a note on nonparametric estimation of bivariate tail dependence

DOI 10.1515/strm-2013-1143 | Stat. Risk Model. 2014; 31(2): 151162

Axel Bcher*A note on nonparametric estimation ofbivariate tail dependence

Abstract: Nonparametric estimation of tail dependence can be based on a stan-dardization of the marginals if their cumulative distribution functions are known.In this paper it is shown to be asymptotically more efficient if the additionalknowledge of the marginals is ignored and estimators are based on ranks. Thediscrepancy between the two estimators is shown to be substantial for the pop-ular Clayton and GumbelHougaard models. A brief simulation study indicatesthat the asymptotic conclusions transfer to finite samples.

Keywords: Asymptotic variance, nonparametric estimation, rank-based infer-ence, tail copula, tail dependence.

AMS (2010): Primary 62G32; Secondary 62G05

||*Corresponding Author: Axel Bcher, Ruhr-Universitt Bochum, Fakultt fr Mathematik, Uni-versittsstrae 150, 44780 Bochum, Germany, andUniversit catholique de Louvain, Institut de statistique, Voie du Roman Pays 20, 1348 Louvain-la-Neuve, Belgium, e-mail: [email protected]

1 IntroductionLet (, ) be a bivariate random vector with joint cumulative distribution func-tion (cdf) and continuous marginal cdfs and . The cdf of (, ) =((), ()), called the copula of (, ), is the only function satisfying the re-lationship (, ) = {(), ()} for all , and therefore characterizesthe stochastic dependence between and. The lower and upper tail copulas of(, ) (or the lower andupper functions of tail dependence) are defined as the fol-lowing directional derivatives of the copula and its associated survival copula(, ) = + 1 + (1 , 1 ) at the point (0, 0):

(, ) = lim

0

Pr(() | () ) = lim0

(, )

,

(, ) = lim

0

Pr(() 1 | () 1 ) = lim0

(, )

,

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152 | A. Bcher

where (, ) [0,]2 \ {(,)}. Note that is the joint cdf of the vector(1 , 1 ) and that it is the copula of the pair (, ). As a consequence,the upper tail copula of (, ) is equal to the lower tail copula of (, ) andvice versa, whence there is no conceptual difference between upper and lowertail dependence. Tail copulas and variants thereof characterize extremal depen-dence of the vector (, ), see [7]. One of the variants of

is given by the stable

tail dependence function (, ) = + (, ). The restriction of to the

unit sphere with respect to the 1-norm, i.e., the function () = (1 , ) =

1 (1 , ) for [0, 1], is called Pickands dependence function, see [32].

Since tail copulas are homogeneous in the sense that(, ) =

(, ) for

all > 0, the Pickands dependence function and the upper tail copula are one-to-one. Similar remarks can be made for lower tail copulas. Of course, the popularcoefficients of tail dependence are included in the concept of tail copulas and aregiven by

=

(1, 1) and

=

(1, 1).

Assuming that (, ) is in the domain of attraction of a bivariate ex-treme value distribution, the nonparametric estimation of tail copulas, or,equivalently, of stable tail dependence functions, based on i.i.d. observations(

1,

1), . . . , (

,

) such that (

,

)

= (, ) has first been addressed by [22].The underlying idea of his estimator (and of several variants) can be summarizedas follows. For the sake of brevity we restrict ourselves to lower tail copulas andwe begin by supposing that the marginal distributions are known to the statis-tician. In that case, a natural estimator for is given by the empirical distribu-tion function of the standardized sample ((

1), (

1)), . . . , ((

), (

)), i.e.,

(, ) = 1

=11{(

) , (

) }. A promising estimator for

is then

given by

(, ) =

(

,

) =

1

=1

1{() /, (

) /},

where = = () is a constant that needs to be chosen by the statisti-

cian. Relaxing the assumption of having knowledge of the marginal distribu-tions we must replace

(, ) by the empirical copula

. With the pseudo-

observations =

(

) = rank(

among

1, . . . ,

)/ and

=

(

) =

rank(among

1, . . . ,

)/, where

and

denote themarginal empirical dis-

tribution functions, the empirical copula is defined as (, ) =

1

=11{

, }. Hence, we can define

(, ) =

(

,

) =

1

=1

1{(

) /,

(

) /}

as a rank-based estimator for. The asymptotics of these estimators (or of slight

variants thereof) have been investigated in several papers. The first weak conver-



A note on nonparametric estimation | 153

gence results dates back to [22], while [10] consider the problem of which rate canbe achieved in this estimation problem. [15] prove weak convergence with respectto weighted metrics, whereas [37] give an alternative proof for the asymptotics bythe functional delta method. [16] and [4] provide some improvements regardingthe imposed smoothness conditions on the tail copulas, which we also adopt inthe present paper.

