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Bias-corrected estimation of stable tail dependence function
Accepted Manuscript Bias-corrected estimation of stable tail dependence function Jan Beirlant, Mikael Escobar-Bach, Yuri Goegebeur, Armelle Guillou PII: DOI: Reference:
S0047-259X(15)00257-2 http://dx.doi.org/10.1016/j.jmva.2015.10.006 YJMVA 4029
To appear in:
Journal of Multivariate Analysis
Received date: 12 February 2015 Please cite this article as: J. Beirlant, M. Escobar-Bach, Y. Goegebeur, A. Guillou, Bias-corrected estimation of stable tail dependence function, Journal of Multivariate Analysis (2015), http://dx.doi.org/10.1016/j.jmva.2015.10.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Bias-corrected estimation of stable tail dependence function Jan Beirlantp1q , Mikael Escobar-Bachp2q , Yuri Goegebeurp2q & Armelle Guilloup3q p1q p2q
Department of Mathematics and LStat, KU Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
p3q
Institut Recherche Math´ematique Avanc´ee, UMR 7501, Universit´e de Strasbourg et CNRS, 7 rue Ren´e Descartes, 67084 Strasbourg cedex, France [email protected]
Abstract We consider the estimation of the stable tail dependence function. We propose a biascorrected estimator and we establish its asymptotic behaviour under suitable assumptions. The finite sample performance of the proposed estimator is evaluated by means of an extensive simulation study where a comparison with alternatives from the recent literature is provided. Keywords: Multivariate extreme value statistics, stable tail dependence function, bias correction. AMS Subject classification: 62G05, 62G20, 62G32.
1
Introduction and notations
Many problems involving extreme events are inherently multivariate. For instance, de Haan and de Ronde (1998) estimate the probability that a storm will cause a sea wall near the town of Petten (the Netherlands) to collapse because of a dangerous combination of sea level and wave height. Other examples can be found in actuarial science, finance, environmental science and geology, to name but a few. A fundamental question that arises when studying more than one variable is that of extremal dependence. Similarly to classical statistics one can summarise 1
extremal dependency in a number of well-chosen coefficients that give a representative picture of the dependency structure. Here, the prime example of such a dependency measure is the coefficient of tail dependence (Ledford and Tawn, 1997). Alternatively, a full characterization of the extremal dependence between variables can be obtained from functions like e.g. the stable tail dependence function, the spectral distribution function or the Pickands dependence function. We refer to Beirlant et al. (2004) and de Haan and Ferreira (2006), and the references therein, for more details. In this paper we will focus on bias-corrected estimation of the stable tail dependence function. For any arbitrary dimension d, let pX p1q , ..., X pdq q be a multivariate vector with continuous marginal cumulative distribution functions (cdfs) F1 , ..., Fd . The stable tail dependence function is defined for each xi
PR
, i 1, ..., d, as
lim tP 1 F1 pX p1q q ¤ t1 x1
Ñ8
t
or
... or
1 Fd pX pdq q ¤ t1 xd
Lpx1, ..., xdq
which can be rewritten as
lim t 1 F F11 p1 t1 x1 q, ..., Fd1 p1 t1 xd q
Ñ8
t
Lpx1, ..., xdq
(1)
where F is the multivariate distribution function of the vector pX p1q , ..., X pdq q. Now, consider a sample of size n drawn from F and an intermediate sequence k
kn, i.e. k Ñ 8
Ñ 8 with k{n Ñ 0. Let us denote x px1, ..., xdq a vector of the positive quadrant Rd pj q and Xk,n the k th order statistic among n realisations of the margins X pj q , j 1, ..., d. The
as n
empirical estimator of L is then given by p k px q L
n 1¸ 1l p1q p1q k i1 tXi ¥Xnrkx1 s
1,n
or
...
or
pdq ¥X pdq nrkxd s
Xi
1,n
u.
The asymptotic behaviour of this estimator was first studied by Huang (1992); see also Drees and Huang (1998), and de Haan and Ferreira (2006). As is common in extreme value statistics, p k pxq is affected by bias, which often complicates its application in the empirical estimator L
practice. This bias-issue will be addressed in the present paper.
2
In the univariate framework there are numerous contributions to the bias-corrected estimation of the extreme value index and tail probabilities. Typically, the bias reduction of estimators for tail parameters is obtained by taking the second order structure of an extreme value model explicitly into account in the estimation stage. We refer here to Beirlant et al. (1999), Feuerverger and Hall (1999), Matthys and Beirlant (2003), and more recently, Gomes et al. (2008) and Caeiro et al. (2009) . In the bivariate framework some attention has been paid to bias-corrected estimation of the coefficient of tail dependence η. Goegebeur and Guillou (2013) obtained the bias correction by a properly weighted sum of two biased estimators, whereas Beirlant et al. (2011) fitted the extended Pareto distribution to properly transformed bivariate observations. Recently, a robust and bias-corrected estimator for η was introduced by Dutang et al. (2014). For what concerns the stable tail dependence function we are only aware of the estimator recently proposed by Foug`eres et al. (2015). p k pxq or a function of it, we need to assume For the sequel, in order to study the behaviour of L
some conditions mentioned below and well-known in the extreme value framework: First order condition: The limit in (1) exists and the convergence is uniform on r0, T sd for T
¡ 0;
Second order condition: There exist a positive function α such that αptq
Ñ 0 as t Ñ 8
and a non null function M such that for all x with positive coordinates ( 1 t 1 F F11 p1 t1 x1 q, ..., Fd1 p1 t1 xd q Lpxq M pxq, tÑ8 αptq
lim
uniformly on r0, T sd for T
(2)
¡ 0;
Third order condition: There exist a positive function β such that β ptq Ñ 0 as t Ñ 8 and a non null function N such that for all x with positive coordinates 1 lim tÑ8 β ptq
#
t 1 F F11 p1 t1 x1 q, ..., Fd1 p1 t1 xd q αptq 3
+
Lpxq M pxq N pxq,
(3)
uniformly on r0, T sd for T
¡ 0. This requires that N is not a multiple of M .
Note that these assumptions imply that the functions α and β are both regularly varying with indices ρ and ρ1 respectively which are non positive. In the sequel we assume that both indices are negative. Remark also that the functions L, M and N have an homogeneity property, that is 1 Lpaxq aLpxq, M paxq a1ρ M pxq and N paxq a1ρρ N pxq for a positive scale parameter a. p k given in PropoIn this paper, based on the process representation for the empirical estimator L
sition 2 in Foug`eres et al. (2015), we introduce a novel bias correction procedure. Indeed, we propose to estimate the bias directly and then to subtract it from our uncorrected kernel estimator. This is an alternative to the approach proposed by Foug`eres et al. (2015) where differences between estimators of L are used to eliminate the bias. Moreover, in the spirit of Gomes et al. (2008), we show that using the present approach the bias is decreased while keeping the asymptotic variance at the level of the uncorrected kernel estimator. These theoretical results are also complemented by improved finite sample behavior.
The remainder of our paper is organised as follows. In the next section we introduce our estimators for Lpxq, as well as for the second order quantities ρ and α, and study their asymptotic properties. The finite sample performance of the proposed bias-corrected estimator and of some estimators from the recent literature are evaluated by a simulation experiment in Section 3. The proofs of all results are given in the Appendix.
2
Estimators and asymptotic properties
Consider now the rescaled version p k,a pxq : a1 L p k paxq L
for a positive scale parameter a. Our first aim is to look at the behaviour of r k pxq : L
k 1 ¸ p k,a pxq K pa j qL j k j 1
4
where aj : that
1, ..., k, and K is a function defined on p0, 1q which is positive and such 0 K puqdu 1. This function is called a kernel function in the sequel. Let ej be a dvector
³1
j k 1, j
with zeros, except for position j where it is one. Theorem 1: Let X1 , ..., Xn be independent multivariate random vectors in Rd with common joint cdf F and continuous marginal cdfs Fj , j
1, ..., d. Assume that the third order condition
(3) holds with negative indices ρ and ρ1 and that the first order partial derivatives of L, say δj L, exist and that δj L is continuous on the set of points tx
P Rd
: xj
¡ 0u. Suppose further that
the function M is continuously differentiable and N continuous. Let K be a kernel such that
? 8. Assuming that the intermediate sequence k satisfies kαpn{kq Ñ 8 and ? kαpn{k qβ pn{k q Ñ 0 as n Ñ 8, we have # + » 1
k k n ? r 1 ¸ 1 ¸ d ρ ÝÑ K puqu du ZL pxq (4) k Lk p xq K paj qLpxq α K pa j qa j M px q k k k ³1 0
K puqu1{2 du
j 1
in Dpr0, T sd q for every T
j 1
1 2
0
¡ 0 where ZL pxq : WL pxq
d ¸
WL pxj ej qδj Lpxq
j 1
with WL a continuous centered Gaussian process with covariance structure ErWL pxqWL pyqs µtRpxq X Rpyqu where Rpxq tu P Rd : there exists j such that 0 ¤ uj
¤ xj u
and µ is the measure defined as µ tRpxqu : Lpxq.
