robust estimation for copula parameter in scomdy models

Robust estimation for copula Parameter inSCOMDY modelsByungsoo Kima and Sangyeol Leea,�,†

In this study, we study the robust estimation for the copula parameter in semiparametric copula-based multivariatedynamic (SCOMDY) models proposed by Chen and Fan (2006). To this end, instead of the pseudo maximumlikelihood estimator in Chen and Fan (2006), we use a minimum density power divergence estimator (MDPDE)proposed by Basu et al. (1998). It is shown that the MDPDE is consistent and asymptotically normal underregularity conditions. We compare the performance between the two estimators when outliers exist through asimulation study.

Keywords: Density-based divergence measures; robust estimation for copula parameter; SCOMDY model; consistency;asymptotic normality.

1. INTRODUCTION

Copulas are widely used to model the multivariate distributions. According to Sklar’s theorem, one can represent any multivariate distributionsby selecting a suitable copula function and marginal distributions. It provides a theoretical foundation for the copula approach in generatingmultivariate distributions from univariate distributions. Gaussian, Student t, Clayton, Frank and Gumbel copulas are mostly commonexamples. See Nelson (1998) and Joe (1997) for the examples and properties of copulas. Since copula functions determine the dependencestructure of multivariate distributions, estimating the copula parameter is an important task to figure out a correct dependence structure.

Chen and Fan (2006) introduced a new class of semiparametric copula-based multivariate dynamic (SCOMDY) models. SCOMDYmodels not only specify the multivariate conditional mean and conditional variance parametrically but also the distribution of the(standardized) innovations semiparametrically as a parametric copula evaluated at the nonparametric univariate marginaldistributions. They are very flexible in capturing a wide range of nonlinear and asymmetric dependence structures and marginalbehaviour of multivariate time series. They cover many commonly used models such as GARCH, VAR and Markov switching models.Chen and Fan provided the asymptotic properties of the copula parameter estimator associated with a SCOMDY model under apossibly misspecified parametric copula of the standardized innovation. In particular, in their study a pseudo maximum likelihoodestimator is employed to estimate the copula parameter.

The objective of this study was to provide a robust estimator for the copula parameter in SCOMDY models. We employ the density-based divergence method proposed by Basu et al. (1998) (abbreviated as BHHJ in the sequel) instead of the Kullback–Leiblerdivergence used in Chen and Fan (2006). It is well known that in comparison with the methods proposed by Beran (1977), Tamuraand Boos (1986), Simpson (1987), Basu and Lindsay (1994) and Cao et al. (1995), the BHHJ method has merit of not requiring anysmoothing methods. In this case, we can avoid the drawbacks and difficulties such as the selection of bandwidth, that necessarilyfollow from the kernel smoothing method. BHHJ demonstrated that the minimum density power divergence estimator (MDPDE) hasstrong robust properties with low loss in the asymptotic efficiency relative to the maximum likelihood estimator. Therefore, theirestimator can be viewed as a good alternative to the maximum likelihood estimator in terms of both the efficiency and robustness.With regard to the references for the MDPDE in time series models, we refer to Lee and Song (2009) and Kim and Lee (2011) whoconsidered the robust estimation of GARCH parameters and covariance matrices of multivariate time series models.

The organization of this article is as follows. In Section 2, we introduce the construction of the robust estimator in SCOMDY modelsusing the BHHJ’s procedure and show the asymptotic properties of the proposed estimator. In Section 3, we conduct a simulationstudy to compare the performance of the proposed estimator with that of the pseudo maximum likelihood estimator when outliersexist. In Section 4, we provide the proofs of the results in Section 2.

