n−1/2 and non-normal, respectively.
The above class of densities, however, does not include the three-parameter Generalized Extreme-Value distribution. Still, after a reparameterization, the formulation does
include the case −1/2 < γ < 0. Smith (1985, p. 88) claims that similar arguments will
work for γ > 0 too, but details are omitted, while the case γ = 0 is not mentioned. It
could be argued that the case γ < 0 treated in Smith (1985) is indeed the most difficult
one. However, we believe that his proof contains a gap which cannot be easily remedied.
Uniformity in certain convergence statements is claimed but not proven, and we doubt
that it can be done with the techniques used in the article. We justify our point of view
at the end of Section 2.
Our contribution consists of a general result on the asymptotic normality of the maximum likelihood estimator for parametric models whose support may depend on the
parameter. We apply the result to the three-parameter GEV family. An interesting side
result concerns the differentiability in quadratic mean of that family; for the Gumbel
case, see Marohn (1994, 2000).
As in Smith (1985), we need to show that certain limit relations hold uniformly over
specific subsets of the parameter space. To do so, we use empirical process machinery
borrowed from van der Vaart (1998). The approach requires exercising control on the
entropy of certain function classes. We obtain such control via carefully formulated
Lipschitz conditions. Checking these conditions for the three-parameter GEV family
turns out to be surprisingly tedious.
The set-up in our paper is that of independent random samples drawn from a distribution within the parametric family itself. For the GEV family, Dombry (2015) considers
the more realistic setting of triangular arrays of block maxima extracted from independent random variables sampled from a distribution in the domain of attraction of a
GEV distribution. He shows consistency of the maximum likelihood estimator, and our
Proposition 3.1 below is not far from being a special case of his Theorem 2. In Bücher
and Segers (2016), we go one step further and also establish asymptotic normality, even

ON THE MLE FOR THE GEV

3

for block maxima extracted from time series. In that paper, however, we do not consider
the full three-parameter GEV family but focus on the two-parameter Fréchet sub-family.
The support of the latter is equal to the positive half-line for all values of the parameter
vector. The complications that motivate the present paper do therefore not arise.
Asymptotic normality of the maximum likelihood estimator in general parametric
families with parameter-dependent support is asserted in Section 2. The proof is given in
Appendix A. We specialize the theory to the GEV family in Section 3 and we provide an
outline for possible applications to other parametric families in Section 4. The reasonings
and calculations needed to work out the results for the GEV family are sufficiently
complicated to fill Appendices B to E.
2. Maximum likelihood estimator of a support-determining parameter
Let (Pθ : θ ∈ Θ) be a family of distributions on some measurable space (X , A), where
Θ ⊂ Rk . Suppose that each Pθ has a density pθ with respect to some common dominating
measure µ. The model (Pθ : θ ∈ Θ) is said to be differentiable in quadratic mean at an
inner point θ0 ∈ Θ if there exists a measurable function `˙θ0 : X → Rk such that
Z n
1
√
√ o2
√
pθ0 +h − pθ0 − hT `˙θ0 pθ0 dµ = o(khk2 ),
h → 0.
(2.1)
2
X
The function `˙θ0 is referred to as the score vector. Its components are square-integrable
with respect to Pθ0 and have mean zero. The covariance matrix Iθ0 = Pθ0 `˙θ0 `˙Tθ0 is called
Fisher information matrix. Differentiability in quadratic mean with invertible Fisher
information matrix implies an asymptotic expansion of the log-likelihood ratio statistic
implying local asymptotic normality of the associated sequence of statistical experiments.
For more on the statistical implications of this property, see for instance Le Cam (1986)
and van der Vaart (1998, Chapters 7–8). In Marohn (2000), local asymptotic normality
is exploited to construct asymptotically efficient tests of the Gumbel hypothesis.
Let X1 , . . . , Xn be an i.i.d. sample from
P Pθ0 . By definition, any global maximizer of
the R̄-valued function θ 7→ Pn log pθ = ni=1 log pθ (Xi ) is called a maximum likelihood
estimator. Here, it is implicitly assumed that the set of global maximizers is nonempty. This is easily satisfied in most situations where Θ is compact. Otherwise, a
compactification argument often works (van der Vaart, 1998, Chapter 5) or one can
restrict attention to local maximizers instead.
√
).
Under “regularity conditions”, the asymptotic distribution of n(θ̂n −θ0 ) is Nk (0, Iθ−1
0
Various sets of sufficient conditions have been proposed in the literature: Cramér’s classical conditions require the existence of and bounds on the third-order derivatives of
θ 7→ `θ (x) = log pθ (x). Theorem 5.39 in van der Vaart (1998) only demands differentiability in quadratic mean plus a Lipschitz condition on θ 7→ `θ (x), for all x in the support
of Pθ0 and all θ in a neighbourhood of θ0 .
However, if the support, {x : pθ (x) > 0}, of Pθ depends on θ, an approach based
on smoothness of θ 7→ `θ (x) is not possible. Indeed, for any neighbourhood of θ0 , we
may find θ in that neighbourhood and x ∈ X such that pθ0 (x) > 0 but pθ (x) = 0
and thus `θ (x) = −∞. For the same reason, empirical processes indexed by θ in a
neighbourhood of θ0 will have unbounded trajectories with probability tending to one.

4

AXEL BÜCHER AND JOHAN SEGERS

But weak convergence of such empirical processes in the space of bounded functions is a
crucial ingredient in the proofs of many theorems on the asymptotics of M-estimators.
A possible way to circumvent these problems consists of replacing the criterion function `θ by


pθ + pθ0
,
(2.2)
mθ = 2 log
2pθ0
a real-valued function on {x : pθ0 (x) > 0} satisfying mθ (x) ≥ −2 log(2) for all such x.
Using Corollary
5.53 in van der Vaart (1998), the function mθ can be used to obtain
√
the OP (1/ n) rate of convergence of the maximum likelihood estimator for the GEV
parameter θ, see Proposition D.1 below. The method of proof does not allow to obtain
the asymptotic distribution of the estimator,
however.
√
Yet, if the rate of convergence OP (1/ n) has been established, then our next proposition provides alternative conditions (on `θ ) which can be used to prove asymptotic
normality of the maximum likelihood estimator. In doing so, we follow the well-known
three-step strategy for deriving the asymptotic distribution of M-estimators (see, e.g.,
van de Geer, 2009): (1) prove consistency, then (2) derive the rate of convergence and
finally (3) derive the exact limiting distribution. In Section 3, the three steps are worked
out for the GEV family by an application of the following proposition for step (3).
The support of Pθ is Sθ = {x : pθ (x) > 0}, a subset of X which may vary with θ ∈ Θ.
Let Uε (θ) = {θ0 ∈ Θ : kθ − θ0 k < ε} and put
\
S̄(ε) = S̄(θ0 , ε) =
Sθ = {x : pθ (x) > 0 for all θ such that kθ − θ0 k < ε}.
θ∈Uε (θ0 )

The set S̄(ε) is the intersection of the supports of the distributions Pθ for all θ in an
ε-ball centered at θ0 . The following proposition claims the asymptotic normality of the
maximum likelihood estimator under a Lipschitz condition on the function θ 7→ `θ (x)
for x and θ in a certain range.
Proposition 2.1. Let (Pθ : θ ∈ Θ) be a parametric model on (X , A), with Θ ⊂ Rk ,
and let θ0 be an inner point of Θ. Let X1 , X2 , . . . be a sequence of independent and
identically distributed random elements in X with common distribution Pθ0 and let θ̂n be
a maximum likelihood estimator based on X1 , . . . , Xn . Assume the following conditions:
(a) The model is differentiable in quadratic mean at θ0 with score vector `˙θ0 and nonsingular Fisher information matrix Iθ0 .
(b) We have

Pθ0 X \ S̄(ε) = o(ε2 ),
ε ↓ 0.
(2.3)
(c) There exist `˙ ∈ L2 (Pθ0 ) and ε0 > 0 such that, for every 0 < ε < ε0 ,
˙
|`θ1 (x) − `θ2 (x)| ≤ `(x)kθ
for all x ∈ S̄(2ε) and θ1 , θ2 ∈ Uε (θ0 ).
1 − θ2 k
√
(d) We have n(θ̂n − θ0 ) = OP (1) as n → ∞.
Then
n
X
√
−1 1
√
n(θ̂n − θ0 ) = Iθ0
`˙θ0 (Xi ) + oP (1)
Nk (0, Iθ−1
),
n → ∞.
0
n
i=1

(2.4)

(2.5)

ON THE MLE FOR THE GEV

5

Condition (b) controls the size of that part of the support of Pθ0 that is not contained
in the support of Pθ for some θ in a neighbourhood of θ0 . Condition (c) is a Lipschitz
condition on θ 7→ `θ (x) in which the range of x and θ is specified in such a way that
pθ0 (x) > 0 for all θ0 in a neighbourhood of θ. In view of (b) and (d), the part of the
range of x that has been left out is asymptotically negligible.
The proof of Proposition 2.1 adapts arguments from the proofs of Theorems 5.23, 5.39
and 7.2 and Lemma 19.31 in van der Vaart (1998). It is given in detail in Appendix A.

The role of uniformity. Theorem 3(i) in Smith (1985) states the asymptotic normality
of the maximum likelihood estimator for a certain class of real-valued parametric families
whose supports depend in a smooth way on the parameter. As our paper may have the
appearance of needlessly repeating known results, we feel obliged to explain our motives.
The matter is not easy to explain, and we invite the reader to consult Smith’s article
while reading this remark. The pages in Smith (1985) we will be needing are 71, 75–76,
80, and 82, and it will be convenient to discuss them in more or less reverse order.
The proof of Theorem 3(i), stated on page 80, comprises only three lines at the bottom
of page 82. The heart of the argument is the claim that the second-order derivatives of
the log-likelihood are asymptotically constant in a sequence of shrinking neighbourhoods
of the true parameter. We agree with the author that if the claim on the second-order
derivatives is true, then the asymptotic normality can be proven too, via a second-order
Taylor expansion of the log-likelihood around the true parameter. Another good example
of this technique is the proof of Theorem 5.41 in van der Vaart (1998).
As a justification of the assertion on the asymptotic constancy of the second-order
derivatives of the log-likelihood, the author refers to the proof of his Theorem 1. Studying
the proof of that theorem, we find the assertion in Lemma 4, items I(i), II(i), and III,
on page 75. Only for item I(i), a proof is given (page 76), while the proof of the other
items is said to be similar. The proof of item I(i) in Lemma 4 relies on Assumption 7
on page 71. This assumption concerns the L1 -continuity of the second-order derivatives
of the log-likelihood of a single observation as a function of the parameter vector. It is
here that we wish to formulate an objection.
After careful reflection, we are convinced that Assumption 7 is insufficient to conclude
that the supremum of the sample averages on line 4 of page 76 converges to zero. Indeed,
the expectation of the supremum of a collection of random variables is in general larger
than the supremum of the expectations of those random variables. In fact, the difference
is fundamental and lies at the heart of what makes proving uniform laws of large numbers
and uniform central limit theorems so challenging. Controlling expectations of suprema
is the topic of maximal inequalities. In van der Vaart (1998, Section 19.6), such control
is exercised by bracketing integrals of the function classes over which suprema are to be
taken.
Bracketing integrals appear in the proof of our Proposition 3.3. We find bounds
on such integrals by relying on the Donsker theory in Chapter 19 in van der Vaart
(1998). The complexity of the function classes under consideration is tempered by
means of uniform Lipschitz conditions. This explains the appearance of our Lipschitz
condition (2.4).