Theorem 1 is a direct application of Proposition 2 in Foug`eres et al. (2015) combining with the homogeneity properties of L and M mentioned above and the equality in distribution between the processes ZL puxq and
?uZ pxq. L
From our Theorem 1, we can easily deduce the following
corollary which gives the asymptotic behaviour of an uncorrected estimator, 5
1 k
r pxq L °k k
j 1
p q , for L.
K aj
Corollary 1: Under the assumptions of Theorem 1, we have
?
#
k
r k px q L
1 k
°k
j 1K
Lpxq α
pa j q
in Dpr0, T sd q for every T
n
k
M p xq
1 k
°k
+
p q ρ d » 1 ÝÑ K puqu p q 0
j 1 K aj aj 1 °k j 1 K aj k
1 2
du
Z L p xq
¡ 0. r pxq L °k k
M p xq K paj q as well as the second order rate parameter ρ. These quantities will be estimated externally with
Now the idea is to remove from
1 k
the same intermediate sequence k
the bias term by estimating the function α
j 1
kn, which is such that k opkq.
n k
This idea was originally
proposed in the univariate framework by Gomes and co-authors (see Caeiro et al., 2009) and has the advantage that the variance of the reduced bias estimator is the same as that of the uncorrected estimator.
First we have to define an estimator for ρ. Similarly to Foug`eres et al. (2015), we propose the following estimator:
∆k,a prx q 1 ρrk px q : 1 log ^0
∆k,a px
log r
at a fixed dvector x , where r
q
(5)
P p0, 1q and
r k paxq L r k px q . ∆k,a pxq : a1 L
Proposition 1. For any fixed dvector x , under the assumptions of Theorem 1, we have
?
³1 1 ρ 12 ZL px q 0 K puqu 2 du a1{2 1 d 1r kα pρrk px q ρq ÝÑ log r M pxq ³1 ρ du aρ 1 . k K p u q u 0 n
qk pxq : αpn{k qM pxq. To this aim consider the following Secondly we study the estimation of α
regression model, inspired from Proposition 2 of Foug`eres et al. (2015): p k,a pxq Lpxq L j
ρ q k p x qa j α 6
εj ,
j
1, . . . , k,
(6)
where ε1 , . . . , εk are random error terms. Straightforward application of least squares estimation to (6), where ρ is fixed at ρrk px q given in (5), leads to
ρrk px q aρrk px q Lp pxq k,aj aj ` rk pxq : ° α ° ρrk px q aρrk px q aρrk px q . k k a `1 j j 1 j ` °k
°k
` 1
j 1
(7)
The aim of the next proposition is to establish the asymptotic behaviour of this estimator. Proposition 2. For any fixed dvector x , under the assumptions of Theorem 1, we have
?
r k p xq α k α
n
k
M px q
ÝÑ d
³1
ρ2 ρ 1 1 rρ 2 M pxq 0 K puqu 2 du a1{2 1 Z L px q ³ log r M px q 1 K puquρ du aρ 1 ρp1 ρqp1 2ρq 1
2p1 ρq Z L p xq ρ
in Dpr0, T sd q for every T
1
0
¡ 0.
We have now all the ingredients to study the behaviour of our bias-corrected estimator for L, namely Lk,k pxq :
r k px q L
ρr px q k k k
1 k
rk pxq k1 α
°k
j 1K
°k
j 1K
pa j q
q
paj qaj ρr px k
.
This leads to our Theorem 2. Theorem 2: Under the assumptions of Theorem 1, satisfied for two intermediate sequences k and k¯ such that k
opk¯q we have » 1 ? d K puqu k Lk,k pxq Lpxq ÝÑ 0
in Dpr0, T sd q for every T
1 2
du
Z L p xq
(8)
¡ 0.
Note that the limiting process is independent of the value of ρ, and that the bias-corrected estimator has the same asymptotic variance as the uncorrected estimator of Corollary 1.
A problem of interest could be now to minimize the variance in our Theorem 2. To this aim, we illustrate in Corollary 2 that if we take a power kernel, that is a kernel of the form
¡ 1{2, the variance of our bias-corrected estimator Lk,k pxq p k pxq, i.e. is decreasing as τ increases, and it can reach the variance of the empirical estimator L
K ptq
pτ
1q tτ 1lttPp0,1qu with τ
7
the variance of ZL pxq (see Proposition 2 in Foug`eres et al., 2015). Corollary 2: Under the assumptions of Theorem 2, for the power kernel K ptq pτ 1q tτ 1lttPp0,1qu with τ
¡ 1{2, we have
?
in Dpr0, T sd q for every T
3
k Lk,k pxq Lpxq
1 d ÝÑ 2 1
τ Z L px q 2τ
¡ 0.
Simulation experiment
In order to evaluate the finite sample behaviour of our estimator Lk,k pxq, we perform a simulation
p k pxq, and two bias-corrected study and we compare our estimator with the empirical one, L
estimators recently proposed by Foug`eres et al. (2015), defined as follows p k,a pxq ∆ p L9 k,a,k pxq : L
p ρpk px q 1q1{ρpk px q pxq
k, a
p k px q ∆ p paxq L p k paxq∆ p px q L k,a k,a
r L k,a,k pxq :
p px q p paxq a∆ ∆ k,a k,a
p k,a pxq is defined similarly as ∆k,a pxq but based on the empirical estimator, that is where ∆ p k,a pxq : a1 L p k paxq L p k p xq ∆ p k,a pxq: and ρpk px q similarly as ρrk px q but based on ∆
ρpk px q :
1
∆ 1 p k,a prx q log p k,a px q ∆ log r
^ 0.
(9)
For simplicity, we focus on R2 and using the homogeneity property, we consider only the estimation of Lpt, 1 tq for t P p0, 1q, corresponding to Pickands dependence function, and the same distributions as in Foug`eres et al. (2015), namely
the Cauchy distribution, for which Lpx, yq px2
the Student(ν) distribution, for which p y {xq1{ν θ ? ? 2 ν 1 Lpx, y q y Fν 1 1θ 8
y 2 q1{2 ;
x Fν
1
px?{yq1{ν θ ?ν 1 θ2
1
where Fν
1
is the cdf of the univariate Student distribution with ν
1 degrees of freedom and
0.5 and ν 2;
the bivariate Pareto of type II model, for which Lpx, yq x y pxp ypq1{p. We set p 3 and we called this model BPII(3);
the Symmetric logistic model, for which Lpx, yq px1{s y1{sqs. We set s 1{3;
the Archimax model with the logistic generator Lpx, yq px2 y2q1{2 and with the mixed generator Lpx, y q px2 y 2 xy q{px y q. θ is the Pearson correlation coefficient. We set θ
For each distribution, we simulate 1000 samples of size 1000. As recommended by Foug`eres et
r 0.4, k 990, and we take x x. Since any stable tail dependence function satisfies maxpt, 1 tq ¤ Lpt, 1 tq ¤ 1, all the estimators have been al. (2015), we use the values a
corrected so that they satisfy these bounds. First, in Figure 1 we give the boxplots of ρpk p0.5, 0.5q and ρrk p0.5, 0.5q for a power kernel with τ
0, 5 and 10 in case of a Studentp2q distribution. It is clear from these boxplots that the esti-
mators for ρ perform reasonably well with respect to the median, but with some volatility, except in case τ
0 where it is reduced. However, as we will see in the other figures, these uncertainties
seem to have fortunately no impact on the performance of our estimators for L. Note also that our estimator ρrk px q is at least as competitive as the one proposed by Foug`eres et al. (2015),
¥ 0. Moreover, to avoid problems in the computation of Lk,a,k pxq due to the fact that ρpk px q can be too close to 0, we set ρpk px q 1 if ρpk px q P r0.1, 0s and its definition given in (9) otherwise. Similarly, for consistency, this modification has been used for ρrk px q. whatever the value of τ
9
Next, we examine the performance of the estimators for Lpxq at specific points in R2 . In Figure 2 and 3 we show the sample mean (left) and the empirical mean squared error (MSE, right) of Lk,k pxq with τ
0 (dashed line), τ 5 (full line) and τ 10 (dotted line) as a function of k, for x p0.5, 0.5q and x p0.2, 0.8q, respectively, on three of the above mentioned distributions for brevity. As is clear from these figures, in general the estimators have for all values of τ a nice stable behaviour close to the true value of Lpxq. This can be expected since our estimators
9
are bias-corrected. In terms of MSE the best performance is at both x positions obtained for τ
5, and therefore in subsequent comparisons of estimators we will focus on this choice for our
estimator. It is noteworthy to observe that the MSE has a very low curvature, so the MSE-values are very close to the minimum of the MSE for a very wide range of values for k. In Figures 4 and 5 we compare the performance of Lk,k pxq (full line) with the two estimators proposed by
r Foug`eres et al. (2015), L9 k,a,k pxq (dotted line) and L k,a,k pxq (dashed line), and the empirical one, p k pxq (dash-dotted line), for x p0.5, 0.5q and x p0.2, 0.8q, respectively. From the plots of the L
r sample means we can clearly see the bias-correcting effect of Lk,k pxq and L k,a,k pxq: compared to
the empirical estimator these estimators have a stable sample path that is close to the true value of Lpxq and this for a wide range of values of k. The estimator L9 k,a,k pxq has some bias-correcting effect, though the gain relative to the empirical estimator is quite distribution dependent. Unlike r L k,a,k pxq, the estimator Lk,k pxq has a very smooth sample path, which can be expected as we
take essentially a weighted sum of the empirical estimator calculated over different values of x. r In terms of MSE, Lk,k pxq has a better performance than L9 k,a,k pxq and L k,a,k pxq. Compared
to the empirical estimator, one can see that it has an MSE value that is at least as good as p k pxq, but it has this low MSE over a very wide range of k-values. that of the minimum MSE for L
Finally, we compare the different estimators for L not at a single point but for the whole function, using the absolute bias and MSE, computed over N Abiaspk q : MSEpk q :
1000 replications as follows:
10 N 1 ¸ 1 ¸ p piq Lk xt N 10
1 10N
p q p q
t 1 i 1 10 N ¸ ¸
L xt
piq
p px t q L px t q L k
2
where txt : ; t 1, ..., 10u and Lppkiq is the estimate based on the ith sample. We p piq by Lpiq , L9 piq and L r piq . In Figure 6 we show the Abiaspk q of the estinaturally replace L k k,k k,a,k k,a,k t 10 , 1
t 10
t 1i 1
mators under consideration as a function of k for the six distributions mentioned above. Figure 7 displays MSEpk q. These global performance measures lead us to the same conclusions as the aforementioned study at specific points. Because of the bias-correction, the estimators Lk,k and r L k,a,k keep the bias low for a wide range of k-values. Overall, the estimator Lk,k has a more r smooth sample path than L9 k,a,k and L k,a,k , and therefore it also outperforms the estimators of
10
Foug`eres et al. (2015) in terms of minimal MSE. Also here, the MSE of our estimator is close to or better than the minimal MSE value of the empirical estimator, and this for a wide range
−2
−1
0
of k-values.
