large sample behavior of the bernstein copula estimator

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Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference 142 (2012) 1189–1197

0378-37

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jspi

Large sample behavior of the Bernstein copula estimator

Paul Janssen a, Jan Swanepoel b, Noel Veraverbeke a,n

a Hasselt University, Center for Statistics, Agoralaan, Gebouw D, 3590 Diepenbeek, Belgiumb North-West University, Potchefstroom Campus, Potchefstroom, South Africa

a r t i c l e i n f o

Article history:

Received 24 February 2011

Received in revised form

4 October 2011

Accepted 29 November 2011Available online 6 December 2011

Keywords:

Asymptotic properties

Bernstein estimator

Copula estimator

Mean squared error

58/$ - see front matter & 2011 Elsevier B.V. A

016/j.jspi.2011.11.020

esponding author.

ail addresses: [email protected] (P. Jan

a b s t r a c t

Bernstein polynomial estimators have been used as smooth estimators for density

functions and distribution functions. The idea of using them for copula estimation has

been given in Sancetta and Satchell (2004). In the present paper we study the

asymptotic properties of this estimator: almost sure consistency rates and asymptotic

normality. We also obtain explicit expressions for the asymptotic bias and asymptotic

variance and show the improvement of the asymptotic mean squared error compared

to that of the classical empirical copula estimator. A small simulation study illustrates

this superior behavior in small samples.

& 2011 Elsevier B.V. All rights reserved.

1. The Bernstein copula estimator

Copulas are functions that couple multivariate distribution functions to their one-dimensional marginal distributionfunctions. Copulas provide a very nice tool to model multivariate data and are therefore very useful in, for example,financial economics. See e.g. Sancetta and Satchell (2004) and Sancetta (2007). The analysis of multivariate survival dataprovides another example where copulas are instrumental. See e.g. Wienke (2011).

Consider a random vector X ¼ ðX1, . . . ,XdÞT with joint cumulative distribution function H and marginal distribution

functions F1, . . . ,Fd. From Sklar’s theorem (Sklar, 1959) there exists a d-variate function C such that

Hðx1, . . . ,xdÞ ¼ CðF1ðx1Þ, . . . ,FdðxdÞÞ:

For a detailed account on copulas we refer to the excellent book by Nelsen (2006), which includes many examples.A variety of parametric models for copulas and marginal distributions has been proposed and the correspondingparametric estimation methods are available. The study of nonparametric estimators of copulas (kernel smoothedempirical copulas) goes back to Deheuvels (1979) and Gaenssler and Stute (1987). In more recent papers kernel basedsmooth versions of the empirical copula have been studied. Omelka et al. (2009) study weak convergence of improvedkernel estimators and include an excellent overview.

Nonparametric estimation of copulas is also the focus of this paper. For simplicity we restrict the presentation to thecase of bivariate data ðd¼ 2Þ. We consider a bivariate random vector (X,Y) with joint distribution function H and marginaldistribution functions F and G, i.e., Hðx,yÞ ¼ PðXrx,YryÞ; FðxÞ ¼ PðXrxÞ and GðyÞ ¼ PðYryÞ. Sklar’s theorem says thatthere exists a copula C on ½0;1�2 such that

Hðx,yÞ ¼ CðFðxÞ,GðyÞÞ:

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ssen), [email protected] (J. Swanepoel), [email protected] (N. Veraverbeke).

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dx.doi.org/10.1016/j.jspi.2011.11.020

P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–11971190

We will assume throughout that F and G are continuous, which implies that C is unique and

Cðu,vÞ ¼HðF�1ðuÞ,G�1

ðvÞÞ:

Given a sample ðX1,Y1Þ, . . . ,ðXn,YnÞ from the bivariate distribution function H, the empirical copula estimator for Cðu,vÞ isgiven by

Cnðu,vÞ ¼HnðF�1n ðuÞ,G

�1n ðvÞÞ

with

Hnðx,yÞ ¼ n�1Xn

i ¼ 1

IðXirx,YiryÞ

having marginals

FnðxÞ ¼Hnðx,þ1Þ¼ n�1Xn

i ¼ 1

IðXirxÞ,

GnðyÞ ¼Hnðþ1,yÞ ¼ n�1Xn

i ¼ 1

IðYiryÞ:

In this paper we consider Bernstein copula estimators, which are defined in terms of Bernstein polynomials. Before we givethe precise definition of Bernstein copula estimators, we review some results on smooth estimators for the distributionfunction of univariate data, since the results obtained in this paper provide extensions of these results to multivariate(bivariate) data.