6

AXEL BÜCHER AND JOHAN SEGERS

3. Maximum likelihood estimator for the GEV family
Let X1 , . . . , Xn be an independent random sample from the Generalized ExtremeValue distribution Gθ0 in (1.1) with unknown parameter θ0 = (γ0 , µ0 , σ0 ), to be estimated. Expressions for the log-density `θ (x) = log pθ (x) and the score vector `˙θ (x) =
(∂γ `θ (x), ∂µ `θ (x), ∂σ `θ (x))T are given in Appendix B.
Consider the maximum
likelihood estimator, θ̂n , defined by maximizing the logP
likelihood θ 7→ n−1 ni=1 `θ (Xi ) = Pn `θ . As argued by Dombry (2015), it is not guaranteed that the log-likelihood attains a unique, global maximum. For that reason, he
rather defines any local maximizer of the log-likelihood over Θ0 = R × R × (0, ∞) as
a maximum likelihood estimator. His main result then states that, for block maxima
constructed from an underlying i.i.d. series, one can always find a strongly consistent
local maximizer, as long as γ0 is strictly larger than −1. Going into his proofs, we see
how that local maximizer is constructed: by restricting the parameter space to some arbitrary compact set Θ ⊂ (−1, ∞) × R × (0, ∞) which contains θ0 as an interior point, it
is guaranteed that the [−∞, ∞)-valued, continuous function θ 7→ Pn `θ attains its global
maximum over Θ. That global maximum is then shown to be a local maximum over
Θ0 = R × R × (0, ∞), eventually, and to be strongly consistent.
In the subsequent parts of this paper, we could also work along Dombry’s lines, redefining maximum likelihood estimators as local rather than global maxima. However,
we prefer to keep track of the above-mentioned construction within his and our proofs.
For the following proposition concerning consistency, we therefore consider the restriction
of the parameter space to an arbitrarily large compact set Θ ⊂ (−1, ∞)×R×(0, ∞) that
contains the true value θ0 ∈ (−1, ∞) × R × (0, ∞) in its interior. In practice, maximizers
produced by numerical algorithms are local maximizers anyway, so that the restriction
of the parameter space to an arbitrarily large compact set is hardly a restriction, at least
if γ0 is known to be larger than −1; see Dombry (2015, Remark 4) for the case γ0 ≤ −1.
Proposition 3.1 (Consistency). Let X1 , X2 , . . . be an independent random sample from
the Gθ0 distribution, with θ0 = (γ0 , µ0 , σ0 ) ∈ (−1, ∞) × R × (0, ∞). For any compact
set Θ ⊂ (−1, ∞) × R × (0, ∞) such that θ0 is in the interior of Θ and for any estimator
sequence θ̂n such that Pn `θ̂n = maxθ∈Θ Pn `θ , such maximizers always existing, we have
θ̂n → θ0 almost surely as n → ∞.
Proof of Proposition 3.1. Since an extreme-value distribution is in its own domain of
attraction, Proposition 3.1 is in fact a combination of Theorem 2 and Proposition 2
in Dombry (2015) with block size sequence m(n) ≡ 1 and with am = σ0 and bm =
µ0 . However, such a choice for m(n) is prohibited in the cited theorem because of its
condition (5), requiring that m(n)/ log n → ∞ as n → ∞. Therefore, we need to pass
through the proof of the theorem and adapt where necessary.
First, the hypothesis m(n) → ∞ is used in Lemmas 2 and 3 to apply the domain-ofattraction condition that F m (am x + bm ) → Ḡγ0 (x) as m → ∞, where Ḡγ = Gγ,0,1 . But
in our case, F = Gγ0 ,µ0 ,σ0 , and thus F (σ0 x + µ0 ) = Ḡγ0 (x) already, without having to
pass through the limit.
Further, as explained in Remark 3 in Dombry (2015), the condition m(n)/ log n → ∞
appears only in the proof of Lemma 4 in the same paper. Writing `¯γ = `γ,0,1 and

ON THE MLE FOR THE GEV

Zi = (Xi − µ0 )/σ0 , the claim of that lemma is that
Z
n
1 X¯
`¯γ0 (z) dḠγ0 (z),
lim
`γ0 (Zi ) =
n→∞ n
R

7

almost surely.

i=1

But the random variables Z1 , Z2 , . . . are independent and identically distributed with
common distribution Ḡγ0 , so that the claim follows from the strong law of large numbers. Note that the random variables Ei = (1 + γZi )−1/γ have a unit Exponential
distribution and that `¯γ0 (Zi ) = (1 + γ) log(Ei ) − Ei , the absolute value of which has a
finite expectation.

To prove the asymptotic normality of the maximum likelihood estimator, we apply
Proposition 2.1. To that end, we need to check a number of conditions, one of which,
differentiability in quadratic mean, is interesting in its own right.
Proposition 3.2 (Differentiability in quadratic mean). The three-parameter GEV family {G(γ,µ,σ) : (γ, µ, σ) ∈ Θ0 = R × R × (0, ∞)} is differentiable in quadratic mean at
θ0 = (γ0 , µ0 , σ0 ) if and only if γ0 > −1/2. In that case, the score vector `˙θ0 (x) is equal
to the gradient of the map θ 7→ `θ (x) at θ = θ0 for x such that σ0 + γ0 (x − µ0 ) > 0 and
equal to 0 otherwise.
The proof of Proposition 3.2 requires showing (2.1). For every real x such that σ0 +
γ0 (x − µ0 ) 6= 0, the order of thepintegrand in (2.1) is o(khk2 ) as h → ∞ due to the
differentiability of the map θ 7→ pθ (x) at θ = θ0 for such x. This asymptotic relation
is true pointwise in x, and it remains to be shown that it can be integrated over x. The
most delicate case occurs when −1/2 < γ0 ≤ −1/3 because in that case, the first-order
√
partial derivatives of the map θ 7→ pθ (x) are not continuous at the boundary situation
σ + γ(x − µ) = 0. A detailed proof is given in Appendix C.
Proposition 3.3 (Asymptotic normality). Let X1 , X2 , . . . be independent and identically distributed random variables with common GEV law Gθ0 , with θ0 = (γ0 , µ0 , σ0 ) ∈
(−1/2, ∞)×R×(0, ∞). Then, for any compact parameter set Θ ⊂ (−1/2, ∞)×R×(0, ∞),
any sequence of maximum likelihood estimators over Θ is strongly consistent and asymptotically normal:
√

n(θ̂n − θ0 ) =

1
√
Iθ−1
0

n

n
X

`˙θ0 (Xi ) + oP (1)

N3 (0, Iθ−1
),
0

n → ∞.

i=1

Proof. Strong consistency follows from Proposition 3.1. The proof of Proposition 3.3
therefore consists of checking the four conditions (a)–(d) of Proposition 2.1. Property
(a) is just Proposition 3.2. The rate of convergence (d) is the topic of Proposition D.1,
the proof of which consists of an application of Corollary 5.53 in van der Vaart (1998) to
the modified criterion function mθ in (2.2); a Lipschitz property of the latter function is
established in Lemma D.3. The remaining points (b) and (c) are treated in Appendix E:
the condition (b) on the support is given in Lemma E.1, while the Lipschitz condition
(c) is given in Lemma E.2.

Remark 3.4. If γ0 = −1/2, then (d/dx)pθ (x) converges to a negative constant as x
increases to the upper endpoint of the support, and the results in Woodroofe (1972)

8

AXEL BÜCHER AND JOHAN SEGERS

suggest that the maximum likelihood estimator is still asymptotically Normal, but at a
rate faster than n−1/2 ; see also Smith (1985, Theorem 3(ii)). For −1 < γ0 < −1/2, the
results in Woodroofe (1974) suggest that the maximum likelihood estimator converges
weakly to a certain non-Gaussian limit and at a rate depending on γ0 ; see also Smith
(1985, Theorem 3(iii)). Since at γ0 ≤ −1/2, the GEV family is not differentiable in
quadratic mean, our Proposition 2.1 does not apply, and we do not pursue the matter
further.
4. Maximum likelihood estimators for other parametric families
The result of Proposition 2.1 is general enough to be applicable for a wide variety of
parametric families with support depending on the parameter. Deriving all the details
may be quite cumbersome and is beyond the scope of the present paper but, nevertheless, we would like to review a couple of parametric families on the real line where
Proposition 2.1 may come in handy.
The Generalized Pareto distribution with parameter θ = (γ, σ) ∈ R × (0, ∞) is defined
by the cdf

1 − e−x/σ
if γ = 0 and x > 0,



1 − (1 + γx/σ)−1/γ if γ 6= 0 and x ∈ S ,
θ
Gθ (x) =

0
if
x
≤
0,



1
if γ < 0 and x ≥ −σ/γ,
where Sθ = (0, ∞) for γ ≥ 0 and Sθ = (0, −σ/γ) for γ < 0. Using the notation in (B.1)
below, its density can be written as
pθ (x) = σ −1 uγ (x/σ)1+γ 1(x > 0, σ + γx > 0),

x ∈ R,

a function which is similar to but simpler than the one of the GEV family. Along
similar lines as for that family, the results from Appendix B can be used to derive
differentiability in quadratic mean at any point (γ0 , σ0 ) ∈ (−1/2, ∞) × (0, ∞) (see also
Marohn, 1994, for the case γ0 = 0). Also, the Lipschitz condition on `θ (Condition (c)
of Proposition 2.1) and on mθ (needed to prove Condition (d) of Proposition 2.1 via an
application of Corollary 5.53 in van der Vaart, 1998; see also the proof of Proposition D.1
below) can be checked similarly to how we proceeded for the GEV family. Asymptotic
normality of the MLE based on high-threshold exceedances was proved with different
techniques in Drees et al. (2004).
Following Smith (1985), we may also consider parametric families of densities on the
real line of the form
pθ (x) = (x − µ)α(φ)−1 g(x − µ; φ),

x ∈ (µ, ∞),

where θ = (µ, φ) ∈ Θ = R × Φ ⊂ R1+q are parameters and where g : (0, ∞) × Φ → [0, ∞)
and α : Φ → (0, ∞) are known, smooth functions. Among other families (see Smith,
1985 for the three-parameter gamma, beta and log-gamma distributions), we obtain for
instance the three-parameter Weibull family by choosing
Φ = (0, ∞)2 ,

g(x; φ1 , φ2 ) = φ1 φ2 exp(−φ2 xφ1 ),

α(φ1 , φ2 ) = φ1 .

ON THE MLE FOR THE GEV

9

Under regularity conditions on α and g, Proposition 2.1 can be applied on the restricted
parameter set Θ2 = R × Φ2 = R × {φ : α(φ) > 2}; this corresponds to case (i) of
Theorem 3 of Smith (1985).
Consider Condition (b) of Proposition 2.1. In Smith (1985), it is assumed that
c(φ) = lim g(x; φ)/α(φ) ∈ (0, ∞)
x↓0

exists (note that c(φ) = φ2 > 0 for the Weibull family), which implies that g(x; φ0 ) may
be bounded by a constant K0 for sufficiently small x. As a consequence, Condition (b)
in Proposition 2.1 is met: clearly, Sθ = (µ, ∞), so that S̄(ε) = (µ0 + ε, ∞), and thus,
since α0 = α(φ0 ) > 2,
Z µ0 +ε

(x − µ0 )α0 −1 g(x − µ0 ; φ0 ) dx
Pθ0 R \ S̄(ε) =
µ0
Z ε
y α0 −1 dy = O(εα0 ) = o(ε2 ),
ε ↓ 0.
≤ K0
0

Let us sketch how to arrive at the other conditions of Proposition 2.1. Since
`θ (x) = {α(φ) − 1} log(x − µ) + log g(x − µ; φ),
Condition (c) of Proposition 2.1 can obviously be deduced from smoothness properties
of the map φ 7→ α(φ) on Φ2 and of the map (x, φ) 7→ h(x, φ) = log g(x; φ) on (0, ∞)×Φ2 .
Note that
α(φ) − 1
|∂µ `θ (x)| ≤
+ |∂x h(x − µ; φ)|
x−µ
and
|∂φ `θ (x)| ≤ |∂φ α(φ) log(x − µ)| + |∂φ h(x − µ; φ)|,
showing that the inequality α(φ) > 2 must be used again to control the integrals of
the squared score functions near the lower endpoint. Condition (a) of Proposition 2.1
can be conveniently shown by using Lemma 7.6 in van der Vaart (1998): if the Fisher
information Iθ = Pθ `˙θ `˙Tθ exists and ispcontinuous in θ, we only need to show that, for
each x ∈ R, the function θ 7→ sθ (x) = pθ (x) on Θ2 is continuously differentiable. Using
the relations ∂µ sθ (x) = ∂φ sθ (x) = 0 for x < µ and the identities
∂µ sθ (x) =

1
sθ (x) ∂µ `θ (x),
2

∂φ sθ (x) =

1
sθ (x) ∂φ `θ (x),
2

for x > µ, this follows from continuous differentiability of α and h, provided that α(φ) >
3. For 2 < α(φ) ≤ 3, ∂µ sθ (x) will not converge to 0 as x ↓ µ; this is similar to the case
−1/2 < γ ≤ −1/3 for the GEV family and may require more sophisticated arguments
(see the proof of Proposition 3.2). Finally, Condition (d) of Proposition 2.1 may be
deduced from Corollary 5.53 in van der Vaart (1998). By following the arguments that
lead to (D.3) in Lemma D.3 (with a = 1/2), the necessary Lipschitz condition on mθ
can again be conveniently reformulated in terms of α and h.