−5
−4
−3
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
rho_hat
● ● ● ● ●
● ● ● ● ●
● ● ● ● ● ● ●
● ● ●
● ● ● ● ● ●
● ● ● ●
● ● ● ●
● ● ● ● ● ● ●
●
tau=0 ●
● ● ●
tau=5
tau=10 ● ●
●
● ● ●
●
●
Figure 1: Boxplots of ρpk p0.5, 0.5q and ρrk p0.5, 0.5q for a power kernel with τ
of a Studentp2q distribution.
4
0, 5 and 10 in case
Appendix: Proofs
4.1
Proof of Proposition 1
The key elements in the proof are the homogeneity of the functions L and M together with the equality in distribution between the processes ZL paxq and
?aZ pxq. L
Thus from our Theorem
1, we deduce that n 1 ∆k,a px q α M p x q
k
k
k ¸
ρ ρ K paj qa 1s j ra
j 1
Z L p x q ? k
»1 0
1 1 K puqu 2 dura 2
1s
oP
?1 k
A similar expression can be obtained for ∆k,a prx q from which we can easily justify the expression
of our estimator ρrk px q given in (5), and deduce that ρrk px q ρ
Z L p x q 1 rρ 2 1 ? log r k α nk M px q 1
³1
2 du a 2 1 0 K puqu ° ρ aρ 1 k 1 j 1 K paj qaj 1
k
from which we can easily deduce our Proposition 1. 11
1
oP
? 1 kα
n k
,
.
0.005 0.004
0.90
MSE
0.003
0.85
0.002
Mean
0.80
0.000
0.001
0.75 0.70 0
200
400
600
800
1000
0
200
400
600
800
1000
600
800
1000
600
800
1000
k
MSE
0.75
0.000
0.001
0.80
0.002
0.85
Mean
0.003
0.90
0.004
0.95
0.005
k
0
200
400
600
800
1000
0
200
400 k
MSE
0.65
0.000
0.001
0.70
0.002
0.75
Mean
0.003
0.80
0.004
0.85
0.005
k
0
200
400
600
800
1000
0
k
200
400 k
12 Figure 2: Mean (left) and MSE (right) of our estimator Lk,k p0.5, 0.5q for different values of τ : τ
0 (dashed line), τ 5 (full line), τ 10 (dotted line).
Three distributions have been
considered: First line: Student(2); Second line: Cauchy; Third line: BPII(3).
0.005 0.004
0.95
0.003
0.90
0.002
MSE
Mean
0.001
0.85
0.000
0.80 0
200
400
600
800
1000
0
200
400
600
800
1000
600
800
1000
600
800
1000
k
MSE 0.001
0.002
0.90
0.000
0.85
Mean
0.003
0.95
0.004
1.00
0.005
k
0
200
400
600
800
1000
0
200
400 k
0.003 0.000
0.75
0.001
0.80
0.002
MSE
0.85
Mean
0.90
0.004
0.005
k
0
200
400
600
800
1000
0
k
200
400 k
13 Figure 3: Mean (left) and MSE (right) of our estimator Lk,k p0.2, 0.8q for different values of τ : τ
0 (dashed line), τ 5 (full line), τ 10 (dotted line).
Three distributions have been
considered: First line: Student(2); Second line: Cauchy; Third line: BPII(3).
0.015
0.90
MSE
0.010
0.85
0.005
Mean
0.80
0.000
0.75 0.70 0
200
400
600
800
1000
0
200
400
0.95
800
1000
600
800
1000
600
800
1000
MSE
0.010
0.90 0.75
0.000
0.80
0.005
0.85
Mean
600 k
0.015
k
0
200
400
600
800
1000
0
200
400 k
MSE
0.65
0.000
0.70
0.005
0.75
Mean
0.010
0.80
0.85
0.015
k
0
200
400
600
800
1000
0
200
k
14 Figure 4: Mean (left) and MSE (right) of our estimator Lk,k at x
400 k
p0.5, 0.5q with τ
5
r (full line) compared to the estimators of Foug`eres et al. (2015): L9 k,a,k (dotted line) and L k,a,k p k (dash-dotted line). Three distributions have been (dashed line) and the empirical estimator L
considered: First line: Student(2); Second line: Cauchy; Third line: BPII(3).
0.015 0.010
0.95 0.90
MSE
Mean
0.005
0.85
0.000
0.80 0
200
400
600
800
1000
0
200
400
1.00
800
1000
600
800
1000
600
800
1000
MSE
0.010
0.95
0.000
0.85
0.005
0.90
Mean
600 k
0.015
k
0
200
400
600
800
1000
0
200
400 k
0.005
MSE
0.85
0.000
0.75
0.80
Mean
0.010
0.90
0.015
k
0
200
400
600
800
1000
0
200
k
15 Figure 5: Mean (left) and MSE (right) of our estimator Lk,k at x
400 k
p0.2, 0.8q with τ
5
r (full line) compared to the estimators of Foug`eres et al. (2015): L9 k,a,k (dotted line) and L k,a,k p k (dash-dotted line). Three distributions have been (dashed line) and the empirical estimator L
considered: First line: Student(2); Second line: Cauchy; Third line: BPII(3).
0.10 0.00
0.02
0.04
Abias
0.06
0.08
0.10 0.08 0.06 Abias 0.04 0.02 0.00 0
200
400
600
800
1000
0
200
400
600
800
1000
600
800
1000
600
800
1000
0.08 0.06 Abias 0.04 0.02 0.00
0.00
0.02
0.04
Abias
0.06
0.08
0.10
k
0.10
k
0
200
400
600
800
1000
0
200
400
0.08 0.06 Abias 0.04 0.02 0.00
0.00
0.02
0.04
Abias
0.06
0.08
0.10
k
0.10
k
0
200
400
600
800
1000
k
16 Figure 6: Absolute bias of our estimator Lk,k with τ
0
200
400 k
5 (full line) compared to the estimators of
r Foug`eres et al. (2015): L9 k,a,k (dotted line) and L k,a,k (dashed line) and the empirical estimator p k (dash-dotted line). Six distributions have been considered: First line: Student(2) (left), L
Cauchy (right); Second line: BPII(3) (left), Symmetric logistic (right); Third line: Archimax mixed (left), Archimax logistic (right).
0.015 0.000
0.005
MSE
0.010
0.015 0.010 MSE 0.005 0.000 0
200
400
600
800
1000
0
200
400
600
800
1000
600
800
1000
600
800
1000
0.010 MSE 0.005 0.000
0.000
0.005
MSE
0.010
0.015
k
0.015
k
0
200
400
600
800
1000
0
200
400
0.010 MSE 0.005 0.000
0.000
0.005
MSE
0.010
0.015
k
0.015
k
0
200
400
600
800
1000
k
0
200
400 k
17 Figure 7: MSE of our estimator Lk,k with τ 5 (full line) compared to the estimators of r Foug`eres et al. (2015): L9 k,a,k (dotted line) and L k,a,k (dashed line) and the empirical estimator p k (dash-dotted line). Six distributions have been considered: First line: Student(2) (left), L
Cauchy (right); Second line: BPII(3) (left), Symmetric logistic (right); Third line: Archimax mixed (left), Archimax logistic (right).
