Semi-Parametric Probability-Weighted Moments
Estimation Revisited
∗
Frederico Caeiro
Universidade Nova de Lisboa, FCT and CMA
M. Ivette Gomes
Universidade de Lisboa, DEIO, CEAUL and FCUL
and
Bj¨orn Vandewalle
Universidade Nova de Lisboa, ISEGI, and CEAUL
January 19, 2012
Abstract
In this paper, for heavy-tailed models and through the use of probability weighted moments
based on the largest observations, we deal essentially with the semi-parametric estimation of the
Value-at-Risk at a level p, the size of the loss occurred with a small probability p, as well as the
dual problem of estimation of the probability of exceedance of a high level x. These estimation
procedures depend crucially on the estimation of the extreme value index, the primary parameter in
Statistics of Extremes, also done on the basis of the same weighted moments. Under regular variation
conditions on the right-tail of the underlying distribution function F , we prove the consistency and
asymptotic normality of the estimators under consideration in this paper, through the usual link of
their asymptotic behaviour to the one of the extreme value index estimator they are based on. The
performance of these estimators, for finite samples, is illustrated through Monte-Carlo simulations.
An adaptive choice of thresholds is put forward. Applications to a real data set in the field of
insurance as well as to simulated data are also provided.
AMS 2000 subject classification. Primary 62G32, 62E20; Secondary 65C05.
Keywords and phrases. Heavy tails, value-at-risk or high quantiles, probability of exceedance of a high level,
semi-parametric estimation.
∗
Research partially supported by National Funds through FCT — Funda¸c˜ao para a Ciˆencia e a Tecnologia, projects
PEst-OE/MAT/UI0006/2011 and PEst-OE/MAT/UI0297/2011, and PTDC/FEDER.
1
1 Introduction, preliminaries and scope of the article
Let X
1
, X
2
, . . . , X
n
be a set of n independent and identically distributed (i.i.d.), or even possibly weakly
dependent and stationary, random variables (r.v.’s), from a population with distribution function (d.f.)
F . Let us arrange them in ascending order, to get the order statistics (o.s.’s) X
1:n
≤ ··· ≤ X
n:n
. Suppose
that we are interested in the estimation of a high quantile of probability 1 − p, or equivalently, in the
estimation of the Value-at-Risk (VaR) at a level p, the size of the loss occurred with a small probability
p, given by
VaR
p
≡ χ
1−p
:= F
←
(1 −p) = inf{x : F (x) ≥ 1 −p}, (1.1)
with the notation F
←
standing thus for the generalized inverse function of F . Moreover, we are also
interested in the estimation of the probability of exceedance of a high level x = x
n
,
p = p
x
:= 1 − F (x) =: F (x). (1.2)
Extreme Value Theory (EVT) provides a great variety of results that enable us to to deal with alternative
approaches in the statistical analysis of extreme events. Those approaches are essentially based on the
well-established limiting results described in the following.
1.1 Main limiting results in EVT
The main limiting result in EVT can be attributed to Gnedenko (1943), who fully characterized the pos-
sible non-degenerate limiting distribution of the linearly normalised maximum, (X
n:n
− b
n
)/a
n
, a
n
> 0,
b
n
∈ R. Such a limit is of the type of the general extreme value distribution (EVD),
EV
γ
(x) :=
(
exp(−(1 + γx)
−1/γ
), 1 + γx > 0 if γ 6= 0
exp(−exp(−x)), x ∈ R if γ = 0.
(1.3)
When such a non-degenerate limit exists, we say that F belongs to the max-domain of attraction of
EV
γ
and denote this by F ∈ D
M
(EV
γ
). The shape parameter γ is related with the heaviness of the
right-tail F = 1 − F and it is often called the extreme value index (EVI).
Another seminal result in the field of EVT is due to Balkema and de Haan (1974) and Pickands
(1975). If we properly scale the excesses over a high threshold u, the limit distribution of those scaled
excesses is the Generalized Pareto distribution (GPD), strongly related with the d.f. EV
γ
(x), in (1.3),
and defined by,
GP
γ
(x) := 1 + ln EV
γ
(x) =
(
1 − (1 + γx)
−1/γ
, 1 + γx > 0, x > 0 if γ 6= 0
1 − exp(−x), x > 0 if γ = 0
(1.4)
(see, for instance, Embrechts et al., 1997, Section 3.4, and Reiss and Thomas, 2007, Section 1.4, for
more details).