For univariate data, Bernstein estimators have been used by Babu et al. (2002) to estimate the univariate distributionand density function. Leblanc (2008) proves the asymptotic normality of the Bernstein estimator of the univariatedistribution function and shows that the estimator outperforms the classical empirical distribution function estimator byderiving an explicit formula for the first order improvement for the asymptotic variance. In Leblanc (2009) it is shown thatthe Chung–Smirnov property holds for the Bernstein estimator. In the present paper we extend both results to theBernstein empirical copula.

Towards the definition of the Bernstein copula estimator, recall that the Bernstein polynomial of order m40corresponding to the copula C is defined as

Bmðu,vÞ ¼Xm

k ¼ 0

Xm

l ¼ 0

Ck

m,‘

m

� �Pk,mðuÞPl,mðvÞ

with

Pk,mðuÞ ¼m

k

� �ukð1�uÞm�k

the binomial probabilities. We have, uniformly in ðu,vÞ 2 ½0;1�2,

limm-1

Bmðu,vÞ ¼ Cðu,vÞ

since C is continuous on ½0;1�2.Note that Bm is a copula by itself, since it satisfies the sufficient conditions given in Theorem 1 of Sancetta and Satchell

(2004).They propose the following Bernstein estimator of order m40 of the copula function C:

Cm,nðu,vÞ ¼Xm

k ¼ 0

Xm

l ¼ 0

Cnk

m,‘

m

� �Pk,mðuÞPl,mðvÞ,

where Cn is the empirical copula estimator. The order of m will depend on n and we will have that m-1 if n-1.In Section 2 we obtain Chung–Smirnov consistency rates for the Bernstein copula estimator.In Section 3 we use a stochastic approximation for the Bernstein copula estimator to study the asymptotic bias and

variance. Based on these results we show that the Bernstein copula estimator outperforms the empirical copula estimator.The asymptotic normality of the smooth estimator follows as an easy consequence.

Section 4 provides a small simulation study in which for a number of examples it is shown that the Bernstein copulaestimator Cm,n outperforms the empirical copula estimator Cn.

Finally, some of the more technical proofs are collected in Appendix.

2. Chung–Smirnov consistency rates

For any function g on ½0;1�2 we define JgJ¼ sup0ru,vr19gðu,vÞ9, the supremum norm. In this section we give uniformstrong consistency rates for the Bernstein copula estimator.

P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197 1191

Theorem 1. If m�mðnÞ-1 and n=ðm log log nÞ-cZ0, then JCm,n�CJ¼Oðn�1=2ðlog log nÞ1=2Þ a.s., n-1.

To prove this theorem we rely on the following lemma (see Appendix for the proof).

Lemma 1.

Qn ¼ JCn�CJ¼ sup0ru,vr1

9HnðF�1n ðuÞ,G

�1n ðvÞÞ�HðF�1

ðuÞ,G�1ðvÞÞ9¼Oðn�1=2ðlog log nÞ1=2

Þ a:s:, n-1:

Proof of Theorem 1.

JCm,n�CJrJCm,n�BmJþJBm�CJ: ð1Þ

Lemma 1 and the fact that Pk,mðuÞ and P‘,mðvÞ, k,‘¼ 0, . . . ,m, are binomial probabilities, imply

JCm,n�BmJ¼ sup0ru,vr1

Xm

k ¼ 0

Xm

‘ ¼ 0

Cnk

m,‘

m

� ��C

k

m

‘

m

� �� Pk,mðuÞP‘,mðvÞ

��

r max0rk,‘rm

Cnk

m,‘

m

� ��C

k

m,‘

m

� ��rJCn�CJ¼Oðn�1=2ðlog log nÞ�1=2

Þ a:s:, n-1: ð2Þ

To obtain an order bound for the second term in the rhs of (1), we use the Lipschitz property of copulas and we write Bu

and Bv to denote binomial random variables with parameters m and u, and m and v respectively. We then have