4.2
Proof of Proposition 2
To prove the proposition, we use the Skorohod construction, meaning that we have to look at the difference D :
rk x k α
?
p qα
n
k
M px q
³1
ρ2 ρ 1 1 rρ 2 M pxq 0 K puqu 2 du a1{2 1 Z L px q ³ log r M px q 1 K puquρ du aρ 1 ρp1 ρqp1 2ρq 1
1
0
2p1 ρ ρq ZLpxq
and to show that it tends almost surely uniformly to 0 as n
Ñ 8.
According to Proposition 2
in Foug`eres et al. (2015) and using (7), we have D
¤
°k °k ρ aρrk px q aρrk px q aρrk px q j 1 `1 aj j j ` kα n M x °k °k ρ r p x q ρ r p x q ρrk px q k k k aj a` j 1 `1 aj ³1 1 1
?
r
pq
r
sr
s
s
ρ 2 1{2 2 2 1 logr r MMppxxqq ³01 K puqu ρ du aaρ 11 ρp1ρ ρqpρ112ρq ZLpxq 0 K puqu du 1 °k °k 2 ρrk px q ρrk px q a r a a s 2 p 1 ρ q j 1 ` 1 j j ` ρ ZLpxq op1q. °k °k ρrk px q ρrk px q ρrk px q s a` raj j 1 `1 aj
The main idea now is to replace everywhere the terms with ρrk px q by the same terms with ρ and to study the difference by the mean value theorem. For instance, if we look at the term k k 1 ¸ ¸ ρrk px q ρrk px q ρrk px q s a r a a j j ` k 2 j 1 `1
appearing several times as a denominator in the bound of D, we can rewrite it as follows: k k 1 ¸ ¸ ρ ρ ρ a raj a ` s k 2 j 1 ` 1 j
k k 1 ¸ ¸ ρrk px q ρrk px q ρrk px q q aρ paρ aρ q a p a a j j j ` ` k 2 j 1 `1 j
ρ2 p1 ρq2p1 2ρq
o
2pρrk pxq ρq 1 k
k ¸
?1 k
2ρq px q ln a a k j
j
1 2pρrk px q ρq k
j 1
k ¸
j 1
ρq px q a k j
k 1 ¸ ρqk px q a ln aj k j 1 j
by the mean value theorem where ρqk px q is an intermediate value between ρrk px q and ρ. Using the same approach for each term combining with Proposition 1 leads to Proposition 2. 18
4.3
Proof of Theorem 2
According to our Theorem 1, we clearly have the following decomposition
?
k Lk,k pxq Lpxq
ρrk px q
paj qaj ρ 1 k j 1 K pa j q k ρ aρr px q s 1 °k ? k ρr pxq K p a qr a j j 1 j j k r k px q k α 1 °k k K p a q j j 1 k ³ 1 du 0 K puqu ZL pxq oP p1q ° k 1 j 1 K paj q k °k ρr px q
ρ 1 ? n k n j 1 K paj qaj k k α k M px q k α M px q 1 °k k j 1 K paj q k
?
k α
n
M px q
k k
α r k p xq
1 k
°k
j 1K
°k
k
k
1 2
k
ρrk px q 12 a
n k OP kα ρrk x
k
³
pq
p p q ρq
k
12 du
1 0K u u 1 °k j 1K k
paj q
Z L px q
oP p1q
by applying our Proposition 2 and again the mean value theorem. Under the assumptions of our Theorem 2, we have
?
k Lk,k pxq Lpxq
?
k α ³
n
k
pq
M px q
du
1 0K u u 1 °k j 1K k
1 2
ρrk px q
paj q
19
k k
Z L px q
α
n k
oP p1q.
M px q
1 k
°k
p q ρ p q
j 1 K aj aj ° k 1 j 1 K aj k
Recall now that the function α is regularly varying with index ρ
0, that is αptq tρ`αptq
where `α is slowly varying at infinity. Thus
?
k Lk,k pxq Lpxq
c a k
n k
kα
k
c a k ³
³
n k
pq
12 du
1 0K u u 1 °k j 1K k
o P p1 q
paj q
OP
Z L p xq
2
ρrk px q
k k
`α p nk q `α p nk q
1
1 k
p q ρ p q p q p q
°k
j 1 K aj aj ° k 1 j 1 K aj k ρ 1 °k j 1 K aj aj k 1 °k j 1 K aj k
o P p1 q
Z L p xq
1 ρq
k k
1 ρ a
k k
paj q
M px q
k k
12 du
pq
1 0K u u 1 °k j 1K k
k k
ρ
kα
k
M px q
ρ
k
2
n k
kα
px q
ln
β
n k
a k
kα
k
n k
`α p nk q `α p nk q β n 1
pρrk pxq ρq
1
O p1 q
k
by the mean value theorem. Theorem 2 now follows under the assumptions since 1 lim tÑ8 β ptq
`α ptcq `α ptq
1 Op1q
(10)
when t Ñ 8 and c cptq Ñ 8. To be convinced by (10), remark that for all x, y
¡ 0 and t ¡ 0
we have with β1 ptq αptqβ ptq and Lt px1q t 1 F F11 p1 t1 xq, ..., Fd1 p1 t1 xq , and
using the fact that Lt px1q xLt{x p1q, that 1 x
"
*
Lt pxy1q Lpxy1q αptqM pxy1q Lt px1q Lpx1q αptqM px1q β 1 pt q β1 ptq Lt{x py1q Lpy1q αpt{xqM py1q Lt{x p1q Lp1q αpt{xqM p1q β1 pt{xq β1 pt{xq β1 pt{xq β1 ptq ( ρ αpt{xq αptqx {β1ptq pM py1q M p1qq .
Taking limits for t Ñ 8 we obtain that 1 pN pxy1q N px1qq x "
tlim Ñ8
pN py1q N p1qq ββ1ptp{txq q 1
1 β ptq
20
αpt{xq α pt q
x ρ
pM py1q M p1qq
*
.
(11)
Since the limit for t Ñ 8 of β1 pt{xq{β1 ptq exists, from (11) the limit for t Ñ 8 of
1 β t
pq
p { q x ρ pq
α t x α t
exists such that, combining with Theorem B.2.1 in de Haan and Ferreira (2006):
αpt{xq `α pt{xq 1 1 xρ xρ lim 1 xρψp1{xq, lim tÑ8 β ptq tÑ8 β ptq αptq `α ptq ρ1 1 with ψ pxq a x ρ 1 and a ¡ 0 (in case a 0 one can redefine `α as `α and the result will follow with a ¡ 0). Note that the positive constant a can in fact be incorporated in the function β, so from now on we take without loss of generality a 1. We have thus, with βr : β`α , lim
Ñ8
`α ptxq `α ptq βrptq
t
Now write, for some function βr where βr `α t `α tx βr t
p q p q ¤ pq
βr ptq βrptq
r β,
|ψpxq|
ψ px q. `α t `α tx r β t
p q p q ψpxq pq
.
By Theorem B.2.18 in de Haan and Ferreira (2006) (see also Drees, 1998) we have for any ε ¡ 0 there is a t0 such that for t, tx ¡ t0 ,
` tx ` t α α βr t
p q p q ¤ p1 κqr|ψpxq| εxρ1 δ s. pq where 0 δ ρ1 and κ ¡ 0. Since both functions in the right-hand side are bounded for x P rx0 , 8q, x0 ¡ 0, result (10) follows. Acknowledgement The authors thank the editor and the reviewers for their helpful comments which led to improvements of our first version of the paper. Also the authors kindly acknowledge Anne-Laure Foug`eres, Laurens de Haan and C´ecile Mercadier for providing the program of their six distributions considered. This work was supported by a research grant (VKR023480) from VILLUM FONDEN and an international project for scientific cooperation (PICS-6416).
References [1] Beirlant, J., Dierckx, G., Goegebeur, Y., Matthys, G., 1999. Tail index estimation and an exponential regression model. Extremes, 2, 177–200. 21
[2] Beirlant, J., Dierckx, G., Guillou, A., 2011. Bias-reduced estimators for bivariate tail modelling. Insurance: Mathematics and Economics, 49, 18–26. [3] Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J., 2004. Statistics of Extremes – Theory and Applications. Wiley. [4] Caeiro, F., Gomes, M.I., Rodrigues, L.H., 2009. Reduced-bias tail index estimators under a third order framework. Communications in Statistics - Theory and Methods, 38, 1019–1040. [5] Drees, H., 1998. On smooth statistical tail functionals. Scandinavian Journal of Statistics, 25, 187–210. [6] Drees, H., Huang, X., 1998. Best attainable rates of convergence for estimators of the stable tail dependence function. Journal of Multivariate Analysis, 64, 25–47. [7] Dutang, C., Goegebeur, Y., Guillou, A., 2014. Robust and bias-corrected estimation of the coefficient of tail dependence. Insurance: Mathematics and Economics, 57, 46–57. [8] Feuerverger, A., Hall, P., 1999. Estimating a tail exponent by modelling departure from a Pareto distribution. Annals of Statistics, 27, 760–781. [9] Foug`eres, A.L., de Haan, L., Mercadier, C., 2015. Bias correction in multivariate extremes. Annals of Statistics, 43, 903–934. [10] Goegebeur, Y., Guillou, A., 2013. Asymptotically unbiased estimation of the coefficient of tail dependence. Scandinavian Journal of Statistics, 40, 174–189. [11] Gomes, M.I., de Haan, L., Rodrigues, L.H., 2008. Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. Journal of the Royal Statistical Society Series B, 70, 31–52. [12] de Haan, L., Ferreira, A., 2006. Extreme Value Theory - An Introduction, Springer. [13] de Haan, L., de Ronde, J., 1998. Sea and wind: multivariate extremes at work. Extremes, 1, 7–45.