2
1.2 Most relevant approaches in the field of Statistics of Univariate Extremes
We shall briefly refer the three most important approaches in the area of Statistics of Univariate Ex-
tremes: the block maxima (BM) method, the peaks-over-threshold (POT) or even the peaks-over-random-
threshold (PORT) methods and the largest observations (LOB) method. For a more detailed review,
with extensive associated references, see Gomes et al. (2008) and Beirlant et al. (2012).
• The first method, the BM method, is of a parametric nature: we work with a sample of maxima of
adequate blocks of observations, and estimate the parameters (λ, δ, γ) of the EVD, EV
γ
((x − λ)/δ),
λ ∈ R, δ > 0, γ ∈ R, with EV
γ
(x) given in (1.3). This method is known to be possibly inefficient,
due to the fact that the loss of information in each block can be catastrophic.
• In the second approach, the POT method, inference is performed through the use of the sample
of excesses over a high deterministic threshold u. The limiting d.f. of these excesses is, up to
a scale factor, the distribution GP
γ
(x), in (1.4), and the method can be of a parametric or a
semi-parametric nature. Note that the high threshold can also be a random value, leading to the
PORT methodology, a terminology recently introduced in Ara´ujo Santos et al. (2006).
• The third approach, the LOB method, is the one we shall consider in this paper. It uses the largest
k observations to make inference about the right tail F = 1 − F , assuming only that F belongs
to a wide sub-domain of D
M
(EV
γ
).
1.3 Estimators under study
Under the largest observations framework, and whenever dealing with heavy-tailed models, the classi-
cal semi-parametric EVI and VaR-estimators are the Hill (Hill, 1975) and Weissman-Hill’s estimators
(Weissman, 1978), with functional expressions
ˆγ
H
k,n
:=
1
k
k
X
i=1
(ln X
n−i+1:n
− ln X
n−k:n
) (1.5)
and
b
Q
H
k,n
(p) := X
n−k:n
c
ˆγ
H
k,n
k
, c
k
≡ c
k
(p) :=
k
np
, k = 1, 2, . . . , n − 1, (1.6)
respectively, which are pseudo-maximum likelihood estimators, consistent in the whole D
+
M
:=
D
M
(EV
γ
)
γ>0
, provided that k is intermediate, i.e. if
k = k
n
→ ∞ and k/n → 0, as n → ∞. (1.7)
In a way dual to (1.6), and given a high level x = x
n
, the probability p = p
x
of exceedance of such a
level can be estimated by
ˆp
H
k,n
(x) :=
k
n
e
C
−1/ˆγ
H
k,n
k
,
e
C
k
≡
e
C
k
(x) :=
x
X
n−k:n
, k = 1, 2, . . . , n − 1. (1.8)
3
Under further adequate restrictions on k, we can guarantee the asymptotic normality of the estimators
ˆγ
H
k,n
,
b
Q
H
k,n
(p) and ˆp
H
k,n
(x), in (1.5), (1.6) and (1.8), respectively. But most of the times, these estimators
exhibit a large variance for small k, a strong bias for moderate k, sample paths with very short stability
regions around the target value and a very peaked mean square error (MSE) structure, as a function of
k. This has led researchers to the search of alternative estimators, with a smaller MSE.
Since heavy-tailed models only have mean value if γ < 1, methods based on sample moments have
been rarely considered when we work with such a type of distributions. But in many practical fields
like in finance or insurance, for example, we usually have a positive EVI smaller than one, and even
smaller than 1/2. In this article, and for the estimation of the above mentioned parameters of extreme
events, we now revisit the use of a probability weighted moments (PWM) method based on the largest
observations, developed in Caeiro and Gomes (2011) for the EVI.
The PWM method is a generalization of the method of moments. It also consists in equating sample
moments with their corresponding theoretical moments, and then solving those equations in order to
obtain estimates of the different parameters under play. The PWM of a r.v. X are defined by
M
p,r,s
:= E(X
p
(F (X))
r
(1 − F (X))
s
),
where p, r and s are any real numbers (Greenwood et al., 1979). When r = s = 0, M
p,0,0
are the
usual noncentral moments of order p. Hosking et al. (1985) advise the use of M
1,r,s
, because then the
relations between parameters and moments have usually a much simpler form. Also, when r and s are
integers, F
r
(1 − F )
s
can be written as a linear combination of powers of F or 1 − F . So it is usual to
work with the particular case,
a
r
:= M
1,0,r
= E(X(1 − F (X))
r
), r ≥ 0,
and the associated estimator,
ˆa
r
=
1
n
n−r
X
i=1
(n − 1 − r)!(n − i)!