JBm�CJ¼ sup0ru,vr1

Xm

k ¼ 0

Xm‘ ¼ 0

Ck

m,‘

m

� ��Cðu,vÞ

� �Pk,mðuÞP‘,mðvÞ

��

r sup0ru,vr1

Xmk ¼ 0

Xm‘ ¼ 0

k

m�u

��þ ‘

m�v

��

� �Pk,mðuÞP‘,mðvÞr sup

0rur1E

Bu

m�u

� �2 !1=2

þ sup0rvr1

EBv

m�v

� �2 !1=2

¼ sup0rur1

uð1�uÞ

m

� �1=2

þ sup0rvr1

vð1�vÞ

m

� �1=2

r1

2m1=2þ

1

2m1=2¼m�1=2 ¼Oðn�1=2ðlog log nÞ1=2

Þ: ð3Þ

The proof follows from (1)–(3). &

Remark 1. It is possible to modify the proof of Theorem 1 slightly and to obtain the same result under stronger conditionson C and weaker conditions on m. Indeed, if C has first order partial derivatives Cu and Cv that are Lipschitz continuous oforder a ð0oar1Þ and if n=ðm1þa log log nÞ-cZ0 then JCm,n�CJ¼Oðn�1=2ðlog log nÞ1=2

Þ a.s., n-1.

3. Asymptotic bias and variance of the Bernstein copula estimator

Towards an asymptotic expression for the bias and the variance of the Bernstein copula estimator, the followingdecomposition is useful:

n1=2ðCm,nðu,vÞ�Cðu,vÞÞ ¼ n1=2ðCm,nðu,vÞ�Bmðu,vÞÞþn1=2ðBmðu,vÞ�Cðu,vÞÞ: ð4Þ

Assuming bounded third order partial derivatives on ð0;1Þ2 for C, we have the following convergence rate of the Bernsteinapproximation:

Bm,nðu,vÞ�Cðu,vÞ ¼Xm

k ¼ 0

Xm

‘ ¼ 0

1

2

k

m�u

� �2

Cuuðu,vÞþ1

2

‘

m�v

� �2

Cvvðu,vÞ

"

þk

m�u

� �‘

m�v

� �Cuvðu,vÞ

�Pk,mðuÞP‘,mðvÞþoðm�1Þ ¼m�1bðu,vÞþoðm�1Þ, ð5Þ

where

bðu,vÞ ¼ 12½uð1�uÞCuuðu,vÞþvð1�vÞCvvðu,vÞ�:

To handle the first term in (4) we rely on the following two lemmas.

Lemma 2. For 0ouo1,

R1,mðuÞ :¼ m�1Xmk ¼ 0

Xm

‘ ¼ kþ1

ðk�muÞPk,mðuÞP‘,mðuÞ ¼m�1=2f�c1ðuÞþoð1Þg,

where

c1ðuÞ ¼uð1�uÞ

4p

� �1=2

:


Proof.

R1,mðuÞ ¼m�1Xmk ¼ 0

Xm‘ ¼ kþ1

ðk�muÞPk,mðuÞP‘,mðuÞ ¼m�1Xm‘ ¼ 1

P‘,mðuÞX‘�1

k ¼ 0

ðk�muÞPk,mðuÞ

¼ �m�1Xm

‘ ¼ 1

P‘,mðuÞXm

k ¼ ‘

ðk�muÞPk,mðuÞ

" #¼�m�1

Xm‘ ¼ 1

P‘,mðuÞ ‘m

‘

� �u‘ð1�uÞm�‘þ1

h i,

where the last equality is an identity in Johnson (1957). Then,

R1,mðuÞ ¼ �m�1ð1�uÞXm

‘ ¼ 1

‘P2‘,mðuÞ ¼ �ð1�uÞ

Xm‘ ¼ 1

‘

m�uþu

� �P2‘,mðuÞ ¼�uð1�uÞ

Xm

‘ ¼ 1

P2‘,mðuÞ�ð1�uÞ

Xm

‘ ¼ 1

‘

m�u

� �P2‘,mðuÞ:

Now, by Lemma 3.1 in Babu et al. (2002), we have

Xm

‘ ¼ 1

P2‘,mðuÞ ¼m�1=2fð4puð1�uÞÞ�1=2

þoð1Þg ð6Þ

and, using the Cauchy–Schwarz inequality and the fact that 0rP‘,mðuÞr1, we also have