22
[14] Huang, X., 1992. Statistics of bivariate extremes. PhD Thesis, Erasmus University Rotterdam, Tinbergen Institute Research series No. 22. [15] Ledford, A.W., Tawn, J.A., 1997. Modelling dependence within joint tail regions. Journal of the Royal Statistical Society Series B, 59, 475–499. [16] Matthys, G., Beirlant, J., 2003. Estimating the extreme value index and high quantiles with exponential regression models. Statistica Sinica 13, 853–880.
23
S0047-259X(15)00257-2 http://dx.doi.org/10.1016/j.jmva.2015.10.006 YJMVA 4029
To appear in:
Journal of Multivariate Analysis
Received date: 12 February 2015 Please cite this article as: J. Beirlant, M. Escobar-Bach, Y. Goegebeur, A. Guillou, Bias-corrected estimation of stable tail dependence function, Journal of Multivariate Analysis (2015), http://dx.doi.org/10.1016/j.jmva.2015.10.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to download Manuscript: Rev-Lbiasred_1.pdf
Click here to view linked References
Bias-corrected estimation of stable tail dependence function Jan Beirlantp1q , Mikael Escobar-Bachp2q , Yuri Goegebeurp2q & Armelle Guilloup3q p1q p2q
Department of Mathematics and LStat, KU Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
p3q
Institut Recherche Math´ematique Avanc´ee, UMR 7501, Universit´e de Strasbourg et CNRS, 7 rue Ren´e Descartes, 67084 Strasbourg cedex, France [email protected]
Abstract We consider the estimation of the stable tail dependence function. We propose a biascorrected estimator and we establish its asymptotic behaviour under suitable assumptions. The finite sample performance of the proposed estimator is evaluated by means of an extensive simulation study where a comparison with alternatives from the recent literature is provided. Keywords: Multivariate extreme value statistics, stable tail dependence function, bias correction. AMS Subject classification: 62G05, 62G20, 62G32.
1
Introduction and notations
Many problems involving extreme events are inherently multivariate. For instance, de Haan and de Ronde (1998) estimate the probability that a storm will cause a sea wall near the town of Petten (the Netherlands) to collapse because of a dangerous combination of sea level and wave height. Other examples can be found in actuarial science, finance, environmental science and geology, to name but a few. A fundamental question that arises when studying more than one variable is that of extremal dependence. Similarly to classical statistics one can summarise 1
extremal dependency in a number of well-chosen coefficients that give a representative picture of the dependency structure. Here, the prime example of such a dependency measure is the coefficient of tail dependence (Ledford and Tawn, 1997). Alternatively, a full characterization of the extremal dependence between variables can be obtained from functions like e.g. the stable tail dependence function, the spectral distribution function or the Pickands dependence function. We refer to Beirlant et al. (2004) and de Haan and Ferreira (2006), and the references therein, for more details. In this paper we will focus on bias-corrected estimation of the stable tail dependence function. For any arbitrary dimension d, let pX p1q , ..., X pdq q be a multivariate vector with continuous marginal cumulative distribution functions (cdfs) F1 , ..., Fd . The stable tail dependence function is defined for each xi
PR
, i 1, ..., d, as
lim tP 1 F1 pX p1q q ¤ t1 x1
Ñ8
t
or
... or
1 Fd pX pdq q ¤ t1 xd
Lpx1, ..., xdq
which can be rewritten as
lim t 1 F F11 p1 t1 x1 q, ..., Fd1 p1 t1 xd q
Ñ8
t
Lpx1, ..., xdq
(1)
where F is the multivariate distribution function of the vector pX p1q , ..., X pdq q. Now, consider a sample of size n drawn from F and an intermediate sequence k
kn, i.e. k Ñ 8
Ñ 8 with k{n Ñ 0. Let us denote x px1, ..., xdq a vector of the positive quadrant Rd pj q and Xk,n the k th order statistic among n realisations of the margins X pj q , j 1, ..., d. The
as n
empirical estimator of L is then given by p k px q L
n 1¸ 1l p1q p1q k i1 tXi ¥Xnrkx1 s
1,n
or
...
or
pdq ¥X pdq nrkxd s
Xi
1,n
u.
The asymptotic behaviour of this estimator was first studied by Huang (1992); see also Drees and Huang (1998), and de Haan and Ferreira (2006). As is common in extreme value statistics, p k pxq is affected by bias, which often complicates its application in the empirical estimator L
practice. This bias-issue will be addressed in the present paper.
2
In the univariate framework there are numerous contributions to the bias-corrected estimation of the extreme value index and tail probabilities. Typically, the bias reduction of estimators for tail parameters is obtained by taking the second order structure of an extreme value model explicitly into account in the estimation stage. We refer here to Beirlant et al. (1999), Feuerverger and Hall (1999), Matthys and Beirlant (2003), and more recently, Gomes et al. (2008) and Caeiro et al. (2009) . In the bivariate framework some attention has been paid to bias-corrected estimation of the coefficient of tail dependence η. Goegebeur and Guillou (2013) obtained the bias correction by a properly weighted sum of two biased estimators, whereas Beirlant et al. (2011) fitted the extended Pareto distribution to properly transformed bivariate observations. Recently, a robust and bias-corrected estimator for η was introduced by Dutang et al. (2014). For what concerns the stable tail dependence function we are only aware of the estimator recently proposed by Foug`eres et al. (2015). p k pxq or a function of it, we need to assume For the sequel, in order to study the behaviour of L
some conditions mentioned below and well-known in the extreme value framework: First order condition: The limit in (1) exists and the convergence is uniform on r0, T sd for T
¡ 0;
Second order condition: There exist a positive function α such that αptq
Ñ 0 as t Ñ 8
and a non null function M such that for all x with positive coordinates ( 1 t 1 F F11 p1 t1 x1 q, ..., Fd1 p1 t1 xd q Lpxq M pxq, tÑ8 αptq
lim
uniformly on r0, T sd for T
(2)
¡ 0;
Third order condition: There exist a positive function β such that β ptq Ñ 0 as t Ñ 8 and a non null function N such that for all x with positive coordinates 1 lim tÑ8 β ptq
#
t 1 F F11 p1 t1 x1 q, ..., Fd1 p1 t1 xd q αptq 3
+
Lpxq M pxq N pxq,
(3)
uniformly on r0, T sd for T
¡ 0. This requires that N is not a multiple of M .
Note that these assumptions imply that the functions α and β are both regularly varying with indices ρ and ρ1 respectively which are non positive. In the sequel we assume that both indices are negative. Remark also that the functions L, M and N have an homogeneity property, that is 1 Lpaxq aLpxq, M paxq a1ρ M pxq and N paxq a1ρρ N pxq for a positive scale parameter a. p k given in PropoIn this paper, based on the process representation for the empirical estimator L
sition 2 in Foug`eres et al. (2015), we introduce a novel bias correction procedure. Indeed, we propose to estimate the bias directly and then to subtract it from our uncorrected kernel estimator. This is an alternative to the approach proposed by Foug`eres et al. (2015) where differences between estimators of L are used to eliminate the bias. Moreover, in the spirit of Gomes et al. (2008), we show that using the present approach the bias is decreased while keeping the asymptotic variance at the level of the uncorrected kernel estimator. These theoretical results are also complemented by improved finite sample behavior.
The remainder of our paper is organised as follows. In the next section we introduce our estimators for Lpxq, as well as for the second order quantities ρ and α, and study their asymptotic properties. The finite sample performance of the proposed bias-corrected estimator and of some estimators from the recent literature are evaluated by a simulation experiment in Section 3. The proofs of all results are given in the Appendix.