In order to control the bias of the estimators one needs to assume a second or-der condition on the speed of convergence in the defining relation for. Supposethat

(, ) (/, /) = (())

for locally uniformly in (, ), where : [0,) [0,) satisfieslim

() = 0. Then for , = () such that(/) = (1) it is well

known that{

(, )

(, )}

(, )

in the space (([0, ]2), ) for each > 0, see, e.g., Theorem 4 in [37] for

easy reference. Here,

denotes a tight centered Gaussian field with covariancestructure given by

Cov{

(, ),

(, )} = ( , ). (1.1)

Note that the definition of

can be extended to the set [0,]2 \ {(,)}.Using the margin-free estimator

, and assuming that the partial derivatives

,1

=

and

,2=

exist and are continuous on (0,)2, we have

{(, )

(, )}

(, ),

where the process

can be expressed as

(, ) =

(, ) ,1(, )

(,) ,2(, )

(, ),

see, in particular, [4] for the imposed smoothness conditions on the partial deriva-tives of

. Similar results hold for the upper tail copula and its corresponding

estimators

(, ) =

1

=1

1{1 () /, 1 (

) /},

(, ) =

1

=1

1{1 (

) /, 1

(

) /}.

Observing that the cdf of is given by 1 () and that (

)

is almost surely equal to 1 (

), where

denotes the empirical cdf



154 | A. Bcher

of 1, . . . ,

, we can conclude that the estimators

and

based on

(1,

1), . . . , (

,

) coincide with the respective estimators

and

based

on (1,

1), . . . , (

,

). Therefore,

{(, )

(, )}

(, ), {(, )

(, )}

(, ),

in (([0, ]2), ), where

and

are defined exactly as before, butwith

replaced by

.

In the case of unknownmarginal distributions, a statistician being interestedin (say, the lower) tail dependence has no choice between the estimators

and

and has to rely on

. The question of interest of this note deals with the case

of having knowledge of the marginal distributions. Then we are confronted withthe question of which estimator to prefer, and observing that

exploits addi-

tional knowledge and has the somewhat easier limiting distribution might sug-gest to use this estimator. We are going to show that this conclusion is mislead-ing: even though

is discarding what appears to be pertinent information, this

estimator is always preferable from an asymptotic point-of-view. This result is in-line with a recent observation by [19] for the empirical copula process, where therank-based estimator for the copula ismore efficient for a broad class of positivelyassociated copulas. For a similar observation regarding the analysis of censoreddata see, e.g., [33].

The remainder of this note is organized as following. In Section 2 we discussthe bias and the asymptotic variance of the two estimators

and

. While the

bias is shown to be (almost) the same for both estimators the variance of is

shown to be substantially smaller. We investigate our findings for the Clayton andGumbel tail copula both theoretically and by means of a small simulation study.

2 Main resultIn the subsequent developments we restrict ourselves to the investigation of lowertail dependence. As mentioned in the Introduction, by passing from (

,

) to

(,

), we easily obtain the same results for the upper tail dependence. We

are going to show that both estimators and

share a comparable bias un-

der usual second order conditions, whereas the variance of is substantially

smaller than the one of . We begin with the discussion of the bias which can

be derived from the decompositions

{(, )

(, )}

= {

(

,

)

(

,

)}




+ {

(

,

)

(, )} ,

{(, )

(, )} (2.1)

= {

(

,

)

(

,

)}

+ {

(

,

)

(, )} .

The first summand in each line is the leading term and converges weakly towards

and

, respectively. Whereas the leading term in the first line is unbiasedfor every, its counterpart in the second line is asymptotically unbiased. The sum-mands on the right-hand side of the preceding decomposition are the same forboth estimators and constitute the term which determines the asymptotic bias.Depending on the second order condition and on the limit behavior of(/)for it may converge to 0 or to some function , or its absolute value mayblow up to.

For these reasons an (asymptotic) comparison of the estimators and

must be based on a discussion of their asymptotic (co)variance. The followingtheorem, in which we abbreviate

by, is our main result.

Theorem 2.1. Suppose that the partial derivatives of the tail copula 1and

2

exist and are continuous on (0,)2. Then

Cov{(, ),

(, )} Cov{

(, ),

(, )}

for all , , , 0. In particular, Var{(, )} Var{

(, )}.