Xm

‘ ¼ 1

‘

m�u

� �P2‘,mðuÞ ¼Oðm�3=4Þ: ð7Þ

Hence

R1,mðuÞ ¼m�1=2f�c1ðuÞþoð1Þg: &

Lemma 3. Assume bounded second order partial derivatives for C on ð0;1Þ2. Then

(i)
n1=2ðCm,nðu,vÞ�Bmðu,vÞÞ ¼ n�1=2Pn
i ¼ 1 YmiþoPð1Þ where

Ymi ¼Xm

k ¼ 0

Xm‘ ¼ 0

I Uirk

m,Vir

‘

m

� ��C

k

m,‘

m

� ��Cu

k

m,‘

m

� �I Uir

k

m

� ��

k

m

� �

�Cvk

m,‘

m

� �I Vir

‘

m

� ��‘

m


and where ðU1,V1Þ, . . . ,ðUn,VnÞ are independent random vectors with distribution function C and with uniform marginals

on ½0;1�.
(ii) The Ymi’s are bounded and have mean zero.
(iii)
VarðYmiÞ ¼ s2ðu,vÞ�m�1=2Vðu,vÞ,
where

s2ðu,vÞ ¼ VarfIðUru,V rvÞ�Cðu,vÞ�Cuðu,vÞ½IðUruÞ�u��Cvðu,vÞ½IðV rvÞ�v�g

¼ Cðu,vÞð1�Cðu,vÞÞþuð1�uÞC2uðu,vÞþvð1�vÞC2

vðu,vÞ�2ð1�uÞCðu,vÞCuðu,vÞ�2ð1�vÞCðu,vÞCvðu,vÞþ2Cuðu,vÞCvðu,vÞ Cðu,vÞ�uv½ �

and

Vðu,vÞ ¼ Cuðu,vÞð1�Cuðu,vÞÞuð1�uÞ

p

� �1=2" #

þ Cvðu,vÞð1�Cvðu,vÞÞvð1�vÞ

p

� �1=2" #

:

Proof of Lemma 3. The proof of (i) is immediate since we have, uniformly in (u,v), the following asymptoticrepresentation (see Fermanian et al., 2004, p. 857):

Cnðu,vÞ�Cðu,vÞ ¼ n�1Xn

i ¼ 1

fIðUiru,VirvÞ�Cðu,vÞ�Cuðu,vÞ½IðUiruÞ�u��Cvðu,vÞ½IðVirvÞ�v�gþoPðn�1=2Þ:

The proof of (ii) is immediate and the technical proof of (iii) is available from Appendix.

Theorem 2. Assume bounded third order partial derivatives for C on ð0;1Þ2.If n1=2m�1-0, then for ðu,vÞ 2 ð0;1Þ2

n1=2ðCm,nðu,vÞ�Cðu,vÞÞ-D N ð0,s2ðu,vÞÞ:


If n1=2m�1-d, 0odo1, then for ðu,vÞ 2 ð0;1Þ2

n1=2ðCm,nðu,vÞ�Cðu,vÞÞ-D N ðdbðu,vÞ,s2ðu,vÞÞ:

Remark 2. Note that the asymptotic variance s2ðu,vÞ in Theorem 2 coincides with the asymptotic variance in the limitdistribution of n1=2ðCnðu,vÞ�Cðu,vÞÞ. However, from Lemma 3(iii) it is clear that the first-order term in the expansion of thevariance of n1=2ðCm,nðu,vÞ�Cðu,vÞÞ is smaller than s2ðu,vÞ (at the price of some bias in case n1=2m�1-d40).

Remark 3. The asymptotic normality result in Theorem 2 can be strengthened to weak convergence of the processn1=2ðCm,nðu,vÞ�Cðu,vÞÞ in the space ‘1ðð0;1Þ2Þ of bounded functions. The limiting process is Gaussian with mean functiondbðu,vÞ and with covariance function

E½fIðUru,V rvÞ�Cðu,vÞ�Cuðu,vÞ½IðUruÞ�u��Cvðu,vÞ½IðV rvÞ�vg�g

�fIðUru0,V rv0Þ�Cðu0,v0Þ�Cuðu0,v0Þ½IðUru0Þ�u0��Cvðu

0,v0Þ½IðV rv0Þ�v0g�

for 0ou,v,u0,v0o1.