2
Estimators and asymptotic properties
Consider now the rescaled version p k,a pxq : a1 L p k paxq L
for a positive scale parameter a. Our first aim is to look at the behaviour of r k pxq : L
k 1 ¸ p k,a pxq K pa j qL j k j 1
4
where aj : that
1, ..., k, and K is a function defined on p0, 1q which is positive and such 0 K puqdu 1. This function is called a kernel function in the sequel. Let ej be a dvector
³1
j k 1, j
with zeros, except for position j where it is one. Theorem 1: Let X1 , ..., Xn be independent multivariate random vectors in Rd with common joint cdf F and continuous marginal cdfs Fj , j
1, ..., d. Assume that the third order condition
(3) holds with negative indices ρ and ρ1 and that the first order partial derivatives of L, say δj L, exist and that δj L is continuous on the set of points tx
P Rd
: xj
¡ 0u. Suppose further that
the function M is continuously differentiable and N continuous. Let K be a kernel such that
? 8. Assuming that the intermediate sequence k satisfies kαpn{kq Ñ 8 and ? kαpn{k qβ pn{k q Ñ 0 as n Ñ 8, we have # + » 1
k k n ? r 1 ¸ 1 ¸ d ρ ÝÑ K puqu du ZL pxq (4) k Lk p xq K paj qLpxq α K pa j qa j M px q k k k ³1 0
K puqu1{2 du
j 1
in Dpr0, T sd q for every T
j 1
1 2
0
¡ 0 where ZL pxq : WL pxq
d ¸
WL pxj ej qδj Lpxq
j 1
with WL a continuous centered Gaussian process with covariance structure ErWL pxqWL pyqs µtRpxq X Rpyqu where Rpxq tu P Rd : there exists j such that 0 ¤ uj
¤ xj u
and µ is the measure defined as µ tRpxqu : Lpxq.
Theorem 1 is a direct application of Proposition 2 in Foug`eres et al. (2015) combining with the homogeneity properties of L and M mentioned above and the equality in distribution between the processes ZL puxq and
?uZ pxq. L
From our Theorem 1, we can easily deduce the following
corollary which gives the asymptotic behaviour of an uncorrected estimator, 5
1 k
r pxq L °k k
j 1
p q , for L.
K aj
Corollary 1: Under the assumptions of Theorem 1, we have
?
#
k
r k px q L
1 k
°k
j 1K
Lpxq α
pa j q
in Dpr0, T sd q for every T
n
k
M p xq
1 k
°k
+
p q ρ d » 1 ÝÑ K puqu p q 0
j 1 K aj aj 1 °k j 1 K aj k
1 2
du
Z L p xq
¡ 0. r pxq L °k k
M p xq K paj q as well as the second order rate parameter ρ. These quantities will be estimated externally with
Now the idea is to remove from
1 k
the same intermediate sequence k
the bias term by estimating the function α
j 1
kn, which is such that k opkq.
n k
This idea was originally
proposed in the univariate framework by Gomes and co-authors (see Caeiro et al., 2009) and has the advantage that the variance of the reduced bias estimator is the same as that of the uncorrected estimator.
First we have to define an estimator for ρ. Similarly to Foug`eres et al. (2015), we propose the following estimator:
∆k,a prx q 1 ρrk px q : 1 log ^0
∆k,a px
log r
at a fixed dvector x , where r
q
(5)
P p0, 1q and
r k paxq L r k px q . ∆k,a pxq : a1 L
Proposition 1. For any fixed dvector x , under the assumptions of Theorem 1, we have
?
³1 1 ρ 12 ZL px q 0 K puqu 2 du a1{2 1 d 1r kα pρrk px q ρq ÝÑ log r M pxq ³1 ρ du aρ 1 . k K p u q u 0 n
qk pxq : αpn{k qM pxq. To this aim consider the following Secondly we study the estimation of α
regression model, inspired from Proposition 2 of Foug`eres et al. (2015): p k,a pxq Lpxq L j
ρ q k p x qa j α 6
εj ,
j
1, . . . , k,
(6)
where ε1 , . . . , εk are random error terms. Straightforward application of least squares estimation to (6), where ρ is fixed at ρrk px q given in (5), leads to
ρrk px q aρrk px q Lp pxq k,aj aj ` rk pxq : ° α ° ρrk px q aρrk px q aρrk px q . k k a `1 j j 1 j ` °k
°k
` 1
j 1
(7)
The aim of the next proposition is to establish the asymptotic behaviour of this estimator. Proposition 2. For any fixed dvector x , under the assumptions of Theorem 1, we have
?
r k p xq α k α
n
k
M px q
ÝÑ d
³1
ρ2 ρ 1 1 rρ 2 M pxq 0 K puqu 2 du a1{2 1 Z L px q ³ log r M px q 1 K puquρ du aρ 1 ρp1 ρqp1 2ρq 1
2p1 ρq Z L p xq ρ
in Dpr0, T sd q for every T
1
0
¡ 0.
We have now all the ingredients to study the behaviour of our bias-corrected estimator for L, namely Lk,k pxq :
r k px q L
ρr px q k k k
1 k
rk pxq k1 α
°k
j 1K
°k
j 1K
pa j q
q
paj qaj ρr px k
.
This leads to our Theorem 2. Theorem 2: Under the assumptions of Theorem 1, satisfied for two intermediate sequences k and k¯ such that k
opk¯q we have » 1 ? d K puqu k Lk,k pxq Lpxq ÝÑ 0
in Dpr0, T sd q for every T
1 2
du
Z L p xq
(8)
¡ 0.
Note that the limiting process is independent of the value of ρ, and that the bias-corrected estimator has the same asymptotic variance as the uncorrected estimator of Corollary 1.
A problem of interest could be now to minimize the variance in our Theorem 2. To this aim, we illustrate in Corollary 2 that if we take a power kernel, that is a kernel of the form
¡ 1{2, the variance of our bias-corrected estimator Lk,k pxq p k pxq, i.e. is decreasing as τ increases, and it can reach the variance of the empirical estimator L
K ptq
pτ
1q tτ 1lttPp0,1qu with τ
7
the variance of ZL pxq (see Proposition 2 in Foug`eres et al., 2015). Corollary 2: Under the assumptions of Theorem 2, for the power kernel K ptq pτ 1q tτ 1lttPp0,1qu with τ
¡ 1{2, we have
?
in Dpr0, T sd q for every T
3
k Lk,k pxq Lpxq
1 d ÝÑ 2 1
τ Z L px q 2τ
¡ 0.
Simulation experiment
In order to evaluate the finite sample behaviour of our estimator Lk,k pxq, we perform a simulation
p k pxq, and two bias-corrected study and we compare our estimator with the empirical one, L
estimators recently proposed by Foug`eres et al. (2015), defined as follows p k,a pxq ∆ p L9 k,a,k pxq : L
p ρpk px q 1q1{ρpk px q pxq
k, a
p k px q ∆ p paxq L p k paxq∆ p px q L k,a k,a
r L k,a,k pxq :
p px q p paxq a∆ ∆ k,a k,a
p k,a pxq is defined similarly as ∆k,a pxq but based on the empirical estimator, that is where ∆ p k,a pxq : a1 L p k paxq L p k p xq ∆ p k,a pxq: and ρpk px q similarly as ρrk px q but based on ∆
ρpk px q :
1
∆ 1 p k,a prx q log p k,a px q ∆ log r
^ 0.