10

AXEL BÜCHER AND JOHAN SEGERS

Appendix A. Proof of Proposition 2.1
For 0 < ε < ε0 and θ ∈ Uε (θ0 ), define a real-valued function `θ,ε on Sθ0 by
`θ,ε (x) = `θ (x) 1{x ∈ S̄(2ε)}.
The Lipschitz condition (2.4) on ` implies that
˙ kθ1 − θ2 k
|`θ1 ,ε (x) − `θ2 ,ε (x)| ≤ `(x)

(A.1)

for all θ1 and θ2 in Uε (θ0 ) and for all x ∈ Sθ0 .
We claim that, for any C > 0, we have
Pn `θ̂n = sup Pn `θ ≥
θ∈Θ

sup

√

Pn `θ ≥

kθ−θ0 k<C/ n

sup

√

kθ−θ0 k<C/ n

Pn `θ,C/√n − oP (1/n).

(A.2)

√
Only the last inequality is non-trivial. Write, for an arbitrary θ with kθ − θ0 k < C/ n,
√
Pn `θ = Pn `θ,C/√n + Pn `θ 1{ · ∈ Sθ0 \ S̄(2C/ n)}
√
sup √ Pn `θ0 1{ · ∈ Sθ0 \ S̄(2C/ n)} .
≥ Pn `θ,C/√n −
kθ0 −θ0 k<C/ n

The supremum on the second line is oP (rn ) for any sequence rn ↓ 0: for all η > 0,
#
"
√
P rn−1
sup √ Pn `θ0 1{ · ∈ Sθ0 \ S̄(2C/ n)} > η
kθ0 −θ0 k<C/ n

√
≤ P{∃ i = 1, . . . , n : Xi ∈ X \ S̄(2C/ n)}
√ 
≤ nPθ0 X \ S̄(2C/ n) = o(1),

n → ∞, (A.3)

by Assumption (2.3). Equation (A.2) has thus been proved.
Now, let us show that, for fixed C > 0 and all converging sequences hn → h ∈ Rk
with khn k < C for all n, we have
√
n(`θ0 +hn /√n,C/√n − `θ0 ,C/√n ) → hT `˙θ0
in L2 (Pθ0 ).
(A.4)
For that purpose, write the left-hand side as
√
√
√
An1 − An2 = n(`θ0 +hn /√n − `θ0 ) − n(`θ0 +hn /√n − `θ0 ) 1{ · ∈ Sθ0 \ S̄(2C/ n)}.
Because of differentiability in quadratic mean, Assumption (a), the term An1 converges
to hT `˙θ0 in Pθ0 -probability; see the proof of Theorem 5.39 in van der Vaart (1998).
Moreover, An2 converges to zero in Pθ0 -probability: for all η > 0, we have
√
Pθ0 (|An2 | > η) ≤ Pθ0 {Sθ0 \ S̄(2C/ n)} = o(1),
n → ∞.
Hence, the convergence in (A.4) holds in Pθ0 -probability. In view of the Lipschitzproperty of `θ,C/√n in (A.1) and the fact that Pθ0 `˙2 < ∞ by Assumption (c), convergence in Pθ0 -probability can be strengthened to L2 (Pθ0 ) convergence by applying the
dominated convergence theorem. Indeed, for any subsequence of the right hand-side
of (A.4), we may choose a sub-subsequence along which the convergence holds almost
surely. The dominated convergence theorem implies convergence in L2 (Pθ0 ) along that
sub-subsequence. The claim follows since the subsequence we have started with was
arbitrary.

ON THE MLE FOR THE GEV

11

Next, we shall show that, for any converging sequence hn → h with khn k < C for all n,

1
n Pθ0 `θ0 +hn /√n,C/√n − `θ0 ,C/√n → − hT Iθ0 h,
n → ∞.
(A.5)
2
√
Recall the empirical process Gn = n(Pn −Pθ0 ). In view of (A.4), computing means
and variances, we have
√
√
√ −`
√ ) − hT `˙ } = o (1),
Bn (hn ) = Gn { n(`
n → ∞. (A.6)
P
θ
θ0 +hn / n,C/ n

θ0 ,C/ n

0

We can rewrite Bn as
Bn (hn ) = n Pn (`θ0 +hn /√n,C/√n − `θ0 ,C/√n ) − Gn hT `˙θ0
− nPθ0 (`θ0 +hn /√n,C/√n − `θ0 ,C/√n )
= n Pn (`θ0 +hn /√n − `θ0 ) − Gn hT `˙θ0

√
− n Pn (`θ0 +hn /√n − `θ0 ) 1{ · ∈ X \ S̄(2C/ n)}

− nPθ0 (`θ0 +hn /√n,C/√n − `θ0 ,C/√n ).
P
n
√
Note that n Pn (`θ0 +hn /√n − `θ0 ) =
i=1 log(pθ0 +hn / n /pθ0 )(Xi ) is a likelihood ratio
statistic. Differentiability in quadratic mean, Assumption (a), implies
1
n Pn (`θ0 +hn /√n − `θ0 ) − Gn hT `˙θ0 = − hT Iθ0 h + oP (1),
n → ∞;
2
see van der Vaart (1998, Theorem 7.2). Moreover, by Assumption (b),
h
i
√
P n Pn (`θ0 +hn /√n − `θ0 ) 1{ · ∈ X \ S̄(2C/ n)} ≥ η
√ 
≤ n Pθ0 X \ S̄(2C/ n) = o(1),
n → ∞.
The last four displays imply (A.5).
We can now follow the lines of the proof of Theorem 5.23 in van der Vaart (1998)
to prove the following reinforcement of (A.6): for any random sequence h̃n such that
kh̃n k < C almost surely for all n, we have
Bn (h̃n ) = oP (1),

n → ∞.

(A.7)

To see this, note that it follows from (A.6) that Bn (h) = oP (1) for any fixed h ∈ Rk with
khk < C. As a consequence, the finite-dimensional distributions of the stochastic process
Bn converge indeed to 0 in probability. It remains to show asymptotic tightness of the
sequence Bn in the space `∞ ({h : khk < C}) equipped with the supremum distance;
note that by the Lipschitz property (A.1), the trajectories of Bn are indeed bounded
almost surely. Obviously, the sequence of linear processes h 7→ Gn hT `˙θ0 = hT Gn `˙θ0 is
asymptotically tight, so that it suffices to show the same property for the processes
√
h 7→ Mn (h) = n Gn (`θ0 +h/√n,C/√n − `θ0 ,C/√n ).
This can be done along the lines of the proof of Lemma 19.31 in van der Vaart (1998),
relying on a result for empirical processes indexed by sequences of function classes. More
precisely, define a sequence of function classes on Sθ0 through
√
Mn = { n(`θ0 +h/√n,C/√n − `θ0 ,C/√n ) : khk < C}.

12

AXEL BÜCHER AND JOHAN SEGERS

By the Lipschitz property (A.1), the functions in Mn are bounded by the envelope
function C `˙ ∈ L2 (Pθ0 ). From Example 19.7 in van der Vaart (1998), we obtain the
following bound on the bracketing number N[ ] ε, Mn , L2 (Pθ0 )): for some constant K > 0
not depending on n, we have, for all sufficiently small ε > 0,

k

˙ L (P ) ,
N[ ] ε, Mn , L2 (Pθ0 ) ≤ K ε−1 (2C 2 ) k`k
2
θ0
where k denotes the dimension of the Euclidean space of which Θ is a subset; here we
used the property that the diameter of the ball {h ∈ Rk : khk < C} is equal to 2C. As
a consequence, if δn ↓ 0, the bracketing integrals converge to zero:
Z δn q


log N[ ] ε, Mn , L2 (Pθ0 ) dε → 0,
n → ∞.
J[ ] δn , Mn , L2 (Pθ0 ) =
0

˙ 2 1{C `˙ > ε√n} → 0 (n → ∞) is satisfied because the
The Lindeberg condition Pθ0 (C `)
envelope function C `˙ belongs to L2 (Pθ0 ). Asymptotic tightness of Bn then follows from
Theorem 19.28 in van der Vaart (1998). Equation (A.7) is thus proven.
To complete the proof of Proposition 2.1, we can now proceed similarly to the proof
of Theorem 5.23 in van der Vaart (1998), with some additional effort needed to get rid
of the constant C. In view of (A.5), the convergence property (A.7) can be rewritten as

1
n Pn `θ0 +h̃n /√n,C/√n − `θ0 ,C/√n = − h̃Tn Iθ0 h̃n + h̃Tn Gn `˙θ0 + oP (1),
2

n → ∞, (A.8)

where h̃n denotes an arbitrary random sequence in Rk with kh̃n k < C.
√
Gn `˙θ0 . For C > 0, let An,C denote the event
Define ĥn1 = n(θ̂n − θ0 ) and ĥn2 = Iθ−1
0
{max(kĥn1 k, kĥn2 k) < C}, and set h̃nj = ĥnj 1(An,C ), for j ∈ {1, 2}. Inserting both
tilde-expressions into (A.8), we get, as n → ∞,
1
n Pn (`θ0 +h̃n1 /√n,C/√n − `θ0 ,C/√n ) = − h̃Tn1 Iθ0 h̃n1 + h̃Tn1 Gn `˙θ0 + oP (1),
2
1
Gn `˙θ0 1(An,C ) + oP (1).
n Pn (`θ0 +h̃n2 /√n,C/√n − `θ0 ,C/√n ) = Gn `˙Tθ0 Iθ−1
0
2
Subtracting the second equation from the first one yields
n Pn (`θ0 +h̃n1 /√n,C/√n − `θ0 +h̃n2 /√n,C/√n ) = −Qn 1(An,C ) + oP (1),


1
˙θ T Iθ ĥn1 − I −1 Gn `˙θ .
where Qn = ĥn1 − Iθ−1
G
`
n
0
0
0
θ0
0
2
We will show that
Qn = oP (1),
n → ∞.

n → ∞,

(A.9)

Since the Fisher information matrix Iθ0 is positive definite, this will imply that
n
√
1 X˙
√
n(θ̂n − θ0 ) − Iθ−1
`θ0 (Xi ) = ĥn1 − Iθ−1
Gn `˙θ0 = oP (1),
0
0
n

n → ∞,

i=1

which is the first part of (2.5). The asymptotic normality then follows from the multivariate central limit theorem.

ON THE MLE FOR THE GEV

13

It remains to show (A.9). Note that Qn ≥ 0, since Iθ0 is positive definite. By the
maximization property (A.2), we have, as n → ∞,
Rn = n Pn `θ̂n ≥ n Pn `θ0 +h̃n2 /√n,C/√n − oP (1)
= n Pn `θ0 +h̃n1 /√n,C/√n + Qn 1(An,C ) − oP (1).
Isolating Qn 1(An,C ), we find
0 ≤ Qn = Qn 1(An,C ) + Qn 1(Acn,C )
≤ Rn − n Pn `θ0 +h̃n1 /√n,C/√n + Qn 1(Acn,C ) + oP (1),
n → ∞.
√
Note that `θ̂n (x) 1{x ∈ S̄(2C/ n)} 1(An,C ) = `θ0 +h̃n1 /√n,C/√n (x) 1(An,C ). Therefore,
Rn = Rn 1(An,C ) + Rn 1(Acn,C )

√
√ 
= n Pn `θ̂n 1{ · ∈ S̄(2C/ n)} + n Pn `θ̂n 1{ · ∈
/ S̄(2C/ n)} 1(An,C ) + Rn 1(Acn,C )
= n Pn `θ0 +h̃n1 /√n,C/√n 1(An,C ) + oP (1) + Rn 1(Acn,C ),

n → ∞;

the oP (1) term appears because of the argument in (A.3). Substituting the expansion
for Rn into the upper bound for Qn yields, as n → ∞,
0 ≤ Qn ≤ n Pn `θ0 +h̃n1 /√n,C/√n 1(An,C ) + Rn 1(Acn,C )
− n Pn `θ0 +h̃n1 /√n,C/√n + Qn 1(Acn,C ) + oP (1)
= {Rn + Qn − n Pn `θ0 +h̃n1 /√n,C/√n } 1(Acn,C ) + oP (1).
Hence, for any ε > 0,

lim sup P(|Qn | > ε) ≤ lim sup P Acn,C ,
n→∞

n→∞

which can be made arbitrary small by increasing C, using Assumption (d). This finishes
the proof of (A.9) and thus of Proposition 2.1.