Because of the representation in Lemma 3(i), it suffices to prove the asymptotic tightness of the process fn�1=2Pn

i ¼ 1 Ym,ig.The Ymi as functions of u and v are uniformly bounded and continuous on ð0;1Þ2 and have non-zero derivatives up to orderm. By Theorem 2.7.1 in van der Vaart and Wellner (2000) such class of functions has finite e bracketing number of orderOðexpðe�2=mÞÞ. This result and the boundedness of the Ymi as functions of u and v guarantee, for m41, the validity of theintegrability condition on the square root of the logarithm of the bracketing number.

An interesting further question, beyond the scope of this paper, is whether the weak convergence on ð0;1Þ2 can beextended to ½0;1�2 as Fermanian et al. (2004) did for the empirical copula process.

Remark 4. The variance expression in Lemma 3(iii) is useful to arrive at an optimal choice of the order m. For theasymptotic mean squared error we have

AMSEðCm,nðu,vÞÞ ¼ AsVarðCm,nðu,vÞÞþðAsBiasðCm,nðu,vÞÞÞ2 ¼ n�1s2ðu,vÞ�m�1=2n�1Vðu,vÞþm�2b2ðu,vÞ:

Minimizing with respect to m gives

m0 �m0ðu,vÞ ¼4b2ðu,vÞ

Vðu,vÞ

!2=3

n2=3

and

AMSEðCm0 ,nðu,vÞÞ ¼ n�1s2ðu,vÞ�3V2ðu,vÞ

16bðu,vÞ

!2=3

n�4=3:

Since AsVarðCnðu,vÞÞ ¼ n�1s2ðu,vÞ and since Cn is biased, it is clear thatAMSEðCm0 ,nðu,vÞÞ is less than AMSEðCnðu,vÞÞ.

4. A Monte–Carlo comparison

We now report on some small sample experiments which compare the empirical copula estimator Cn with theBernstein copula estimator Cm0 ,n.

The quality of the estimators is firstly evaluated locally by empirically calculating their mean L1-norm, defined by

RðC ,CyÞ ¼ EðLðC ,CyÞÞ,

where

LðC ,CyÞ ¼ 9C ðu,vÞ�Cyðu,vÞ9

with C ¼ Cn or C ¼ Cm0 ,n where m0 ¼m0ðu,vÞ, and Cy a given family of copulas.Secondly they are evaluated globally by calculating their mean discrete L2-norm, given by

LðC ,CyÞ ¼1

ðn�1Þ2

Xn

i ¼ 1

Xn

j ¼ 1

Ci

n,

j

n

� ��Cy

i

n,

j

n

� �� 28<:

9=;

1=2

with C ¼ Cn or C ¼ Cm0 ,n where m0 ¼m0ðði=nÞ,ðj=nÞÞ, i,j¼ 1, . . . ,n�1.Data were generated from the following copulas (see e.g. Nelsen, 2006):

1.
The Farlie–Gumbel–Morgenstern copula (FGMC),
Cyðu,vÞ ¼ uvþyuvð1�uÞð1�vÞ, �1ryr1:

In this case, Spearman’s rho, rS, is given by rS ¼ y=3.


2.
The Cuadras–Auge copula (CAC),
Cyðu,vÞ ¼ fminðu,vÞgyðuvÞ1�y, 0ryr1:

Now, rS ¼ 3y=ð4�yÞ.
3. The Plackett copula (PC): with y40 and ya1,
Cyðu,vÞ ¼½1þðy�1ÞðuþvÞ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½1þðy�1ÞðuþvÞ�2�4uvyðy�1Þ

q2ðy�1Þ

:

We now have that rS ¼ ððyþ1Þ=ðy�1ÞÞ�ðð2y log yÞ=ðy�1Þ2Þ.

Each entry in Tables 1–4 is based on 20 000 independent trials for sample sizes n¼20, 30, 40. The standard errors of theMonte-Carlo estimates were found to be negligibly small and are not reported in the table. All calculations were doneusing double precision arithmetic in FORTRAN and routines from the IMSL library.