(9)
For simplicity, we focus on R2 and using the homogeneity property, we consider only the estimation of Lpt, 1 tq for t P p0, 1q, corresponding to Pickands dependence function, and the same distributions as in Foug`eres et al. (2015), namely
the Cauchy distribution, for which Lpx, yq px2
the Student(ν) distribution, for which p y {xq1{ν θ ? ? 2 ν 1 Lpx, y q y Fν 1 1θ 8
y 2 q1{2 ;
x Fν
1
px?{yq1{ν θ ?ν 1 θ2
1
where Fν
1
is the cdf of the univariate Student distribution with ν
1 degrees of freedom and
0.5 and ν 2;
the bivariate Pareto of type II model, for which Lpx, yq x y pxp ypq1{p. We set p 3 and we called this model BPII(3);
the Symmetric logistic model, for which Lpx, yq px1{s y1{sqs. We set s 1{3;
the Archimax model with the logistic generator Lpx, yq px2 y2q1{2 and with the mixed generator Lpx, y q px2 y 2 xy q{px y q. θ is the Pearson correlation coefficient. We set θ
For each distribution, we simulate 1000 samples of size 1000. As recommended by Foug`eres et
r 0.4, k 990, and we take x x. Since any stable tail dependence function satisfies maxpt, 1 tq ¤ Lpt, 1 tq ¤ 1, all the estimators have been al. (2015), we use the values a
corrected so that they satisfy these bounds. First, in Figure 1 we give the boxplots of ρpk p0.5, 0.5q and ρrk p0.5, 0.5q for a power kernel with τ
0, 5 and 10 in case of a Studentp2q distribution. It is clear from these boxplots that the esti-
mators for ρ perform reasonably well with respect to the median, but with some volatility, except in case τ
0 where it is reduced. However, as we will see in the other figures, these uncertainties
seem to have fortunately no impact on the performance of our estimators for L. Note also that our estimator ρrk px q is at least as competitive as the one proposed by Foug`eres et al. (2015),
¥ 0. Moreover, to avoid problems in the computation of Lk,a,k pxq due to the fact that ρpk px q can be too close to 0, we set ρpk px q 1 if ρpk px q P r0.1, 0s and its definition given in (9) otherwise. Similarly, for consistency, this modification has been used for ρrk px q. whatever the value of τ
9
Next, we examine the performance of the estimators for Lpxq at specific points in R2 . In Figure 2 and 3 we show the sample mean (left) and the empirical mean squared error (MSE, right) of Lk,k pxq with τ
0 (dashed line), τ 5 (full line) and τ 10 (dotted line) as a function of k, for x p0.5, 0.5q and x p0.2, 0.8q, respectively, on three of the above mentioned distributions for brevity. As is clear from these figures, in general the estimators have for all values of τ a nice stable behaviour close to the true value of Lpxq. This can be expected since our estimators
9
are bias-corrected. In terms of MSE the best performance is at both x positions obtained for τ
5, and therefore in subsequent comparisons of estimators we will focus on this choice for our
estimator. It is noteworthy to observe that the MSE has a very low curvature, so the MSE-values are very close to the minimum of the MSE for a very wide range of values for k. In Figures 4 and 5 we compare the performance of Lk,k pxq (full line) with the two estimators proposed by
r Foug`eres et al. (2015), L9 k,a,k pxq (dotted line) and L k,a,k pxq (dashed line), and the empirical one, p k pxq (dash-dotted line), for x p0.5, 0.5q and x p0.2, 0.8q, respectively. From the plots of the L
r sample means we can clearly see the bias-correcting effect of Lk,k pxq and L k,a,k pxq: compared to
the empirical estimator these estimators have a stable sample path that is close to the true value of Lpxq and this for a wide range of values of k. The estimator L9 k,a,k pxq has some bias-correcting effect, though the gain relative to the empirical estimator is quite distribution dependent. Unlike r L k,a,k pxq, the estimator Lk,k pxq has a very smooth sample path, which can be expected as we
take essentially a weighted sum of the empirical estimator calculated over different values of x. r In terms of MSE, Lk,k pxq has a better performance than L9 k,a,k pxq and L k,a,k pxq. Compared
to the empirical estimator, one can see that it has an MSE value that is at least as good as p k pxq, but it has this low MSE over a very wide range of k-values. that of the minimum MSE for L
Finally, we compare the different estimators for L not at a single point but for the whole function, using the absolute bias and MSE, computed over N Abiaspk q : MSEpk q :
1000 replications as follows:
10 N 1 ¸ 1 ¸ p piq Lk xt N 10
1 10N
p q p q
t 1 i 1 10 N ¸ ¸
L xt
piq
p px t q L px t q L k
2
where txt : ; t 1, ..., 10u and Lppkiq is the estimate based on the ith sample. We p piq by Lpiq , L9 piq and L r piq . In Figure 6 we show the Abiaspk q of the estinaturally replace L k k,k k,a,k k,a,k t 10 , 1
t 10
t 1i 1
mators under consideration as a function of k for the six distributions mentioned above. Figure 7 displays MSEpk q. These global performance measures lead us to the same conclusions as the aforementioned study at specific points. Because of the bias-correction, the estimators Lk,k and r L k,a,k keep the bias low for a wide range of k-values. Overall, the estimator Lk,k has a more r smooth sample path than L9 k,a,k and L k,a,k , and therefore it also outperforms the estimators of
10
Foug`eres et al. (2015) in terms of minimal MSE. Also here, the MSE of our estimator is close to or better than the minimal MSE value of the empirical estimator, and this for a wide range
−2
−1
0
of k-values.
−5
−4
−3
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
rho_hat
● ● ● ● ●
● ● ● ● ●
● ● ● ● ● ● ●
● ● ●
● ● ● ● ● ●
● ● ● ●
● ● ● ●
● ● ● ● ● ● ●
●
tau=0 ●
● ● ●
tau=5
tau=10 ● ●
●
● ● ●
●
●
Figure 1: Boxplots of ρpk p0.5, 0.5q and ρrk p0.5, 0.5q for a power kernel with τ
of a Studentp2q distribution.
4
0, 5 and 10 in case
Appendix: Proofs
4.1
Proof of Proposition 1
The key elements in the proof are the homogeneity of the functions L and M together with the equality in distribution between the processes ZL paxq and
?aZ pxq. L
Thus from our Theorem
1, we deduce that n 1 ∆k,a px q α M p x q
k
k
k ¸
ρ ρ K paj qa 1s j ra
j 1
Z L p x q ? k
»1 0
1 1 K puqu 2 dura 2
1s
oP
?1 k
A similar expression can be obtained for ∆k,a prx q from which we can easily justify the expression
of our estimator ρrk px q given in (5), and deduce that ρrk px q ρ
Z L p x q 1 rρ 2 1 ? log r k α nk M px q 1
³1
2 du a 2 1 0 K puqu ° ρ aρ 1 k 1 j 1 K paj qaj 1
k
from which we can easily deduce our Proposition 1. 11
1
oP
? 1 kα
n k
,
.
0.005 0.004
0.90
MSE
0.003
0.85
0.002
Mean
0.80
0.000
0.001
0.75 0.70 0
200
400
600
800
1000
0
200
400
600
800
1000
600
800
1000
600
800
1000
k
MSE
0.75
0.000
0.001
0.80
0.002
0.85
Mean
0.003
0.90
0.004
0.95
0.005
k
0
200
400
600
800
1000
0
200
400 k
MSE
0.65
0.000
0.001
0.70
0.002
0.75
Mean
0.003
0.80
0.004
0.85
0.005
k
0
200
400
600
800
1000
0
k
200
400 k
12 Figure 2: Mean (left) and MSE (right) of our estimator Lk,k p0.5, 0.5q for different values of τ : τ
0 (dashed line), τ 5 (full line), τ 10 (dotted line).
Three distributions have been
considered: First line: Student(2); Second line: Cauchy; Third line: BPII(3).
0.005 0.004
0.95
0.003
0.90
0.002
MSE
Mean
0.001
0.85
0.000
0.80 0
200
400
600
800
1000
0
200
400
600
800
1000
600
800
1000
600
800
1000
k
MSE 0.001
0.002
0.90
0.000
0.85
Mean
0.003
0.95
0.004
1.00
0.005
k
0
200
400
600
800
1000
0
200
400 k
0.003 0.000
0.75
0.001
0.80
0.002
MSE
0.85
Mean
0.90
0.004
0.005
k
0
200
400
600
800
1000
0
k
200
400 k
13 Figure 3: Mean (left) and MSE (right) of our estimator Lk,k p0.2, 0.8q for different values of τ : τ
0 (dashed line), τ 5 (full line), τ 10 (dotted line).
Three distributions have been
considered: First line: Student(2); Second line: Cauchy; Third line: BPII(3).
0.015
0.90
MSE
0.010
0.85
0.005
Mean
0.80
0.000
0.75 0.70 0
200
400
600
800
1000
0
200
400
0.95
800
1000
600
800
1000
600
800
1000
MSE
0.010
0.90 0.75
0.000
0.80
0.005
0.85
Mean
600 k
0.015
k
0
200
400
600
800
1000
0
200
400 k
MSE
0.65
0.000
0.70
0.005
0.75
Mean
0.010
0.80
0.85
0.015
k
0
200
400
600
800
1000
0
200
k
14 Figure 4: Mean (left) and MSE (right) of our estimator Lk,k at x
400 k
p0.5, 0.5q with τ
5
r (full line) compared to the estimators of Foug`eres et al. (2015): L9 k,a,k (dotted line) and L k,a,k p k (dash-dotted line). Three distributions have been (dashed line) and the empirical estimator L
considered: First line: Student(2); Second line: Cauchy; Third line: BPII(3).
0.015 0.010
0.95 0.90
MSE
Mean
0.005
0.85
0.000
0.80 0
200
400
600
800
1000
0
200
400
1.00
800
1000
600
800
1000
600
800
1000
MSE
0.010
0.95
0.000
0.85
0.005
0.90
Mean
600 k
0.015
k
0
200
400
600
800
1000
0
200
400 k
0.005
MSE
0.85
0.000
0.75
0.80
Mean
0.010
0.90
0.015
k
0
200
400
600
800
1000
0
200
k
15 Figure 5: Mean (left) and MSE (right) of our estimator Lk,k at x
400 k
p0.2, 0.8q with τ
5
r (full line) compared to the estimators of Foug`eres et al. (2015): L9 k,a,k (dotted line) and L k,a,k p k (dash-dotted line). Three distributions have been (dashed line) and the empirical estimator L
considered: First line: Student(2); Second line: Cauchy; Third line: BPII(3).
0.10 0.00
0.02
0.04
Abias
0.06
0.08
0.10 0.08 0.06 Abias 0.04 0.02 0.00 0
200
400
600
800
1000
0
200
400
600
800
1000
600
800
1000
600
800
1000
0.08 0.06 Abias 0.04 0.02 0.00
0.00
0.02
0.04
Abias
0.06
0.08
0.10
k
0.10
k
0
200
400
600
800
1000
0
200
400
0.08 0.06 Abias 0.04 0.02 0.00
0.00
0.02
0.04
Abias
0.06
0.08
0.10
k
0.10
k
0
200
400
600
800
1000
k
16 Figure 6: Absolute bias of our estimator Lk,k with τ
0
200
400 k
5 (full line) compared to the estimators of
r Foug`eres et al. (2015): L9 k,a,k (dotted line) and L k,a,k (dashed line) and the empirical estimator p k (dash-dotted line). Six distributions have been considered: First line: Student(2) (left), L
Cauchy (right); Second line: BPII(3) (left), Symmetric logistic (right); Third line: Archimax mixed (left), Archimax logistic (right).