Appendix B. Density and score functions of the GEV family
The density, log-density, score functions and Fisher information matrix of the GEV
family can of course be found in many articles and textbooks. Here we present some
of these objects in a form which is convenient for later analysis. In the notation of
Section 2, the state space (X , A, µ) is the real line equipped with its Borel sets and the
Lebesgue measure. The probability density function of the GEV distribution Pθ with
parameter θ = (γ, µ, σ) ∈ R × R × (0, ∞) is given by
pθ (x) = σ −1 e−u uγ+1 1(1 + γz > 0),

x ∈ R,

where
x−µ
,
σ
 Z
u = uγ (z) = exp −
z = zµ,σ (x) =

0

z

1
dt
1 + γt



(
(1 + γz)−1/γ
=
e−z

if γ 6= 0,
if γ = 0.

(B.1)

The expression uγ (z) is convex and decreasing in z and is increasing in γ. See Figure 1
for graphs of the functions z 7→ uγ (z) for γ ∈ {−0.5, 0, 0.5}.

14

AXEL BÜCHER AND JOHAN SEGERS
uγ(z) = (1 + γz)(−1 γ)

uγ(z)

γ = 0.5
γ=0
γ = − 0.5

1
−2

−1

0

1

2

z

Figure 1. Graph of z 7→ uγ (z) in (B.1) for γ ∈ {−0.5, 0, 0.5} and z ∈ [−2, 2].
The support, Sθ , of the distribution is (defined as) the open interval


(−∞, µ − σ/γ)
Sθ = {x : pθ (x) > 0} = {x ∈ R : σ + γ(x − µ) > 0} = R


(µ − σ/γ, ∞)

if γ < 0,
if γ = 0,
if γ > 0.

The log-density is given by
`θ (x) = − log(σ) − u + (γ + 1) log(u),

x ∈ Sθ ,

Rz

with log(u) = − 0 (1 + γt)−1 dt. The partial derivatives of the map θ 7→ `θ (x) at θ such
that pθ (x) > 0 are given by
z
∂γ `θ (x) = (1 − u) ∂γ log(u) −
,
(B.2)
1 + γz
γ+1−u
,
(B.3)
∂µ `θ (x) =
σ(1 + γz)
(1 − u)z − 1
∂σ `θ (x) =
,
(B.4)
σ(1 + γz)
with ∂γ log(u) given by

 
1 1
z


Z z
log(1 + γz) −
if γ 6= 0,

t
γ γ
1 + γz
0 ≤ ∂γ log(u) =
dt =
(B.5)
2
2

0 (1 + γt)

z
if γ = 0.
2
For γ > −1/2, these partial derivatives have expectation zero and finite second moments
with respect to Pθ . For such θ, the Fisher information matrix Iθ ∈ R3×3 is equal to the
covariance matrix of the score vector
`˙θ = (∂γ `θ , ∂µ `θ , ∂σ `θ )T ,
the three entries of which are viewed as elements of L2 (Pθ ). Explicit expressions for Iθ
are given in Prescott and Walden (1980), but we will not be needing those here. The

ON THE MLE FOR THE GEV

15

only properties of Iθ that are relevant for our current study are that Iθ is symmetric,
positive definite and non-singular and that the map θ 7→ Iθ is continuous.
We continue with a number of properties of the GEV densities and score functions.
Lemma B.1. Let γ and z be such that 1 + γz > 0. Let u = uγ (z) be as in (B.1). If
γz > 0, then
 2
z


,



2




z2


(B.6)
∂γ log(u) ≤
,


z
1
1 + γz


log(1 + γz)
≤


γ
1 + γz 

1




log(1 + γz).
γ2
If −1 < γz < 0, then also
z2
.
1 + γz
As a consequence, for any γ and z such that 1 + γz > 0, we have
∂γ log(u) ≤

0 ≤ ∂γ log(u) ≤

z2
.
1 + γz

(B.7)

(B.8)

Proof. If γ = 0 or z = 0, the stated inequalities are clearly satisfied, so suppose that
γ 6= 0 and z 6= 0. Substituting γt = v in (B.5), we find
Z γz
1
v
∂γ log(u) = 2
dv.
γ 0 (1 + v)2
If γz > 0, since v 7→ v/(1 + v) is increasing in v ≥ 0 and v 7→ 1/(1 + v) is decreasing in
v ≥ 0, we obtain the bounds in the first display of the lemma: not only
Z γz
1
z2
∂γ log(u) ≤ 2
v dv = ,
γ 0
2
but also

Z
z
1 γz 1
z2
dv ≤
.
1 + γz γ 0 1 + v
1 + γz
If γz < 0, then we have γz = − |γz| and
Z |γz|
Z
1
v
|z| |γz|
1
|z|2
dv
≤
dv
=
.
∂γ log(u) = 2
γ 0
(1 − v)2
|γ| 0
(1 − v)2
1 − |γz|
∂γ log(u) ≤



Lemma B.2. Let θ and x be such that 1 + γz > 0, where z = (x − µ)/σ. We have
( −1
σ (1 + |γ|) u1+γ if z ≤ 0,
|∂µ `θ (x)| ≤
σ −1 (1 + |γ|) uγ
if z ≥ 0.
Proof. We have (1 + γz)−1 = uγ and thus ∂µ `θ (x) = σ −1 (γ + 1 − u)uγ . If z ≤ 0, then
u ≥ 1, so that |γ + 1 − u| ≤ u + |γ| ≤ (1 + |γ|)u. If z ≥ 0, then 0 < u ≤ 1, so that
|γ + 1 − u| ≤ 1 + |γ|.


16

AXEL BÜCHER AND JOHAN SEGERS

Lemma B.3. Let θ and x be such that 1 + γz > 0, where z = (x − µ)/σ. We have

z+1


if z ≥ 0,
|∂σ `θ (x)| ≤ σ(1 + γz)

σ −1 1 + u log(u) umax(γ,0) if z ≤ 0.
Proof. If z ≥ 0, then 0 < u ≤ 1, so that |(1 − u)z − 1| ≤ z +1, yielding the stated bound.
If z ≤ 0, then u ≥ 1, so that |(1 − u)z − 1| ≤ u |z| + 1. Further, (1 + γz)−1 = uγ as
well as
Z u
Z u
uγ − 1
|z|
γ−1
max(γ,0)
t−1 dt = umax(γ,0) log(u).
(B.9)
t
dt ≤ u
=
=
1 + γz
γ
1
1
u|z|+1
The stated bound now follows from |∂σ `θ (x)| ≤ σ −1 (1+γz)
.



Lemma B.4. Let θ and x be such that 1 + γz > 0, where z = (x − µ)/σ.
If γ ≥ 0 and z ≥ 0, then, with γ −2 log(1 + γz) and γ −1 being defined as +∞ for γ = 0,




 2
1
z 1
.
(B.10)
,
log(1 + γz) , min z,
|∂γ `θ (x)| ≤ max min
2 γ2
γ
If γ ≤ 0 and z ≥ 0, then
|∂γ `θ (x)| ≤

max(z 2 , z)
.
1 + γz

(B.11)

If γ ≥ 0 and z ≤ 0, then
|∂γ `θ (x)| ≤ u1+γ max{(log u)2 , log u}.

(B.12)

|∂γ `θ (x)| ≤ u max{(log u)2 , log u}.

(B.13)

If γ ≤ 0 and z ≤ 0,
Proof. If γ = 0, then u = e−z and |∂γ `θ (x)| = (1−e−z )z 2 /2−z, and all stated inequalities
are satisfied. In the remainder of the proof, we assume therefore that γ 6= 0.
Suppose first that z ≥ 0, so that 0 < u ≤ 1. Since |a − b| ≤ max(a, b) for nonnegative
numbers a, b, we have


z
|∂γ `θ (x)| ≤ max ∂γ log(u),
.
1 + γz
Use (B.6) to find (B.10) and use (B.7) to find (B.11).
Next suppose z ≤ 0, so that u ≥ 1. If γ < 0, then, by (B.6) and (B.9),
1
|z|
∂γ log(u) ≤ (− ) log(1 + γz)
≤ (log u)2 .
γ
1 + γz
We find, again using (B.9), as stated in (B.13), that


|z|
|∂γ `θ (x)| ≤ max u ∂γ log(u),
≤ u max{(log u)2 , log u}.
1 + γz
Finally, suppose z ≤ 0 and γ > 0. By (B.7) and (B.9)

2
2
|z|
z2
≤ u−γ uγ log(u) = uγ (log u)2 .
∂γ log(u) ≤
= (1 + γz)
1 + γz
1 + γz

ON THE MLE FOR THE GEV

We obtain, again using (B.9),

|∂γ `θ (x)| ≤ max u ∂γ log(u),

|z|
1 + γz



17

≤ u1+γ max{(log u)2 , log u},

which is (B.12).



Lemma B.5. Fix a ∈ [1/2, 1). For any x ∈ R, the function θ 7→ paθ (x) is continuously
differentiable on (−a/(1 + a), ∞) × R × (0, ∞) and the partial derivatives are continuous
in (x, θ) ∈ R × (−a/(1 + a), ∞) × R × (0, ∞).
Proof. Fix θ0 = (γ0 , µ0 , σ0 ) with γ0 > −a/(1 + a) and choose x0 ∈ R.
If σ0 + γ0 (x0 − µ0 ) > 0, then also σ + γ(x − µ) > 0 for all (x, θ) in a neighbourhood
of (x0 , θ0 ) and thus paθ (x) = σ −a e−au ua(γ+1) for such (x, θ). All functions arising in
the expression of paθ (x) are continuously differentiable in the three parameters and are
continuous in (x, θ). Since
∂θk paθ (x) = a paθ (x) ∂θk `θ (x),
the formulas for the score function in (B.2), (B.3), and (B.4) imply that


z
a −ua a(γ+1)
a
∂γ pθ (x) = (1 − u) ∂γ log(u) −
e
u
,
1 + γz σ a
γ + 1 − u a −ua a(γ+1)
∂µ paθ (x) =
e
u
,
σ(1 + γz) σ a
(1 − u)z − 1 a −ua a(γ+1)
e
u
.
∂σ paθ (x) =
σ(1 + γz) σ a
If σ0 + γ0 (x0 − µ0 ) < 0, then also σ + γ(x − µ) < 0 for all (x, θ) in a neighbourhood of
(x0 , θ0 ) and thus paθ (x) = 0 for all such (x, θ), whence the partial derivatives vanish too.
The difficult case is σ0 + γ0 (x0 − µ0 ) = 0, that is, if γ0 6= 0 and x0 = µ0 − σ0 /γ0 . We
need to show that, for every k ∈ {1, 2, 3},
lim
(x,θ)→(x0 ,θ0 )
σ+γ(x−µ)>0

∂θk paθ (x) = 0

where (θ1 , θ2 , θ3 ) = (γ, µ, σ). Recall u = (1 + γz)−1/γ with z = (x − µ)/σ.
First, suppose that γ0 > 0. Then u → ∞ as (x, θ) → (x0 , θ0 ) and convergence to zero
is assured by the exponential factor e−au in each of the three partial derivatives.
Second, suppose that γ0 < 0. Then u → 0 as (x, θ) → (x0 , θ0 ). Using the bound in
(B.8), we see that the limit behaviour of the three partial derivatives is dominated by
the factor
− a −a−1
(1 + γz) γ
as 1 + γz → 0. The exponent must be positive eventually: −(a/γ0 ) − a − 1 > 0. But
this is equivalent to −a/(1 + a) < γ0 < 0.

Appendix C. Differentiability in quadratic mean of the GEV family
√
Proof of Proposition 3.2. Let sθ = pθ . We distinguish between three cases: γ0 > −1/3,
−1/2 < γ0 ≤ −1/3, and γ0 ≤ −1/2.