Proof. The assertion is trivial if any of the variables equals zero, hence suppose, , , > 0. Multiplying out we have

Cov{(, ),

(, )} Cov{

(, ),

(, )} =

4

=1

,

where=

(, , , ) is defined as

1=

1(, ){

1(, )( ) ( , )},

2=

1(, ){

2(, )(, ) ( , )},

3=

2(, ){

2(, )( ) (, )},

4=

2(, ){

1(, )(, ) (, )}.

We only show that 1and

2are non-positive, since, for symmetry reasons,

3

and4canbe treated analogously. Let us first consider

1and suppose that .



156 | A. Bcher

Note that(, )/ =(1, /) togetherwithmonotonicity of(1, ) implies thatthe function (, )/ is non-increasing. Hence, for all , > 0,

1(, ) (, ), (2.2)

which implies that1 0 for . On the other hand, for > , we can use (2.2)

and homogeneity and monotonicity of to see that

1(, ) (, )/ = (, /) (, ),

whence, again,1 0.

Now consider 2. For , the assertion

2 0 simply follows by bound-

edness of the partial derivative, i.e., 0 2 1. For > , note that the same

argumentation as for (2.2) implies that 2(, ) (, ) for any , > 0. Hence,

we get2 0 upon noting that(, ) .

The superiority of the rank-based estimator from the point of view of its (asymp-totic) variance can be explained, at least partially, as follows. For /(or, with a similar argumentation, for /), the function

(, ) =

1

=11(

/) is a non-random step functionwith jumps of size 1 in the

points /, where = 1, . . . , . Therefore, its variance is zero. On the other hand,the variance of its counterpart

(, ) = 1

=11((

) /) is easily

seen to be given by /2{/(1/)} = /2/ for all [0, /], whichexplains the superiority of

for those (, ) for which at least one coordinate

is larger than /. For continuity reasons, this argument shows that should

be preferable at least for all large (, ). Theorem 2.1 shows theoretically that itactually is preferable for all (, ).

As already mentioned in the Introduction a result similar to that in Theo-rem 2.1 has recently been shownby [19] for the estimation of copulas. Even thoughcopulas and tail copulas are closely related (the latter being directional deriva-tives of the former), the problems of comparing their asymptotic covariances areslightly different and the corresponding results do not imply each other. Amongother things, this is due to the fact that the copula estimator

based on known

marginals converges to a Gaussian limiting fieldwith covariance

Cov{(, ),

(, )} = ( , ) (, )(, ),

as opposed to the tail copula setting in (1.1), where only one summand appears.Additionally, note that the results for copulas in [19] only hold for a subclass ofcopulas, namely for all copulas that are left tail decreasing in both arguments,whereas the tail copula-pendant holds for every tail copula.

Finally, as a consequence of Theorem 2.1 and the functional delta methodand by the same arguments as in Proposition 3 in [19], we can conclude that any




real-valued estimator = () that is a non-decreasing and sufficiently smooth

functional of is preferable to the competitor = (

) from an asymptotic

variance point-of-view. We omit the details of this observation.

3 IllustrationIn this sectionwe are going to illustrate the results of the previous section at handof two popular exemplary models, both asymptotically and for finite sample sizesby means of MonteCarlo simulations. We consider the following two underlyingcopulas. The bivariate Clayton copula is defined as

Cl(, ) = (

+

1)

1/

, (, ) [0, 1]2,

where > 0. The Clayton copula is lower tail dependent, with lower tail cop-ula given by

(, ) = lim

0

1{()

+ ()

1}

1/

= lim0

(

+

)1/

= (

+ )1/

,

The first order partial derivatives ofare calculated as

,1(, ) = (

+

)(1+)/

(1+)

,

,2(, ) = (

+

)(1+)/

(1+)

.

The bivariate GumbelHougaard copula (also known as logistic copula) is de-fined as

Gu(, ) = exp [ {( log )

+ ( log )

}1/

] , (, ) [0, 1]2,

where 1. SinceGu

is an extreme-value copulawithPickands-dependencefunction() = { + (1 )}1/, it is upper tail dependent for any > 1withupper tail copula given by

(, ) = + (

+

)1/

,

see, e.g., [21]. The first order partial derivatives ofare calculated as

,1(, ) = 1 (

+

)(1)/

1

,

,2(, ) = 1 (

+

)(1)/

1

.



158 | A.Bcher

0.0

0.5

1.0

0.00.51.00.0

0.2

0.4

0.0

0.5

1.0

0.00.51.00.0

0.2

0.4

0.0

1.0

0.51.00.0

0.5

1.0

Fig. 3.1: The graphs of Var{(, )} (left, unknown marginals), of Var{

(, )} (middle, known

marginals) and of the relative efficiencyVar{(, )}/ Var{

(, )} (right) for the Clayton tail

copula with parameter = 1 and for (, ) [0, 1]2.