The results in Tables 1–3 illustrate the good local performance of the Bernstein copula estimator compared to theclassical empirical copula estimator for small to moderate sample sizes. It is clear that

RðCm0 ,n,CyÞoRðCn,CyÞ

everywhere except for 4 out of the 45 cases, viz. the CAC when n¼ 30;40 and ðu,vÞ 2 fð0:50,0:50Þ,ð0:75,0:75Þg.The superior behavior of the Bernstein copula estimator is even more evident from Table 4. As far as global performance

is concerned, the results in the table show that the strict inequality above holds everywhere.

Table 1Monte–Carlo estimates of expected L1-norms for the empirical copula and Bernstein copula estimators.

Copula (u,v) RðCn ,CyÞ RðCm0 ,n ,CyÞ

FGMC ðy¼ 0:9, rS ¼ 0:3, n¼ 20Þ (0.50, 0.50) 0.044 0.027

(0.25, 0.25) 0.036 0.015

(0.25, 0.75) 0.031 0.016

(0.75, 0.25) 0.031 0.016

(0.75, 0.75) 0.036 0.016

FGMC ðy¼ 0:9, rS ¼ 0:3, n¼ 30Þ (0.20, 0.50) 0.037 0.024

(0.25, 0.25) 0.031 0.012

(0.25, 0.75) 0.028 0.011

(0.75, 0.25) 0.028 0.010

(0.75, 0.75) 0.033 0.017

FGMC ðy¼ 0:9, rS ¼ 0:3, n¼ 40Þ (0.50, 0.50) 0.031 0.018

(0.25, 0.25) 0.026 0.012

(0.25, 0.75) 0.020 0.011

(0.75, 0.25) 0.020 0.011

(0.75, 0.75) 0.026 0.015



CAC ðy¼ 0:364, rS ¼ 0:3, n¼ 20Þ (0.50, 0.50) 0.047 0.045

(0.25, 0.25) 0.036 0.020

(0.25, 0.75) 0.031 0.012

(0.75, 0.25) 0.031 0.012

(0.75, 0.75) 0.041 0.041

CAC ðy¼ 0:364, rS ¼ 0:3, n¼ 30Þ (0.50, 0.50) 0.038 0.046

(0.25, 0.25) 0.030 0.020

(0.25, 0.75) 0.030 0.007

(0.75, 0.25) 0.030 0.007

(0.75, 0.75) 0.031 0.049

CAC ðy¼ 0:364, rS ¼ 0:3, n¼ 40Þ (0.50, 0.50) 0.031 0.048

(0.25, 0.25) 0.026 0.023

(0.25, 0.75) 0.022 0.006

(0.75, 0.25) 0.022 0.006

(0.75, 0.75) 0.026 0.053



PC ðy¼ 2:524, rS ¼ 0:3, n¼ 20Þ (0.50, 0.50) 0.043 0.027

(0.25, 0.25) 0.034 0.017

(0.25, 0.75) 0.031 0.009

(0.75, 0.25) 0.031 0.009

(0.75, 0.75) 0.034 0.019

PC ðy¼ 2:524, rS ¼ 0:3, n¼ 30Þ (0.50, 0.50) 0.036 0.025

(0.25, 0.25) 0.030 0.015

(0.25, 0.75) 0.029 0.009

(0.75, 0.25) 0.029 0.009

(0.75, 0.75) 0.032 0.017

PC ðy¼ 2:524, rS ¼ 0:3, n¼ 40Þ (0.50, 0.50) 0.031 0.018

(0.25, 0.25) 0.025 0.014

(0.25, 0.75) 0.021 0.008

(0.75, 0.25) 0.021 0.008

(0.75, 0.75) 0.025 0.017


Copula n RðCn ,CyÞ RðCm0 ,n ,CyÞ

FGMC ðy¼ 0:9, rS ¼ 0:3Þ 20 0.036 0.022

30 0.029 0.016

40 0.025 0.014

CAC ðy¼ 0:364, rS ¼ 0:3Þ 20 0.037 0.023

30 0.030 0.022

40 0.026 0.022

PC ðy¼ 2:524, rS ¼ 0:3Þ 20 0.036 0.022

30 0.029 0.017

40 0.025 0.014


In light of the asymptotic results derived above and the outcome of the simulation study, we therefore recommend theBernstein copula technique as an effective way for estimating copulas nonparametrically.