18

AXEL BÜCHER AND JOHAN SEGERS

Case γ0 > −1/3. Lemma B.5 implies that for all x ∈ R, the function θ 7→ sθ (x) is
continuously differentiable in a neighbourhood of θ0 . Differentiability in quadratic mean
then follows from an application of Lemma 7.6 in van der Vaart (1998). Note that the
existence and the continuity of the Fisher information matrix in a neighbourhood of θ0
has been derived in Prescott and Walden (1980).
√
Case −1/2 < γ0 ≤ −1/3. Recall sθ = pθ . The conditions of Lemma 7.6 in van der
Vaart (1998) are no longer fulfilled, but an adaptation of that proof still yields differentiability in quadratic mean.
Since the unit ball in R3 is compact, it suffices to show that, if hn → h in R3 and if
tn ↓ 0 as n → ∞, then
2
Z 
sθ0 +tn hn (x) − sθ0 (x) 1 T ˙
− hn `θ0 (x) sθ0 (x) dx = 0.
lim
n→∞ R
tn
2
It is sufficient to show the same equality with hTn `˙θ0 (x) sθ0 (x) replaced by hT `˙θ0 (x) sθ0 (x);
to see why, use the elementary inequality (a+b)2 ≤ 2a2 +2b2 , the fact that the score vector has Pθ0 -square integrable components, and the assumption that hn → h as n → ∞.
For every x ∈ R except x = µ0 − σ0 /γ0 , the integrand of the resulting integral converges
to zero by differentiability of the map θ 7→ sθ (x) at θ = θ0 . What remains is to show
convergence of the integral itself. To this end, we apply Proposition 2.29 in van der
Vaart (1998) with p = 2; the condition to check is that
2

Z 
Z 
1 T ˙
sθ0 +tn hn (x) − sθ0 (x) 2
h `θ0 (x) sθ0 (x) dx.
(C.1)
dx ≤
lim sup
tn
n→∞
R 2
R
R
The right-hand side in (C.1) is equal to (1/4)hT ( `˙θ0 `˙Tθ0 pθ0 )h = (1/4)hT Iθ0 h. We need
to control the left-hand side.
Write hn = (hn1 , hn2 , hn3 )T and θn = θ0 + tn hn = (γn , µn , σn ). Without loss of
generality, assume that n is large enough such that −1/2 < γn < 0. The upper endpoints
of the supports of the GEV distributions with parameter vectors θ0 , θn , (γ0 , µn , σn ) and
(γn , µ0 , σ0 ) are equal to µ0 − σ0 /γ0 , µn − σn /γn , µn − σn /γ0 and µ0 − σ0 /γn , respectively.
Let αn and βn be the minimum and the maximum of these four endpoints, respectively.
Write
sθ +t h − sθ0
.
fn = 0 n n
tn
Since fn (x) = 0 for x > βn , we have
Z βn
Z
Z αn
2
2
fn (x)2 dx.
fn (x) dx +
fn (x) dx =
R

−∞

αn

We will show that the limit superior of the first term on the right-hand side is bounded
by the right-hand side of (C.1), while the second term on the right-hand side converges
to zero.
To bound the integral of fn (x)2 from −∞ to αn , we can proceed as in the proof of
Lemma 7.6 in van der Vaart (1998). See in particular the display on top of page 96. Each
point x ∈ (−∞, αn ) is an element of the support of sθ for each θ on the line segment connecting θ0 and θn ; this was the reason for introducing the additional parameter vectors
(γ0 , µn , σn ) and (γn , µ0 , σ0 ) in the previous paragraph. Let ṡθ (x) denote the gradient

ON THE MLE FOR THE GEV

19

of the map θ 7→ sθ (x), for x in the support of Pθ . Note that ṡθ (x) = (1/2)sθ (x) `˙θ (x).
Since the map u 7→ sθ0 +utn hn (x), for u ∈ [0, 1], is continuously differentiable, we have
R1
fn (x) = u=0 ṡθ0 +utn hn (x)T hn du. Applying the Cauchy–Schwarz inequality and the
Fubini theorem, we find
Z αn Z 1
Z αn
2
2
ṡθ0 +utn hn (x)T hn du dx
fn (x) dx ≤
−∞

x=−∞
Z 1

≤

1
4

u=0

hTn Iθ0 +utn hn hn du.

u=0

By continuity of the map θ 7→ Iθ , the right-hand side converges to (1/4)hT Iθ0 h.
Rβ
To show that αnn fn (x)2 dx converges to zero, observe that

fn (x)2 ≤ 2t−2
n pθn (x) + pθ0 (x) .
Moreover, there exists c0 > 0 such that αn ≥ βn − c0 tn for all sufficiently large n. As a
consequence, it is sufficient to show that, whenever θ̃n → θ0 and un ↓ 0, we have
lim u−2
n Pθ̃n [ω̃n − un , ω̃n ) = 0,

n→∞

(C.2)

with ω̃n the upper endpoint of the support of Pθ̃n . Writing θ̃n = (γ̃n , µ̃n , σ̃n ), we have
" 
#

µ̃n − σ̃n /γ̃n − un − µ̃n −1/γ̃n
Pθ̃n [ω̃n − un , ω̃n ) = 1 − exp − 1 + γ̃n
σ̃n
n
o
= 1 − exp − (|γ̃n | un /σ̃n )1/|γ̃n |
≤ (|γ̃n | un /σ̃n )1/|γ̃n | .
On the last line, we used the elementary inequality 1 − exp(−z) ≤ z for all z ∈ R. Since
1/ |γ̃n | → 1/ |γ0 | > 2, equation (C.2) follows.
Case γ0 ≤ −1/2. We will show that for h = (0, t, 0)T with t ↓ 0, we have
lim inf t−2 Pθ0 +h {pθ0 = 0} > 0.
t↓0

(C.3)

Here, {pθ0 = 0} is short-hand for {x ∈ R : pθ0 (x) = 0} = [µ0 − σ0 /γ0 , ∞). Restricting
the integral on the left-hand side in (2.1) to the set {pθ0 = 0} then shows that the
asymptotic relation in (2.1) cannot hold.
We show (C.3). The upper endpoint of Pθ0 +h is equal to µ0 +t−σ0 /γ0 , which is larger
than the one of Pθ0 . The mass assigned by Pθ0 +h to the difference of the two supports
is given by
#
" 

(µ0 − σ0 /γ0 ) − µ0 − t −1/γ0
Pθ0 +h [µ0 − σ0 /γ0 , ∞) = 1 − exp − 1 + γ0
σ0
n
o
= 1 − exp − (|γ0 | t/σ0 )1/|γ0 | .
Since 0 < 1/ |γ0 | ≤ 2 and since 1 − exp(−u) = (1 + o(1)) u as u → 0, the inequality (C.3)
follows.


20

AXEL BÜCHER AND JOHAN SEGERS

Appendix D. Rate of convergence
√
To apply Proposition 2.1, the OP (1/ n) rate of convergence of the maximum likelihood estimator needs to be established first.
Proposition D.1 (Rate of convergence). Let X1 , X2 , . . . be independent and identically distributed random variables with common GEV law Gθ0 , with θ0 = (γ0 , µ0 , σ0 ) ∈
(−1/2, ∞)×R×(0, ∞). Then, for any compact parameter set Θ ⊂ (−1/2, ∞)×R×(0, ∞),
√
any sequence of maximum likelihood estimators over Θ satisfies θ̂n − θ0 = OP (1/ n) as
n → ∞.
Proof of Proposition D.1. We apply Corollary 5.53 in van der Vaart (1998) to the function mθ in (2.2). To do so, we need to check a number of things:
• θ̂n = θ0 + oP (1) as n → ∞: this is okay by Proposition 3.1.
• Pn mθ̂n ≥ Pn mθ0 − OP (n−1 ): By concavity of the logarithm, we have


p
Pn mθ̂n ≥ Pn log θ̂n + Pn log 1 ≥ 0 = Pn mθ0 .
pθ0
• There exists ṁ ∈ L2 (Pθ0 ) such that |mθ1 (x) − mθ2 (x)| ≤ ṁ(x)kθ1 − θ2 k for Pθ0 almost all x and all θ1 and θ2 in a neighbourhood of θ: this Lipschitz condition
is the topic of Lemma D.3.
• The map θ 7→ Pθ0 mθ admits a second-order Taylor expansion at the point of maximum θ0 with non-singular second derivative: this is established in Lemma D.4.
√

The cited corollary now yields n(θ̂n − θ0 ) = OP (1) as n → ∞, as required.
Lemma D.2 (Relative errors). Let (µ0 , σ0 ) ∈ R × (0, ∞) and ε ∈ (0, 1]. Let (µ, σ) ∈
R × (0, ∞) be such that |µ − µ0 | /σ0 ≤ ε and |σ/σ0 − 1| ≤ ε. Let x ∈ R and write
z0 = (x − µ0 )/σ0 and z = (x − µ)/σ. If |z0 | ≥ 2, then |z/z0 − 1| ≤ 2ε.
Proof. Since (1 + b)(1 − c) − 1 = b − c + bc for b, c ∈ R, we have



(µ − µ0 )/σ0
σ0
z
−1 = 1+
−1 1−
−1
z0
σ
(x − µ0 )/σ0
σ0
σ0
µ − µ0
ε ε2
µ − µ0
≤
−1 +
+
−1
≤ε+ +
≤ 2ε.
σ
2σ0
σ
2σ0
2
2



Lemma D.3 (Lipschitz condition). Let mθ be as in (2.2) with pθ the GEV density
function. For fixed θ0 = (γ0 , µ0 , σ0 ) ∈ (−1/2, ∞) × R × (0, ∞), there exists ṁ such that
Pθ0 ṁ2 < ∞ and such that
|mθ1 − mθ2 | 1{pθ0 >0} ≤ ṁ kθ1 − θ2 k

(D.1)

for all θ1 and θ2 in a neighborhood of θ0 .
Proof of Lemma D.3. The function ṁ can be constructed along the following lines.
First, fix a ∈ [1/2, 1), to be determined later in terms of γ0 . Since |log x − log y| ≤
|x − y| / min(x, y) and since the map z 7→ |(x + z)a − (y + z)a | is decreasing, we have,

ON THE MLE FOR THE GEV

21

on {x : pθ0 (x) > 0},
1
|mθ1 − mθ2 | = |log(pθ1 + pθ0 ) − log(pθ2 + pθ0 )|
2
1
= |log{(pθ1 + pθ0 )a } − log{(pθ2 + pθ0 )a }|
a
1
|(pθ1 + pθ0 )a − (pθ2 + pθ0 )a |
≤ ×
a min{(pθ1 + pθ0 )a , (pθ2 + pθ0 )a }
≤

pa − pa
1
× θ1 a θ2 .
a
pθ0

Suppose we can find a nonnegative function ṗa such that, for some neighbourhood V of
θ0 , we have, for each k ∈ {1, 2, 3},
sup |∂θk paθ (x)| ≤ ṗa (x),

θ∈V

x ∈ {pθ0 > 0}.

(D.2)

Then we find, on {pθ0 > 0},
1
3 ṗa
|mθ1 − mθ2 | ≤
kθ1 − θ2 k.
2
a paθ0
Hence, the Lipschitz condition (D.1) is satisfied, with ṁ = (6/a) ṗa p−a
θ0 , provided that
Z
2
Pθ0 [(ṗa p−a
ṗ2a (x) pθ1−2a
(x) dx < ∞.
(D.3)
θ0 ) ] =
0
{pθ0 >0}

We will split the domain {pθ0 > 0} into certain intervals and we will use the previous
construction on each of these intervals separately, with possibly different values of a.
For bounded intervals, a simplication may occur. Recall Lemma B.5 and let a ∈ [1/2, 1)
be large enough such that γ0 > −a/(1 + a). Let V be a compact neighbourhood of θ0
within (−a/(1 + a), ∞) × R × (0, ∞) and let I be a bounded interval. Since continuous
functions are uniformly bounded on compacta, we have
sup

max |∂θk paθ (x)| < ∞.