Figure 3.1 visualizes a first asymptotic comparison of the two competitive estima-tors: depicted are the graphs of the asymptotic variances of the estimators

and on the unit cube [0, 1]2 for the Clayton lower tail copula with parameter

= 1, which corresponds to a coefficient of tail dependence of 0.5. The differenceis seen to be substantial, especially close to the axis. Analogous pictures for theGumbel upper tail copula are nearly indistinguishable from the Clayton case andare hence omitted for the sake of brevity.

In order to investigate the influence of the parameter we restrict ourselvesto the estimation of the coefficients of tail dependence. The calculations aboveshow that

=

(1, 1) = 21/ for the Clayton copula and

= 2 21/ for the

Gumbel copula. The competitive estimators are =

(1, 1) and

=

(1, 1)

(resp. =

(1, 1) and

=

(1, 1)), and a careful calculation reveals that

their asymptotic variances are given by

Var{

(1, 1)} = 21/

, Var{

(1, 1)} = 21/

3

222/

+1

223/

for the Clayton lower tail copula and by

Var{

(1, 1)} = 2 21/

, Var{

(1, 1)} = 21/

+3

222/

1

223/

for the Gumbel tail copula. In Figure 3.2, these variances are plotted as a func-tion of . It can be seen that, in both models, the rank-based estimator becomessubstantially better with increasing degree of tail dependence.

Finally, we investigate the finite-sample performance of the estimators for

and , respectively, by means of a small simulation study. To this end, we simu-

late i.i.d. samples of size= 1,000 from theClayton copulawith parameter = 0.5and from theGumbel copulawith parameter = log(2)/ log(7/4) 1.24 such that




0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Theta(Clayton)

Asy

mpt

otic

Var

ianc

e / R

elat

ive

Effi

cien

cy

Var(rankbased)Var(known marginals)Relative Efficiency

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Theta(Gumbel)

Asy

mpt

otic

Var

ianc

e / R

elat

ive

Effi

cien

cy

Var(rankbased)Var(known marginals)Relative Efficiency

Fig. 3.2: Asymptotic VarianceVar{(1, 1)} (dashed line) andVar{

(1, 1)} (solid line) and

relative efficiencyVar{(1, 1)}/ Var{

(1, 1)} (dotted line) as a function of for the Clayton tail

copula (left figure) and the GumbelHougaard (logistic) tail copula (right figure).

20 40 60 80 100 120 140

0.00

00.

005

0.01

00.

015

0.02

0

Squared BiasVarianceMSE

20 40 60 80 100 120 140

0.00

00.

005

0.01

00.

015

0.02

0

Squared BiasVarianceMSE

Fig. 3.3: Squared Bias (dashed lines), Variance (dotted lines) and MSE (solid lines) of theestimators (1, 1) (black lines) and (1, 1) (gray lines) as a function of the parameter , for= 0.25 in the Clayton tail copula model (left figure) and for

= 0.25 in the

GumbelHougaard (logistic) tail copula model (right figure).

the coefficient of tail dependence are = 0.25 and

= 0.25, respectively. Our

objective is the estimation of (or

) and we investigate both the squared bias

and the asymptotic variance as a function of the parameter . As usual in extremevalue theory, larger values of result in a larger bias, whereas the variance de-creases in . The results, which are based on 10 000 repetitions, are plotted inFigure 3.3. We clearly see the expected superiority of the rank-based versions inVariance andMean Squared Error. The bias of both estimators is indeed compara-ble as indicated by decomposition (2.2) at the beginning of this Section. Note also



160 | A. Bcher

that the difference in variance entails a different optimal choice of for the twoestimators, which is seen to be slightly larger for the estimator based on knownmarginals. This discrepancy reveals that there is no perfect global answer to thequestion of where the tail begins.

Acknowledgement: The author is grateful to two unknown referees and toStanislav Volgushev for discussions and suggestions concerning this manuscript.This work has been supported in parts by the Collaborative Research Center Sta-tistical modeling of nonlinear dynamic processes (SFB 823) of the German Re-search Foundation (DFG) and by the IAP research network Grant P7/06 of the Bel-gian government (Belgian Science Policy), which is gratefully acknowledged.

Received January 16, 2013; revised July 7, 2013; accepted August 4, 2013.

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A note on nonparametric estimation of bivariate tail dependence1 Introduction2 Main result3 Illustration

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a note on nonparametric estimation of bivariate tail dependence

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