0.015 0.000
0.005
MSE
0.010
0.015 0.010 MSE 0.005 0.000 0
200
400
600
800
1000
0
200
400
600
800
1000
600
800
1000
600
800
1000
0.010 MSE 0.005 0.000
0.000
0.005
MSE
0.010
0.015
k
0.015
k
0
200
400
600
800
1000
0
200
400
0.010 MSE 0.005 0.000
0.000
0.005
MSE
0.010
0.015
k
0.015
k
0
200
400
600
800
1000
k
0
200
400 k
17 Figure 7: MSE of our estimator Lk,k with τ 5 (full line) compared to the estimators of r Foug`eres et al. (2015): L9 k,a,k (dotted line) and L k,a,k (dashed line) and the empirical estimator p k (dash-dotted line). Six distributions have been considered: First line: Student(2) (left), L
Cauchy (right); Second line: BPII(3) (left), Symmetric logistic (right); Third line: Archimax mixed (left), Archimax logistic (right).
4.2
Proof of Proposition 2
To prove the proposition, we use the Skorohod construction, meaning that we have to look at the difference D :
rk x k α
?
p qα
n
k
M px q
³1
ρ2 ρ 1 1 rρ 2 M pxq 0 K puqu 2 du a1{2 1 Z L px q ³ log r M px q 1 K puquρ du aρ 1 ρp1 ρqp1 2ρq 1
1
0
2p1 ρ ρq ZLpxq
and to show that it tends almost surely uniformly to 0 as n
Ñ 8.
According to Proposition 2
in Foug`eres et al. (2015) and using (7), we have D
¤
°k °k ρ aρrk px q aρrk px q aρrk px q j 1 `1 aj j j ` kα n M x °k °k ρ r p x q ρ r p x q ρrk px q k k k aj a` j 1 `1 aj ³1 1 1
?
r
pq
r
sr
s
s
ρ 2 1{2 2 2 1 logr r MMppxxqq ³01 K puqu ρ du aaρ 11 ρp1ρ ρqpρ112ρq ZLpxq 0 K puqu du 1 °k °k 2 ρrk px q ρrk px q a r a a s 2 p 1 ρ q j 1 ` 1 j j ` ρ ZLpxq op1q. °k °k ρrk px q ρrk px q ρrk px q s a` raj j 1 `1 aj
The main idea now is to replace everywhere the terms with ρrk px q by the same terms with ρ and to study the difference by the mean value theorem. For instance, if we look at the term k k 1 ¸ ¸ ρrk px q ρrk px q ρrk px q s a r a a j j ` k 2 j 1 `1
appearing several times as a denominator in the bound of D, we can rewrite it as follows: k k 1 ¸ ¸ ρ ρ ρ a raj a ` s k 2 j 1 ` 1 j
k k 1 ¸ ¸ ρrk px q ρrk px q ρrk px q q aρ paρ aρ q a p a a j j j ` ` k 2 j 1 `1 j
ρ2 p1 ρq2p1 2ρq
o
2pρrk pxq ρq 1 k
k ¸
?1 k
2ρq px q ln a a k j
j
1 2pρrk px q ρq k
j 1
k ¸
j 1
ρq px q a k j
k 1 ¸ ρqk px q a ln aj k j 1 j
by the mean value theorem where ρqk px q is an intermediate value between ρrk px q and ρ. Using the same approach for each term combining with Proposition 1 leads to Proposition 2. 18
4.3
Proof of Theorem 2
According to our Theorem 1, we clearly have the following decomposition
?
k Lk,k pxq Lpxq
ρrk px q
paj qaj ρ 1 k j 1 K pa j q k ρ aρr px q s 1 °k ? k ρr pxq K p a qr a j j 1 j j k r k px q k α 1 °k k K p a q j j 1 k ³ 1 du 0 K puqu ZL pxq oP p1q ° k 1 j 1 K paj q k °k ρr px q
ρ 1 ? n k n j 1 K paj qaj k k α k M px q k α M px q 1 °k k j 1 K paj q k
?
k α
n
M px q
k k
α r k p xq
1 k
°k
j 1K
°k
k
k
1 2
k
ρrk px q 12 a
n k OP kα ρrk x
k
³
pq
p p q ρq
k
12 du
1 0K u u 1 °k j 1K k
paj q
Z L px q
oP p1q
by applying our Proposition 2 and again the mean value theorem. Under the assumptions of our Theorem 2, we have
?
k Lk,k pxq Lpxq
?
k α ³
n
k
pq
M px q
du
1 0K u u 1 °k j 1K k
1 2
ρrk px q
paj q
19
k k
Z L px q
α
n k
oP p1q.
M px q
1 k
°k
p q ρ p q
j 1 K aj aj ° k 1 j 1 K aj k
Recall now that the function α is regularly varying with index ρ
0, that is αptq tρ`αptq
where `α is slowly varying at infinity. Thus
?
k Lk,k pxq Lpxq
c a k
n k
kα
k
c a k ³
³
n k
pq
12 du
1 0K u u 1 °k j 1K k
o P p1 q
paj q
OP
Z L p xq
2
ρrk px q
k k
`α p nk q `α p nk q
1
1 k
p q ρ p q p q p q
°k
j 1 K aj aj ° k 1 j 1 K aj k ρ 1 °k j 1 K aj aj k 1 °k j 1 K aj k
o P p1 q
Z L p xq
1 ρq
k k
1 ρ a
k k
paj q
M px q
k k
12 du
pq
1 0K u u 1 °k j 1K k
k k
ρ
kα
k
M px q
ρ
k
2
n k
kα
px q
ln
β
n k
a k
kα
k
n k
`α p nk q `α p nk q β n 1
pρrk pxq ρq
1
O p1 q
k
by the mean value theorem. Theorem 2 now follows under the assumptions since 1 lim tÑ8 β ptq
`α ptcq `α ptq
1 Op1q
(10)
when t Ñ 8 and c cptq Ñ 8. To be convinced by (10), remark that for all x, y
¡ 0 and t ¡ 0
we have with β1 ptq αptqβ ptq and Lt px1q t 1 F F11 p1 t1 xq, ..., Fd1 p1 t1 xq , and
using the fact that Lt px1q xLt{x p1q, that 1 x
"
*
Lt pxy1q Lpxy1q αptqM pxy1q Lt px1q Lpx1q αptqM px1q β 1 pt q β1 ptq Lt{x py1q Lpy1q αpt{xqM py1q Lt{x p1q Lp1q αpt{xqM p1q β1 pt{xq β1 pt{xq β1 pt{xq β1 ptq ( ρ αpt{xq αptqx {β1ptq pM py1q M p1qq .
Taking limits for t Ñ 8 we obtain that 1 pN pxy1q N px1qq x "
tlim Ñ8
pN py1q N p1qq ββ1ptp{txq q 1
1 β ptq
20
αpt{xq α pt q
x ρ
pM py1q M p1qq
*
.
(11)
Since the limit for t Ñ 8 of β1 pt{xq{β1 ptq exists, from (11) the limit for t Ñ 8 of
1 β t
pq
p { q x ρ pq
α t x α t
exists such that, combining with Theorem B.2.1 in de Haan and Ferreira (2006):
αpt{xq `α pt{xq 1 1 xρ xρ lim 1 xρψp1{xq, lim tÑ8 β ptq tÑ8 β ptq αptq `α ptq ρ1 1 with ψ pxq a x ρ 1 and a ¡ 0 (in case a 0 one can redefine `α as `α and the result will follow with a ¡ 0). Note that the positive constant a can in fact be incorporated in the function β, so from now on we take without loss of generality a 1. We have thus, with βr : β`α , lim
Ñ8
`α ptxq `α ptq βrptq
t
Now write, for some function βr where βr `α t `α tx βr t
p q p q ¤ pq
βr ptq βrptq
r β,
|ψpxq|
ψ px q. `α t `α tx r β t
p q p q ψpxq pq
.
By Theorem B.2.18 in de Haan and Ferreira (2006) (see also Drees, 1998) we have for any ε ¡ 0 there is a t0 such that for t, tx ¡ t0 ,
` tx ` t α α βr t
p q p q ¤ p1 κqr|ψpxq| εxρ1 δ s. pq where 0 δ ρ1 and κ ¡ 0. Since both functions in the right-hand side are bounded for x P rx0 , 8q, x0 ¡ 0, result (10) follows. Acknowledgement The authors thank the editor and the reviewers for their helpful comments which led to improvements of our first version of the paper. Also the authors kindly acknowledge Anne-Laure Foug`eres, Laurens de Haan and C´ecile Mercadier for providing the program of their six distributions considered. This work was supported by a research grant (VKR023480) from VILLUM FONDEN and an international project for scientific cooperation (PICS-6416).
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