2. MDPDE FOR COPULA PARAMETER IN SCOMDY MODELS

Suppose that fðY 0t ; X 0tÞ0 : t ¼ 1; 2; . . . ; ng is a vector stochastic process where Yt is an d-dimensional stochastic process and Xt is a

vector of predetermined or exogenous variables distinct from the Y’s. Let F t�1 denote the sigma field generated byfYt�1; Yt�2; . . . ; Xt; Xt�1; . . .g. Let us consider the SCOMDY model proposed by Chen and Fan (2006):

aSeoul National University�Correspondence to: Sangyeol Lee, Department of Statistics, Seoul National University Seoul 151-742, Korea.†E-mail: [email protected]

Original Article

First version received September 2012 Published online in Wiley Online Library: 6 December 2012

(wileyonlinelibrary.com) DOI: 10.1111/jtsa.12013

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Yt ¼ ltðh01Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiHtðh0Þ

p�t; ð1Þ

where

ltðh01Þ ¼ ðl1tðh01Þ; l2tðh01Þ; . . . ; ldtðh01ÞÞ0 ¼ EðYt j F t�1Þ

is the true conditional mean of Yt given F t�1, and

Htðh0Þ ¼ diagðh1tðh0Þ; h2tðh0Þ; . . . ; hdtðh0ÞÞ

with

hitðh0Þ ¼ hitðh01; h02Þ ¼ E½ðYit � litðh01ÞÞ2 j F t�1�; i ¼ 1; 2; . . . ; d;

is the true conditional variance of Yit given F t�1. We set h0 ¼ ðh001; h002Þ0 2 H1 � H2 � Rp, where h01 and h02 do not have

common elements. The multivariate innovations f�t ¼ ð�1t; �2t; . . . ; �dtÞ0 : t ¼ 1; 2; . . . ; ng in eqn (1) are i.i.d random vectors withE(�it) ¼ 0 and Eð�2

itÞ ¼ 1, independent of F t�1 for i ¼ 1,2, . . . ,d. Moreover, �t ¼ ð�1t; �2t; . . . ; �dtÞ0 has a joint distribution functionF0ð�Þ ¼ C0ðF0

1ð�1Þ; . . . ; F0dð�dÞÞ, where F0

i is the true marginal distribution of �it with density functions f 0i for i ¼ 1, . . . ,d and

C0ðu1; . . . ; udÞ ¼ C0ðu1; . . . ; ud : a0Þ : ½0; 1�d ! ½0; 1� is the true copula function with copula parameter a0 2 A � Ra. We assumethat x 7! f 0

i ðxÞ and x 7! xf 0i ðxÞ are uniformly continuous and bounded on R. Throughout this article, E0[Æ] denotes the expectation

taken under the true copula distribution C0ðF01ð�Þ; . . . ; F0

dð�Þ : a0Þ and �tðhÞ ¼ ½HtðhÞ��1=2ðYt � ltðh1ÞÞ denotes the innovationfunction.

We assume that the dynamic part of the SCOMDY model satisfies the conditions as follows:

D1. fðY 0t ; X 0tÞ0 : t ¼ 1; 2; . . . ; ng is stationary beta mixing with decaying rate fbtg such that

P1t¼1 bt < 1;

D2. The estimator hn of h0 satisfiesffiffiffinpðhn � h0Þ ¼ Opð1Þ as n ! 1;

D3. �itðhÞ ¼ hitðhÞ�1=2ðYit � litðh1ÞÞ is continuously differentiable in a neighborhood of h0;

D4. E0 1ffiffiffiffiffiffiffiffiffiffihitðh0Þp @

@h litðh01Þ��

�� < 1 and E0 1hitðh0Þ

@@h hitðh0Þ

�� < 1 for i ¼ 1,2, . . . ,d;

REMARK 1. Conditions (D1)–(D4) are found in Chen and Fan (2006) and guarantee eqn (5) below.