(D.4)

(x,θ)∈I×V k∈{1,2,3}

Hence, on any bounded interval of the support of Pθ0 on which p1−2a
is integrable, we
θ0
−a
may choose ṁ equal to a constant times pθ0 . Such intervals need therefore not be looked
into further. For a = 1/2, a choice which will occur often, the condition that p1−2a
is
θ0
integrable on bounded intervals is trivially satisfied.
To further control the partial derivatives of paθ , we will use the identity
∂θk paθ (x) = a paθ (x) ∂θk `θ (x),

k ∈ {1, 2, 3}.

paθ

We will seek bounds for the functions
and |∂θk `θ | separately.
To facilitate writing, let us say that positive functions A and B of (x, θ) satisfy
A.B
if there exists c(θ0 ) > 0 such that A(x, θ) ≤ c(θ0 ) B(x, θ) for all (x, θ) in the proper range.
Here c(θ0 ) is a positive constant whose value may depend on θ0 . For instance, for all
θ = (γ, µ, σ) in a compact neighbourhood of θ0 within R × R × (0, ∞), we have 1/σ . 1.

22

AXEL BÜCHER AND JOHAN SEGERS

The relation . is transitive (and reflexive, but not anti-symmetric) and behaves well
under multiplication of positive quantities.
I. Case γ0 > 0. Fix θ0 = (γ0 , µ0 , σ0 ) ∈ (0, ∞) × R × (0, ∞) and set a = 1/2 in (D.2)
and (D.3). Fix ε ∈ (0, 1/4] and consider the following neighbourhood Vε of θ0 :


σ0
σ0
,
.
Vε = [(1 − ε)γ0 , (1 + ε)γ0 ] × [µ0 − εσ0 , µ0 + εσ0 ] ×
1+ε 1−ε
The support of Pθ0 is (µ0 − σ0 /γ0 , ∞). In view of (D.4), it is sufficient to construct
ṗ1/2 (x) for x ≥ µ0 + 2σ0 , i.e., z0 = (x − µ0 )/σ0 ≥ 2. For such x and for θ ∈ Vε , we have
pθ (x) > 0 as well as |z/z0 − 1| ≤ 2ε by Lemma D.2. In particular, 1 ≤ z0 /2 ≤ z ≤ 2z0 .
Put u = (1 + γz)−1/γ ; note that 0 < u < 1. By Lemmas B.2, B.3 and B.4, all three
score functions ∂θk `θ (x) can be bounded in absolute value by a multiple of 1 + log(1/u),
uniformly in θ ∈ Vε and for all x ≥ µ0 + 2σ0 . Further, the density can be bounded by
pθ (x) =

1 −u γ+1
e u
. uγ+1 .
σ

Hence
pθ (x) |∂θk `θ (x)| . (1 + log(1/u))uγ+1 . uγ+1/2 = (1 + γz)−1−1/2γ .
Since z > z0 /2 and 0 < γ0 /2 < γ < 2γ0 , the supremum over θ ∈ Vε of the previous
upper bound is easily seen to be integrable over z0 ≥ 2.
II. Case γ0 ∈ (−1/2, 0). Fix ε ∈ (0, 1/12) sufficiently small such that −1/2 < γ0 − 2ε <
γ0 + 2ε < 0. Consider the following neighbourhood Vε of θ0 :


σ0
σ0
,
.
Vε = [γ0 − ε, γ0 + ε] × [µ0 − εσ0 , µ0 + εσ0 ] ×
1+ε 1−ε
The support of Pθ0 is (−∞, µ0 + σ0 / |γ0 |). We split this set into two intervals:
(−∞, µ0 − 2σ0 ],

(µ0 − 2σ0 , µ0 + σ0 / |γ0 |).

II.1. The interval (µ0 − 2σ0 , µ0 + σ0 / |γ0 |). We follow the construction leading to (D.2)
and (D.3). To this end, we choose a = a(γ0 ) ∈ (1/2, 1) in such a way that
|γ0 |
1
<a<
.
1 − |γ0 |
2(1 − |γ0 |)
Define
ṗa (x) = sup max |∂θk paθ (x)| .
θ∈V k∈{1,2,3}

Our choice of a entails that γ0 > −a/(1 + a). Hence, in view of (D.4), the function ṗa is
bounded on the interval (µ0 − 2σ0 , µ0 + σ0 / |γ0 |). We need to show that the integral in
(D.3) is finite when we restrict the domain to (µ0 −2σ0 , µ0 +σ0 / |γ0 |). It is then sufficient
to show that the function p1−2a
is integrable on that set. Write down the integral and
θ0

ON THE MLE FOR THE GEV

23

make a change of variable z0 = (x − µ0 )/σ0 to obtain that
Z 1/|γ0 |
Z µ0 +σ0 /|γ0 |
n
o
1−2a
σ02a exp −(1 − 2a)(1 + γ0 z0 )−1/γ0
pθ0 (x) dx =
−2

µ0 −2σ0

× (1 + γ0 z0 )(1−2a)(1/|γ0 |−1) dz0
Z

1/|γ0 |

.

(1 + γ0 z0 )(1−2a)(1/|γ0 |−1) dz0 .

−2

The proportionality constant in the last inequality only depends on θ0 . The right-hand
side is finite since the exponent is larger than −1 by our choice of a.
II.2. The interval (−∞, µ0 − 2σ0 ]. We construct ṁ(x) for x ≤ µ0 − 2σ0 , i.e., for z0 =
(x − µ0 )/σ0 ≤ −2. We will again use the construction leading to (D.2) and (D.3), this
time choosing a = 1/2. In that case, it is sufficient to construct ṗ1/2 such that it is
square-integrable on (−∞, µ0 − 2σ0 ) with respect to the Lebesgue measure.
For x ≤ µ0 −2σ0 and for θ ∈ Vε we have pθ (x) > 0 and, by Lemma D.2, |z/z0 − 1| ≤ 2ε
and therefore 2z0 ≤ z ≤ z0 /2 ≤ −1.
Since z is negative, we have u ≥ 1. The density satisfies
1
pθ (x) = e−u uγ+1 . e−u u.
σ
By Lemmas B.2, B.3 and B.4, the three score functions ∂θk `θ (x) can all be bounded by
a multiple of u2 , the proportionality constant neither depending on x ≤ µ0 − 2σ0 nor on
θ ∈ Vε . Hence, for all k ∈ {1, 2, 3}, all θ ∈ Vε and all x ≤ µ0 − 2σ0 ,
pθ (x) |∂θk `θ (x)|2 . e−u u5 . e−u/2
(using that supu≥1 um e−u/2 < ∞ for any scalar m). Since γ ≥ γ0 − ε and z ≤ z0 /2 < 0,
we have u ≥ {1 + (γ0 − ε)z0 /2}−1/(γ0 −ε) . Inserting this bound into the last display yields
a function which is integrable over z0 ∈ (−∞, −2).
III. Case γ0 = 0. For fixed ε ∈ (0, 1/6], consider the following compact neighbourhood
of θ0 = (0, µ0 , σ0 ):


σ0
σ0
Vε = [−ε, ε] × [µ0 − σ0 ε, µ0 + σ0 ε] ×
,
.
1+ε 1−ε
Partition this set in two pieces, according to the sign of γ:
Vε,+ = {θ ∈ Vε : γ ≥ 0},

Vε,− = {θ ∈ Vε : γ ≤ 0}.

The support of the Gumbel distribution is R, which we will decompose into three intervals:
(−∞, µ0 − σ0 /ε],
[µ0 − σ0 /ε, µ0 + σ0 /ε],
[µ0 + σ0 /ε, ∞).
The middle interval is bounded. By (D.4) with a = 1/2, we only need to consider the
cases z0 ≥ 1/ε and z0 ≤ −1/ε, where z0 = (x − µ0 )/σ0 . Put z = (x − µ)/σ and note
that |z/z0 − 1| ≤ 2ε by Lemma D.2.
III.1. Case z0 ≥ 1/ε. We have z0 ≥ 1/ε ≥ 6 and 4 ≤ (1 − 2ε)z0 ≤ z ≤ (1 + 2ε)z0 .
Moreover, 0 < u < 1. We will write the supremum over θ ∈ Vε as the maximum of the
suprema over θ ∈ Vε,+ and θ ∈ Vε,− .

24

AXEL BÜCHER AND JOHAN SEGERS

III.1.1. Case θ ∈ Vε,+ . In this case, always 1 + γz > 1. The bounds on the scores in
Lemmas B.2, B.3, and B.4 imply that
|∂γ `θ (x)| . z 2 . z0 2 ,

|∂µ `θ (x)| . 1,

|∂σ `θ (x)| . z . z0 .

Further, we have
pθ (x) . (1 + γz)−1/γ−1 ≤ (1 + γz)−1/γ ≤ (1 + εz)−1/ε ≤ {1 + ε(1 − 2ε)z0 }−6 .
It follows that, for each k ∈ {1, 2, 3},
sup pθ (x) |∂θk `θ (x)|2 . {1 + ε(1 − 2ε)z0 }−6 z04 .

θ∈Vε,+

The right-hand side is integrable over z0 ∈ [1/ε, ∞).
III.1.2. Case θ ∈ Vε,− . If −ε ≤ γ ≤ −1/z, then 1 + γz ≤ 0 and thus pθ (x) = 0. So
suppose γ > −1/z. By Lemmas B.2, B.3, and B.4, the scores can be bounded as follows:
max |∂θk `θ (x)| .

k∈{1,2,3}

z2
.
1 + γz

(D.5)

Moreover, the density is bounded by
pθ (x) . (1 + γz)−1/γ−1 .
Hence, for k ∈ {1, 2, 3}, since γ 7→ (1 + γz)−1/γ is increasing in γ on {γ : 1 + γz},
sup pθ (x) |∂θk `θ (x)|2 . z 4 (1 + γz)−1/γ−3

θ∈Vε,+

= z 4 {(1 + γz)−1/γ }1+3γ ≤ z 4 e−(1+3γ)z ≤ z 4 e−z/2 . e−z/4 ,
as supz≥0 z 4 e−z/4 < ∞. Bounding z in e−z/4 from below by z0 /2 yields a function which
is integrable over z0 ∈ [1/ε, ∞).
III.2. Case z0 ≤ −1/ε. We have z0 ≤ −1/ε ≤ −6 and, by Lemma D.2, (1 + 2ε)z0 ≤ z ≤
(1 − 2ε)z0 ≤ −4. Moreover, u > 1.
III.2.1. Case θ ∈ Vε,+ . If γ ≥ 1/ |z|, then pθ (x) = 0. Assume 0 ≤ γ < 1/ |z|, so that
1 + γz > 0. By (B.12),
|∂γ `θ (x)| ≤ u1+γ {(log u)2 + log u)} . u1+2ε .

(D.6)

Further, by Lemmas B.2 and B.3,
|∂µ `θ (x)| . u1+γ ≤ u1+ε ,

|∂σ `θ (x)| . u1+γ {log(u) + 1} . u1+2ε .

(D.7)

The density is bounded by
pθ (x) . e−u u1+γ .
In total, since supu≥1 um e−u/2 < ∞ for any scalar m,
pθ (x) |∂θk `θ (x)|2 . e−u u(1+γ)+2(1+2ε) . e−u u3+5ε . e−u/2 .
Since γ 7→ (1 + γz)−1/γ is increasing in γ, a lower bound for u is given by u ≥ e−z =
e|z| ≥ e|z0 |/2 . Plugging this bound into e−u/2 yields a function which is integrable over
z0 ∈ (−∞, −1/ε].

ON THE MLE FOR THE GEV

25

III.2.2. Case θ ∈ Vε,− . By Lemmas B.2, B.3 and B.4,
|∂γ `θ (x)| . u1+ε ,

|∂µ `θ (x)| . u1+ε ,

|∂σ `θ (x)| . u log(u).

The density is bounded by
pθ (x) . e−u u1+γ .
We get
sup pθ (x) |∂θk `θ (x)|2 . e−u u3+3ε . e−u/2 .

θ∈Vε,−

Further, u can be bounded from below by (1 + (−ε)(1 − 2ε)z0 )−1/(−ε) . Inserting this
into e−u/2 yields an integrable function in z0 ∈ (−∞, −1/ε].

Lemma D.4. Let Pθ denote the three-parameter GEV distribution with parameter θ and
let mθ be as in (2.2). For fixed θ0 , the map θ 7→ Pθ0 mθ attains a unique maximum over
(−1/2, ∞) × R × (0, ∞) at θ = θ0 . If moreover γ0 > −1/2, then whenever ht → h in Rk
as t → 0, we have

1
t → 0.
t−2 Pθ0 mθ0 +tht − mθ0 → − hT Iθ0 h,
4
Proof. The fact that θ0 is the unique maximizer of the map θ 7→ Pθ0 follows from
Lemma 5.35 in van der Vaart (1998) applied to the mixture densities (pθ + pθ0 )/2.
The second-order Taylor expansion can now be shown along similar lines as in the
proof of Theorem 5.39 in van der Vaart (1998). In fact, the same technique was used to
show (A.5) in the proof of Proposition 2.1.