Acknowledgment

The authors thank Dr. James Allison for his important help with the Monte–Carlo section.This work was supported by the IAP Research Network P6/03 of the Belgian State (Belgian Science Policy).The second author thanks the National Research Foundation of South Africa for financial support. The third author

acknowledges support from research Grant MTM2008-03129 of the Spanish Ministerio de Ciencia e Innovacion.

Appendix

Proof of Lemma 1. Let F n and Gn denote the marginal empirical distribution functions of the uniform ½0;1� randomvariables Ui ¼ FðXiÞ and Vi ¼ GðYiÞ, i¼ 1, . . . ,n and let Hn denote their bivariate empirical distribution function. Then usingthe identities (see e.g. Swanepoel, 1986) F

�1

n ðuÞ ¼ FðF�1n ðuÞÞ and G

�1

n ðvÞ ¼ GðG�1n ðvÞÞ, it is easy to show that Qn can be

rewritten as

Qn ¼ sup0ru,vr1

9HnðF�1

n ðuÞ,G�1

n ðvÞÞ�Cðu,vÞ9:

Now,

Qnr sup0ru,vr1

9HnðF�1

n ðuÞ,G�1

n ðvÞÞ�CðF�1

n ðuÞ,G�1

n ðvÞÞ9þ sup0rur1

9F�1

n ðuÞ�u9þ sup0rvr1

9G�1

n ðvÞ�v9¼Qn1þQn2þQn3,

where we used the Lipschitz property of C.


From Kiefer (1961) it follows that Qn1 ¼Oðn�1=2ðlog log nÞ1=2Þ a.s., n-1, and for the marginal quantiles we have that Qn2

and Qn3 are both Oðn�1=2ðlog log nÞ1=2Þ a.s., n-1 (see e.g. Swanepoel, 1986). This proves that Qn ¼Oðn�1=2ðlog log nÞ1=2

Þ

a.s., n-1.

Proof of Lemma 3(iii).

VarðYmiÞ ¼X9

j ¼ 1

Vmjðu,vÞ,

where

Vmjðu,vÞ ¼Xm

k ¼ 0

Xm

‘ ¼ 0

Xm

k0 ¼ 0

Xm‘0 ¼ 0

Emjðk,‘,k0,‘0ÞPk,mðuÞP‘,mðvÞPk0 ,mðuÞP‘0 ,mðvÞ

and

Em1ðk,‘,k0,‘0Þ ¼ Ck4k0

m,‘4‘0

m

� ��C

k

m,‘

m

� �C

k0

m,‘0

m

� �,

Em2ðk,‘,k0,‘0Þ ¼ Cuk

m,‘

m

� �Cu

k0

m,‘0

m

� �k4k0

m�

k

m

k0

m

,

Em3ðk,‘,k0,‘0Þ ¼ Cvk

m,‘

m

� �Cv

k0

m,‘0

m

� �‘4‘0

m�‘

m

‘0

m

,

Em4ðk,‘,k0,‘0Þ ¼ �Cuk0

m,‘0

m

� �C

k4k0

m,‘

m

� ��C

k

m,‘

m

� �k0

m

,

Em5ðk,‘,k0,‘0Þ ¼ �Cvk0

m,‘0

m

� �C

k

m,‘4‘0

m

� ��C

k

m,‘

m

� �‘0

m

,

Em6ðk,‘,k0,‘0Þ ¼ �Cuk

m,‘

m

� �C

k4k0

m,‘0

m

� ��C

k0

m,‘0

m

� �k

m

,

Em7ðk,‘,k0,‘0Þ ¼ �Cvk

m,‘

m

� �C

k0

m,‘4‘0

m

� ��C

k0

m,‘0

m

� �‘

m

,

Em8ðk,‘,k0,‘0Þ ¼ Cuk

m,‘

m

� �Cv

k0

m,‘0

m

� �C

k

m,‘0

m

� ��

k

m

‘0

m

,

Em9ðk,‘,k0,‘0Þ ¼ Cvk

m,‘

m

� �Cu

k0

m,‘0

m

� �C

k0

m,‘

m

� ��‘

m

k0

m

:

Direct calculations give

V1mðu,vÞ ¼ Cðu,vÞð1�Cðu,vÞÞþ2Cuðu,vÞR1,mðuÞþ2Cvðu,vÞR1,mðvÞþoðm�1=2Þ, ð8Þ

V2mðu,vÞ ¼ uð1�uÞC2uðu,vÞþ2C2

uðu,vÞR1,mðuÞþoðm�1=2Þ,

V3mðu,vÞ ¼ vð1�vÞC2vðu,vÞþ2C2

vðu,vÞR1,mðvÞþoðm�1=2Þ,

V4mðu,vÞ ¼�ð1�uÞCðu,vÞCuðu,vÞ�2C2uðu,vÞR1,mðuÞþoðm�1=2Þ,

V5mðu,vÞ ¼�ð1�vÞCðu,vÞCvðu,vÞ�2C2vðu,vÞR1,mðvÞþoðm�1=2Þ,

V6mðu,vÞ ¼�ð1�uÞCðu,vÞCuðu,vÞ�2C2uðu,vÞR1,mðuÞþoðm�1=2Þ,

V7mðu,vÞ ¼�ð1�vÞCðu,vÞCvðu,vÞ�2C2vðu,vÞR1,mðvÞþoðm�1=2Þ,

V8mðu,vÞ ¼ Cuðu,vÞCvðu,vÞ½Cðu,vÞ�uv�þoðm�1=2Þ,

V9mðu,vÞ ¼ Cuðu,vÞCvðu,vÞ½Cðu,vÞ�uv�þoðm�1=2Þ:

Here are the details for V1mðu,vÞ in (8). The others follow in a similar way.

V1mðu,vÞ ¼Xm

k ¼ 0

Xm

‘ ¼ 0

Ck

m,‘

m

� �P2

k,mðuÞP2‘,mðvÞ


�Xmk ¼ 0

Xm

‘ ¼ 0

Ck

m,‘

m


!2

þXmk ¼ 0

Xm

‘ ¼ 0

Xmk0 ¼ 0,k0ak

Xm

‘0 ¼ 0,‘0a‘

Ck4k0

m,‘4‘0

m

� �Pk,mðuÞP‘,mðvÞPk0 ,mðuÞP‘0 ,mðvÞ

þXm

k ¼ 0

Xm‘ ¼ 0

Xm

k0 ¼ 0,k0ak

Ck4k0

m,‘

m

� �Pk,mðuÞPk0 ,mðuÞP

2‘,mðvÞþ

Xmk ¼ 0

Xm

‘ ¼ 0

Xm‘0 ¼ 0,‘0a‘

Ck

m,‘4‘0

m

� �P2

k,mðuÞP‘,mðvÞP‘0 ,mðvÞ: ð9Þ

The second term in (9) equals B2mðu,vÞ. After Taylor expansion of Cððk=mÞ,ð‘=mÞÞ around (u,v) the first term in (9) becomes

Cðu,vÞSmðuÞSmðvÞþOðSmðuÞImðvÞþSmðvÞImðuÞÞ,

where SmðuÞ ¼Pm

k ¼ 0 P2k,mðuÞ and ImðuÞ ¼

Pmk ¼ 0 ðk=mÞ�u

�� P2k,mðuÞ.

An expansion of the third term in (9) gives that it is equal to

Cðu,vÞð1�SmðuÞÞð1�SmðvÞÞþ2Cuðu,vÞð1�SmðvÞÞR1,mðuÞþ2Cvðu,vÞð1�SmðuÞÞR1,mðvÞþOðR2,mðuÞþR2,mðvÞÞ,

where R2,mðuÞ ¼m�2Pm

k ¼ 0

Pm‘ ¼ kþ1ðk�muÞ2Pk,mðuÞP‘,mðuÞrm�1uð1�uÞ ¼Oðm�1Þ.

Similarly we obtain that the fourth and fifth term in (9) are equal to respectively

Cðu,vÞSmðvÞð1�SmðuÞÞþ2Cuðu,vÞSmðvÞR1,mðuÞþCvðu,vÞð1�SmðuÞÞOðImðuÞÞ

and

Cðu,vÞSmðuÞð1�SmðvÞÞþ2Cvðu,vÞSmðuÞR1,mðvÞþCuðu,vÞð1�SmðvÞÞOðImðvÞÞ:

Using Lemma 2, together with (6) and (7), gives that (8) holds.Finally use Lemma 2 to arrive at the expression for VarðYmiÞ.

References

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large sample behavior of the bernstein copula estimator

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