(n − 1)!(n − i − r)!
X
i:n
=
1
n
n
X
i=1
(n − i)(n − i − 1) . . . (n − i − r + 1)
(n − 1)(n − 2) . . . (n − r)
X
i:n
. (1.9)
For γ < 1 and for d.f.’s like the EVD, EV
γ
((x − λ)/δ), with EV
γ
(x) given in (1.3), the Pareto d.f.,
P
γ
(x; δ) = 1 − (x/δ)
−1/γ
, x > δ, (1.10)
and the GPD, GP
γ
(x/δ), with GP
γ
(x) defined in (1.4), the PWM have simple expressions, which allow
a straightforward estimation of the EVI, γ. For the EVD, see Hosking et al. (1985) and the improved
versions in Diebolt et al. (2007, 2008). As an example, the Pareto PWM (PPWM) and the generalized
Pareto PWM (GPPWM) estimators of γ are valid for γ < 1, and given by
ˆγ
P P W M
= 1 −
ˆa
1
ˆa
0
− ˆa
1
and ˆγ
GP P W M
= 1 −
2ˆa
1
ˆa
0
− 2ˆa
1
, (1.11)
4
respectively, where ˆa
0
and ˆa
1
are given in (1.9). The estimator ˆγ
GP P W M
, in (1.11), was introduced and
studied in Hosking and Wallis (1987).
We shall consider in this paper, the PPWM estimators of VaR
p
and p
x
, the parameters respec-
tively defined in (1.1) and (1.2), associated with the PPWM EVI-estimators studied in Caeiro and
Gomes (2011). Those estimators are semi-parametric in nature and, for comparison with the equiv-
alent estimators based on the Hill EVI-estimator, in (1.5), are based on the top k + 1 largest o.s.’s,
X
n−k:n
≤ X
n−k+1:n
≤ ··· ≤ X
n:n
. Under such a framework, the estimators ˆa
0
and ˆa
1
, in (1.9), should
be replaced by,
ˆa
0
(k) :=
1
k + 1
k+1
X
i=1
X
n−i+1:n
and ˆa
1
(k) :=
1
k + 1
k+1
X
i=1
i
k + 1
X
n−i+1:n
,
respectively. The PPWM EVI, VaR and p-estimators, based on the largest values are
ˆγ
P P W M
k,n
:= 1 −
ˆa
1
(k)
ˆa
0
(k) − ˆa
1
(k)
, (1.12)
ˆ
Q
P P W M
k,n
(p) :=
ˆa
0
(k) ˆa
1
(k)
ˆa
0
(k) − ˆa
1
(k)
k
np
ˆγ
P P W M
k,n
(1.13)
and
ˆp
P P W M
k,n
(x) :=
k
n
x(ˆa
0
(k) − ˆa
1
(k))
ˆa
0
(k)ˆa
1
(k)
−1/ˆγ
P P W M
k,n
, (1.14)
respectively, with k = 1, 2, . . . , n − 1, and are consistent whenever γ < 1.
De Haan and Ferreira (2006) considered, also for γ < 1, the semi-parametric GPPWM EVI-
estimator, based on the sample of excesses over the high random level X
n−k:n
, i.e.,
ˆγ
GP P W M
k,n
:= 1 −
2ˆa
?
1
(k)
ˆa
?
0
(k) − 2ˆa
?
1
(k)
, (1.15)
with k = 1, 2, . . . , n −1, and ˆa
?
s
(k) :=
P
k
i=1
(i/k)
s
(X
n−i+1:n
−X
n−k:n
)/k , s = 0, 1. For a finite-sample
comparison between the PPWM EVI-estimators in (1.12) and the GPPWM EVI-estimators in (1.15),
see Caeiro and Gomes (2011).
1.4 Scope of the article
In Section 2, after reviewing a few results already available in the literature, we state a lemma and a
theorem related with the asymptotic properties of the PPWM-estimators, defined in (1.13) and (1.14), of
the above mentioned parameters of extreme events, the Value-at-Risk at the level p and the probability
p
x
of exceedance of a high level x, defined in (1.1) and (1.2), respectively. The performance of these
estimators, for finite samples, is illustrated, in Section 3, through a Monte-Carlo simulation study. In
Section 4, we put forward an adaptive choice of thresholds, again on the basis of bootstrap computer-
intensive methods. Applications to a real data set in the field of insurance as well as to a simulated
data set are provided in Section 5.
5