Appendix E. Remaining steps for the proof of Proposition 3.3
In view of the outline of the proof given right after the statement of the Proposition 3.3,
it remains to check the condition on the support in (2.3) and the Lipschitz property (2.4).
This is the content of the present section.
Lemma E.1. For any θ0 ∈ (−1/2, ∞) × R × (0, ∞), the three-parameter GEV family
satisfies Condition (2.3).
Proof. First, consider θ0 ∈ Θ0,− = (−∞, 0) × R × (0, ∞) such that γ0 > −1/2. Let
ω(θ) = µ − σ/γ = µ + σ/ |γ| and ω0 = ω(θ0 ). Since θ 7→ ω(θ) is increasing in each
component of θ ∈ Θ0,− , we have
h− (ε) := inf{ω(θ) : θ ∈ Uε (θ0 )} = ω(γ0 − ε, µ0 − ε, σ0 − ε)
whence S̄(ε) = (−∞, h− (ε)). The function t 7→ h− (t) is continuously differentiable on
a neighbourhood of t = 0 and with negative derivative at t = 0. Hence, we can find
constants 0 < b1 < c1 and t1 > 0 such that
ω0 − t · c1 σ0 / |γ0 | ≤ h− (t) ≤ ω0 − t · b1 σ0 / |γ0 | ,
As a consequence of (E.1), for sufficiently small ε > 0,
X \ S̄(ε) ⊂ [ω0 − εc1 σ0 / |γ0 | , ∞).

t ∈ [0, t1 ].

(E.1)

26

AXEL BÜCHER AND JOHAN SEGERS

Therefore, using the substitution u = {1 + γ0 (x − µ0 )/σ0 }1/|γ0 | ,
Z (εc1 )1/|γ0 |
Z ω0
e−u du = O(ε1/|γ0 | ) = o(ε2 )
pθ0 (x) dx =
Pθ0 (X \ S̄(ε)) ≤
ω0 −c1 εσ0 /|γ0 |

0

as ε ↓ 0, since −1/2 < γ0 < 1.
Now, consider θ0 ∈ Θ0,+ = (0, ∞) × R × (0, ∞), i.e., γ0 > 0. Then, for any ε <
min(σ0 , γ0 ) and any θ ∈ Uε (θ0 ), we have
ω(θ) = µ −

σ
σ0 − ε
≤ µ0 + ε −
=: h+ (ε).
γ
γ0 + ε

whence S̄(ε) = (h+ (ε), ∞). The function t 7→ h+ (ε) is continuously differentiable on a
neighbourhood of t = 0, with positive derivative at t = 0. As a consequence, we can find
constants 0 < b2 < c2 and t2 > 0 such that
ω0 + t · b2 σ0 /γ0 ≤ h+ (t) ≤ ω0 + t · c2 σ0 /γ0 ,

t ∈ [0, t2 ].

(E.2)

Hence, for sufficiently small ε > 0, we obtain that
X \ S̄(ε) ⊂ (−∞, ω0 + ε · c2 σ0 /γ0 ].
Therefore, using the substitution u = {1 + γ0 (x − µ0 )/σ0 }−1/γ0 ,
Z ω0 +εc2 σ0 /γ0
Z ∞
Pθ0 (X \ S̄(ε)) ≤
pθ0 (x) dx =
e−u du = exp{−(c2 ε)−1/γ0 },
(c2 ε)−1/γ0

ω0

which clearly is of the order o(ε2 ) as ε ↓ 0, since γ0 > 0.
Finally, consider θ0 ∈ Θ0 with γ0 = 0. Then, for any ε < σ0 /2, we have
ω(θ) = µ −

σ0 − ε
σ
≤ µ0 + ε −
=: h0,+ (ε),
γ
ε

θ ∈ Uε (θ0 ) ∩ Θ0,+ ,

ω(θ) = µ +

σ0 − ε
σ
≥ µ0 − ε +
=: h0,− (ε),
|γ|
ε

θ ∈ Uε (θ0 ) ∩ Θ0,− .

as well as

As a consequence, for sufficiently small ε, S̄(ε) = (h0,+ (ε), h0,− (ε)). Clearly, we can find
constants d1 , d2 > 0 such that h0,+ (ε) ≤ −d1 /ε and h0,− (ε) ≥ d2 /ε for all sufficiently
small ε. Hence,
Z −d1 /ε
Z ∞
Pθ0 (X \ S̄(ε)) ≤
pθ0 (x) dx +
pθ0 (x) dx
−∞


= exp − exp

d2 /ε



µ0
d1
+
σ0 ε σ0






d2
µ0
+ 1 − exp − exp −
+
,
σ0 ε σ0

which can be easily seen to be of the order o(ε2 ) as ε ↓ 0, since 1 − exp(−y) = (1 + o(1))y
as y → 0.

Lemma E.2. For any θ0 ∈ (−1/2, ∞) × R × (0, ∞), the three-parameter GEV family
satisfies Condition (2.4) with Θ = (−1/2, ∞) × R × (0, ∞) (and hence also with any
compact subset containing θ0 in its interior).

ON THE MLE FOR THE GEV

27

Proof. Throughout, let k · k denote the maximum norm on R3 . The proof will be split
up in three cases, according to the sign of γ0 . For some suitable ε0 > 0 to be specified
below, and for any x ∈ Sθ0 , define a set of admissible parameter vectors θ as

Θ0 (x) = θ ∈ Uε0 (θ0 ) : if θ0 ∈ Θ satisfies kθ0 − θ0 k ≤ 2kθ − θ0 k, then x ∈ Sθ0 .
Note that, if θ ∈ Θ0 (x), then the entire ball with center θ0 and radius kθ − θ0 k is a
subset of Θ0 (x). In other words, Θ0 (x) is the union of all neighbourhoods Uε (θ0 ) for
those ε ∈ (0, ε0 ] such that x ∈ Sθ0 for every θ0 ∈ U2ε (θ0 ). [The multiplicative constant 2
is not essential and could have been replaced by an arbitrary constant c0 > 1.]
It follows from the definition of Θ0 (x) that θ 7→ `θ (x) is continuously differentiable on
Θ0 (x). Hence, we may define
˙
`(x)
= 3 sup{k`˙θ (x)k : θ ∈ Θ0 (x)},

x ∈ Sθ0 ,

(E.3)

and, by the mean-value theorem, immediately obtain that
∀x ∈ Sθ0 : ∀θ1 , θ2 ∈ Θ0 (x) :

˙ kθ1 − θ2 k.
|`θ1 (x) − `θ2 (x)| ≤ `(x)

(E.4)

[The constant 3 appears because of the use of the max-norm.] It can be seen easily that
(E.4) implies (2.4). The proof will be finished once we will have constructed ε0 and will
have showed that the function in (E.3) is square-integrable with respect to Pθ0 .
I. Case γ0 ∈ (0, ∞). Write ω0 = µ0 − σ0 /γ0 and recall that Sθ0 = (ω0 , ∞). For x ∈ Sθ0 ,
write z0 = z0 (x) = (x − µ0 )/σ0 . Note that z0 > −1/γ0 .
For a parameter vector θ with γ > 0, we have Sθ = (ω(θ), ∞) where ω(θ) = µ − σ/γ.
A monotonicity and differentiability argument similar to that yielding (E.2) shows that
there exist positive constants b0 and t0 with b0 t0 ≤ 1/2 and t0 < min(σ0 , γ0 )/2 such that
for all t ∈ [0, t0 ],

σ0 − t  ≤ ω0 + (σ0 /γ0 ) tb0 (4/3),
h+ (t) = sup ω(θ) = µ0 + t −
γ0 + t  ≥ ω + (σ /γ ) tb .
θ∈Ut (θ0 )
0

0

0

0

Partition the interval Sθ0 into three sub-intervals:
Sθ0 = (ω0 , ω0 + (σ0 /γ0 ) t0 b0 ) ∪ [ω0 + (σ0 /γ0 ) t0 b0 , µ0 + 2σ0 ] ∪ (µ0 + 2σ0 , ∞).
Choose 0 < ε0 < min{1/4, t0 /2, σ0 /(2γ0 )} small enough such that
σ/σ0 , γ/γ0 ∈ [5/6, 7/6],

(σ/γ)/(σ0 /γ0 ) ∈ [3/4, 4/3],

for all θ ∈ Uε0 (θ0 ).

˙
We will provide an upper bound for `(x)
in (E.3) for each x in each piece of the partition
separately.
First, suppose that x ∈ Sθ0 is such that x = ω0 + (σ0 /γ0 )y with 0 < y < t0 b0 .
If θ is such that t := kθ − θ0 k satisfies t < ε0 ≤ t0 /2 and 2tb0 ≥ y, then h+ (2t) ≥
ω0 + (σ0 /γ0 )(2tb0 ) ≥ x and thus θ 6∈ Θ0 (x). As a consequence, Θ0 (x) ⊂ Uy/(2b0 ) (θ0 ).
But for θ ∈ Uy/(2b0 ) (θ0 ), we have ω(θ) ≤ ω0 + (σ0 /γ0 )(y/2)(4/3) = ω0 + (σ0 /γ0 )(2/3)y
and thus x − ω(θ) ≥ (σ0 /γ0 )(1/3)y. For such θ, writing z = (x − µ)/σ, we find, using
y = 1 + γ0 z0 , that 1 + γz = (γ/σ)(x − ω(θ)) ≥ (γ/σ)(σ0 /γ0 )(1/3)y ≥ (1/4)(1 + γ0 z0 ).
In addition, x < ω0 + (σ0 /γ0 )(1/2) = µ0 − (σ0 /γ0 )(1/2) < µ0 − ε0 ≤ µ and thus z < 0.
Apply (B.3), (B.4) and (B.12) to arrive at a Pθ0 square-integrable bound.

28

AXEL BÜCHER AND JOHAN SEGERS

Second, suppose that x ∈ Sθ0 is such that ω0 +(σ0 /γ0 ) t0 b0 ≤ x ≤ µ0 +2σ0 . The partial
derivatives of `θ (x) with respect to the three components of θ being continuous functions
of (θ, x) in the compact domain {θ : kθ − θ0 k ≤ ε0 } × [ω0 + (σ0 /γ0 ) t0 b0 , µ0 + 2σ0 ], they
are also uniformly bounded and thus square-integrable with respect to Pθ0 .
Third, suppose that x ∈ Sθ0 is such that z0 ≥ 2. Let θ ∈ Uε0 (θ0 ) and write z =
(x − µ)/σ. By Lemma D.2 we have |(z/z0 ) − 1| ≤ 1/2 and thus z0 /2 ≤ z ≤ 3z0 /2.
The inequalities in Lemmas B.2, B.3 and B.4 then combine into a Pθ0 square-integrable
˙
upper bound for `.
II. Case γ0 ∈ (−1/2, 0). Write ω0 = µ0 + σ0 / |γ0 | and recall that Sθ0 = (−∞, ω0 ). For
x ∈ Sθ0 , write z0 = z0 (x) = (x − µ0 )/σ0 . Note that z0 < 1/ |γ0 |.
For a parameter vector θ with γ < 0, we have Sθ = (−∞, ω(θ)) where ω(θ) = µ+σ/ |γ|.
A monotonicity and differentiability argument similar to that yielding (E.1) shows that
there exist positive constants b0 and t0 with b0 t0 ≤ 1/2 and t0 < min(σ0 , |γ0 |)/2 such
that for all t ∈ [0, t0 ],

σ0 − t  ≤ ω0 − (σ0 / |γ0 |) tb0 ,
h− (t) = inf ω(θ) = µ0 − t −
γ0 − t  ≥ ω − (σ / |γ |) tb (4/3).
θ∈Ut (θ0 )
0

0

0

0

Partition the interval Sθ0 into three sub-intervals:
Sθ0 = (−∞, µ0 − 2σ0 ) ∪ [µ0 − 2σ0 , ω0 − (σ0 / |γ0 |)t0 b0 ] ∪ (ω0 − (σ0 / |γ0 |)t0 b0 , ω0 ).
Choose 0 < ε0 < min{1/4, t0 /2, σ0 /(2 |γ|0 ), σ0 /7} small enough such that
σ/σ0 , γ/γ0 ∈ [5/6, 7/6],

(σ/γ)/(σ0 /γ0 ) ∈ [3/4, 4/3],

for all θ ∈ Uε0 (θ0 ).

˙
We will provide an upper bound for `(x)
in (E.3) for each x in each piece of the partition
separately.
First, consider x ∈ (ω0 − (σ0 / |γ0 |)t0 b0 , ω0 ), which can be rewritten as x = ω0 −
(σ0 / |γ0 |)y for some 0 < y < t0 b0 . Then Θ0 (x) ⊂ Uy/(2b0 ) (θ0 ), for, if θ ∈ Uε0 (θ0 ) ⊂
Ut0 /2 (θ0 ) satisfies t = kθ − θ0 k ≥ y/(2b0 ), then there exists θ0 with kθ0 − θ0 k ≤ 2t such
that ω(θ0 ) = h− (2t) ≤ ω0 − (σ0 / |γ0 |)2tb0 ≤ x. Now, for θ ∈ Uy/(2b0 ) (θ0 ), we have
ω(θ) − x ≥ ω0 − σ0 / |γ0 | (y/2)(4/3) − x = ω0 − (2/3)(ω0 − x) − x = (1/3)(ω0 − x). In
addition, x > ω0 − σ0 / |γ0 | /2 = µ0 + σ0 / |γ0 | > µ0 + ε0 > µ, whence z = (x − µ)/σ > 0.
Therefore, as a consequence of Lemmas B.2, B.3 and B.4, we obtain the upper bound
σ
1
(4/3)σ0
1
1
=
≤
.
k`˙θ (x)k .
1 + γz
|γ| ω(θ) − x
(1/3) |γ0 | ω0 − x
Since γ0 ∈ (−1/2, 0) implies 1/ |γ0 | > 2, the bound can be seen to be square-integrable
with respect to Pθ0 .
Second, consider x ∈ [µ0 − 2σ0 , ω0 − (σ0 / |γ0 |)t0 b0 ]. The partial derivatives of `θ (x)
with respect to the three components of θ being continuous functions of (θ, x) in the
compact domain {θ : kθ − θ0 k ≤ ε0 } × [µ0 − 2σ0 , ω0 − (σ0 / |γ0 |)t0 b0 ], they are also
uniformly bounded and thus square-integrable with respect to Pθ0 .
Third, consider x ∈ (−∞, µ0 − 2σ0 ). Let θ ∈ Uε0 (θ0 ) and write z = (x − µ)/σ.
By Lemma D.2, we have |(z/z0 ) − 1| ≤ 1/2 and thus 3z0 /2 ≤ z ≤ z0 /2 ≤ −1. The
inequalities in Lemmas B.2, B.3 and B.4 then combine into a Pθ0 square-integrable
˙
upper bound for `.

ON THE MLE FOR THE GEV

29

III. Case γ0 = 0. Recall that Sθ0 = R. Find t0 > 0 small enough such that (1 + t0 )t0 ≤
σ0 /4. Then we have
(3/4)σ0 /t ≤ (σ0 /t) − t − 1 ≤ σ0 /t,

for t ∈ (0, t0 ].

It follows that, for all t ∈ (0, t0 ],
σ0 − t
h0,+ (t) =
sup
ω(θ) = µ0 + t −
t
θ∈Ut (θ0 )∩Θ0,+



σ0 − t
h0,− (t) =
inf
ω(θ) = µ0 − t +
t
θ∈Ut (θ0 )∩Θ0,−



≤ µ0 − (3/4)σ0 /t,
≥ µ0 − σ0 /t,

(E.5)

≤ µ0 + σ0 /t,
≥ µ0 + (3/4)σ0 /t.

(E.6)

Choose 0 < ε0 < min{1/4, t0 /2} small enough such that σ/σ0 ∈ [5/6, 7/6] whenever
θ ∈ Uε0 (θ0 ). Partition the real line into three sub-intervals:
R = (−∞, µ0 − σ0 /ε0 ) ∪ [µ0 − σ0 /ε0 , µ0 + σ0 /ε0 ] ∪ (µ0 + σ0 /ε0 , ∞).
˙
We will provide an upper bound for `(x)
in (E.3) for each x in each piece of the partition
separately. Write x = µ0 + σ0 z0 .
III.1 – Case x < µ0 − σ0 /ε0 . We have z0 < −1/ε0 . For any θ ∈ Θ0 (x) ⊂ Uε0 (θ0 ), we
have x < µ0 − σ0 /ε0 < µ0 − ε0 ≤ µ; note that ε20 < t20 < σ0 . Therefore, z = zθ (x) =
(x − µ)/σ < 0 and thus u = uγ (z) > 1, see Figure 1.
We claim that Θ0 (x) ⊂ U1/(2|z0 |) (θ0 ). To prove this, it suffices to show that we can
find a point θ0 such that kθ0 − θ0 k = 2/(2 |z0 |) = 1/ |z0 | and such that x ∈
/ Sθ0 . But this
is easy: just set θ0 = (1/ |z0 | , µ0 , σ0 ) and note that 1 + (1/ |z0 |)z0 = 1 − 1 = 0.
Let θ ∈ Θ0 (x) ⊂ U1/(2|z0 |) (θ0 ). Write z = (x−µ)/σ. Our choice of ε0 and the fact that
|z0 | > 1/ε0 > 2 imply that |z/z0 − 1| ≤ 1/2; see Lemma D.2. Consider three subcases:
γ > 0, γ = 0, and γ < 0.
• Suppose γ > 0. By (E.5), we have ω(θ) ≤ µ0 −(3/4)σ0 (2 |z0 |) = µ0 +(3/2)σ0 z0 <
µ0 + σ0 z0 = x. Hence, x ∈ Sθ . As γ > 0 and z < 0, the bounds in (D.6) and
(D.7) then imply that k`˙θ (x)k . u1+2ε0 . We need to bound u from above. Recall
that uγ (z) is decreasing in z and increasing in γ. Moreover, (3/2)z0 ≤ z < 0 and
0 < γ < 1/(2 |z0 |). We find
−2|z0 |

−2|z0 |
1 3
z0
= 1 − (3/4)
= 16−z0 .
1≤u≤ 1+
2 |z0 | 2
For arbitrary ρ > 1, the function z0 7→ ρ−z0 is integrable with respect to the
standard Gumbel density z0 7→ exp(−e−z0 ) e−z0 .
• Suppose γ = 0. The expressions for the score functions are ∂γ `θ (x) = (1 −
e−z )z 2 /2 − z, ∂µ `θ (x) = (1 − e−z )/σ, and ∂σ `θ (x) = ((1 − e−z )z − 1)/σ. Since
(3/2)z0 ≤ z ≤ (1/2)z0 < 0 for all θ considered, these expressions can be easily
bounded by Pθ0 -square integrable functions.
• Suppose γ < 0. Since z < 0, the components of the score vector can be bounded
by a multiple of u max{z, (log u)2 }. But since uγ (z) is increasing in γ and since
γ < 0, we can bound u by its value at γ = 0, which is e−z . Now continue as for
the case γ = 0.

30

AXEL BÜCHER AND JOHAN SEGERS

III.2 – Case µ0 −σ0 /ε0 ≤ x ≤ µ0 +σ0 /ε0 . This case is trivial, since the three components
of the score vector `˙θ (x) are continuous and thus uniformly bounded on the closure of
the bounded domain {(x, θ) : |x − µ0 | ≤ σ0 /ε0 , θ ∈ Θ0 (x)}.
III.3 – Case x > µ0 + σ0 /ε0 . This case is partially similar to the case x < µ0 − σ0 /ε0 .
For θ ∈ Θ0 (x) ⊂ Uε0 (θ0 ), we have x > µ0 + ε0 > µ and thus z = (x − µ)/σ > 0 and
0 < u = uγ (z) < 1. Moreover, z0 > 1/ε0 .
We claim that Θ0 (x) ⊂ U1/(2z0 ) (θ0 ). Indeed, the point θ0 = (−1/z0 , µ0 , σ0 ) is such
that kθ − θ0 k = 1/z0 = 2/(2z0 ) and still x ∈
/ Sθ0 .
Let θ ∈ Θ0 (x) ⊂ U1/(2z0 ) (θ0 ) and write z = (x−µ)/σ. Again, we have |z/z0 − 1| ≤ 1/2.
Consider three subcases: γ > 0, γ = 0, and γ < 0.
• Suppose γ > 0. Since 0 < u < 1 and 1 + γz > 1, all three components of
the score vector can be bounded by a constant multiple of z 2 and thus by a
constant multiple of z02 . Now it suffices to observe that all moments of the
Gumbel distribution are finite.
• Suppose γ = 0. Then apply the same reasoning as for the subcase γ = 0 in the
case III.1 above.
• Suppose γ < 0. By (E.6), we have ω(θ) ≥ µ0 + (3/4)σ0 (2z0 ) = µ0 + (3/2)σ0 z0 ≤
µ0 + σ0 z0 = x, so that x ∈ Sθ . Since 0 > γ > −1/(2z0 ) ≥ −3/(4z) > −1/z, the
bound in (D.5) applies. We need to find a square-integrable bound on z 2 /(1+γz).
But this is immediate, since the denominator is larger than 1 − 3/4 = 1/4 and
the numerator is smaller than a multiple of z02 .

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.
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).
References
Bücher, A. and J. Segers (2016). Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time series. Bernoulli . To appear.
Cramér, H. (1946). Mathematical Methods of Statistics. Princeton: Princeton University
Press.
Dombry, C. (2015). Existence and consistency of the maximum likelihood estimators
for the extreme value index within the block maxima framework. Bernoulli 21 (1),
420–436.
Drees, H., A. Ferreira, and L. de Haan (2004). On maximum likelihood estimation of
the extreme value index. The Annals of Applied Probability 14 (3), 1179–1201.

ON THE MLE FOR THE GEV

31

Fisher, R. A. and L. H. C. Tippett (1928). On the estimation of the frequency distributions of the largest or smallest member of a sample. Proceedings of the Cambridge
Philosophical Society 24, 180–190.
Gumbel, E. J. (1958). Statistics of extremes. New York: Columbia University Press.
Hosking, J. R. M. (1985). Algorithm as 215: Maximum-likelihood estimation of the parameters of the generalized extreme-value distribution. Journal of the Royal Statistical
Society. Series C (Applied Statistics) 34 (3), 301–310.
Hosking, J. R. M., J. R. Wallis, and E. F. Wood (1985). Estimation of the generalized
extreme-value distribution by the method of probability-weighted moments. Technometrics 27 (3), 251–261.
Jenkinson, A. F. (1955). The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quarterly Journal of the Royal Meteorological
Society 81 (348), 158–171.
Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series
in Statistics. New York: Springer-Verlag.
Marohn, F. (1994). On testing the Exponential and Gumbel distribution. In Extreme
Value Theory and Applications, pp. 159–174. Kluwer Academic Publishers.
Marohn, F. (2000). Testing extreme value models. Extremes 3 (4), 363–384.
Prescott, P. and A. T. Walden (1980). Maximum likelihood estimation of the parameters
of the generalized extreme-value distribution. Biometrika 67 (3), 723–724.
Smith, R. L. (1985). Maximum likelihood estimation in a class of nonregular cases.
Biometrika 72 (1), 67–90.
van de Geer, S. (2009). Empirical Processes in M-Estimation. Cambridge Series in
Statistical and Probabilistic Mathematics. Cambridge University Press.
van der Vaart, A. (1998). Asymptotic Statistics. Cambridge: Cambridge University
Press.
von Mises, R. (1936). La distribution de la plus grande de n valeurs. In Selected Papers,
Volume II, pp. 271–294. Providence, R.I., 1954: American Mathematical Society.
Woodroofe, M. (1972). Maximum likelihood estimation of a translation parameter of a
truncated distribution. The Annals of Mathematical Statistics 43 (1), 113–122.
Woodroofe, M. (1974). Maximum likelihood estimation of translation parameter of
truncated distribution II. The Annals of Statistics 2 (3), 474–488.
Ruhr-Universität Bochum, Fakultät für Mathematik, Universitätsstr. 150, 44780 Bochum,
Germany
E-mail address: axel.buecher@rub.de
Université catholique de Louvain, Institut de Statistique, Biostatistique et Sciences
Actuarielles, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium
E-mail address: johan.segers@uclouvain.be