As the true copula function C 0 is usually unknown, we consider a class of candidate copulas fCðu1; . . . ; ud : aÞ : a 2 A � Rag toplay a role of the true copula. It is possible that C 0 does not belong to the given family and the copula model under consideration ismisspecified. Thus, we introduce the pseudo true copula parameter ak by using a family of density power divergences (cf. BHHJ,1998) as follows:

ak ¼ argmina2A

R½0;1�d cðu : aÞ1þk � ð1þ 1

kÞc0ðu : a0Þcðu : aÞk þ 1k c0ðu : a0Þ1þk

n odu1 � � � dud; k > 0,R

½0;1�d c0ðu : a0Þ log c0ðu : a0Þ � log cðu : aÞf gdu1; � � � ; dud; k ¼ 0,

(

where c(u:a) denotes the copula density function associated with the candidate copula function C(u:a) and c0ðu : a0Þ is the truecopula density for u ¼ ðu1; . . . ; udÞ 2 ½0; 1�d . Note that if k ¼ 0, the above density power divergence is the same as theKullback–Leibler divergence. If cðu1; . . . ; ud : aÞ correctly specifies the true copula density, it holds that ak ¼ a0. Since the k > 1case can cause a great loss of efficiency for some basic models as described in BHHJ, in this study, we focus on the case of0 < k � 1.

To propose the robust estimator for the copula parameter, we a priori need to estimate the model parameter h0 and the truemarginal distributions F0

1 ; . . . ; F0d . First, we assume that the estimator hn of h0 satisfies (D2). For instance, we can estimate the

parameter h01 by the OLS and the parameter h02 by the QMLE (cf. Chen and Fan 2006). Given the parameter estimatorhn ¼ ðh01;n; h02;nÞ

0, we estimate F0i by using the rescaled empirical distribution of residual f�itðhnÞ : t ¼ 1; 2; . . . ; ng:

FinðxÞ ¼1

nþ 1

Xn

t¼1

I �itðhnÞ � x� �

; i ¼ 1; 2; . . . ; d:

Then, based on hn and F1n; . . . ; Fdn, we propose the MDPDE for ak as follows:

ak;n ¼ argmina2A

Hk;nðaÞ;

where Hk;nðaÞ ¼ 1n

Pnt¼1 VkðF1nð�1tðhnÞÞ; . . . ; Fdnð�dtðhnÞÞ : aÞ and

VkðF1nð�1tðhnÞÞ; . . . ; Fdnð�dtðhnÞÞ : aÞ ¼R½0;1�d cðu1; . . . ; ud : aÞ1þkdu1; . . . ; dud � ð1þ 1

kÞcðF1nð�1tðhnÞÞ; . . . ; Fdnð�dtðhnÞÞ : aÞk; k > 0,

� log cðF1nð�1tðhnÞÞ; . . . ; Fdnð�dtðhnÞÞ : aÞ; k ¼ 0.

(

Below, for notational convenience, we use the symbols:

ROBUST ESTIMATION FOR COPULA PARAMETER

J. Time Ser. Anal. 2013, 34 302–314 � 2012 Wiley Publishing Ltd. wileyonlinelibrary.com/journal/jtsa

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VkðaÞðu1; . . . ; ud : aÞ ¼ @

@aVkðu1; . . . ; ud : aÞ;

VkðaaÞðu1; . . . ; ud : aÞ ¼ @2

@a@a0Vkðu1; . . . ; ud : aÞ;

VkðaiÞðu1; . . . ; ud : aÞ ¼ @2

@a@uiVkðu1; . . . ; ud : aÞ;

�tðhnÞ ¼ �t ¼ ð�1t; . . . ; �dtÞ; �tðh0Þ ¼ �0t ¼ ð�0

1t; . . . ; �0dtÞ:

To establish the asymptotic properties of the MDPDE, we assume that the following conditions hold:

A1. ak is unique and is an interior point of the compact parameter space A � Ra;

A2. cðu1; . . . ; ud : aÞ is continuous in ðu1; . . . ; ud; aÞ 2 ð0; 1Þd � A;

A3. For any D0 2 (0,1/2), there exist b0 2 (0,1) and M0 > 0 such that

supa2A

cðu1; . . . ; ud : aÞ � M0f^di¼1uig�b0 for ^d

i¼1 ui � D0

and

supa2A

cðu1; . . . ; ud : aÞ � M0f1� _di¼1uig�b0 for _d

i¼1 ui 1� D0;

A4. For 1 � i,j � a, @2

@ai@ajVkðu1; . . . ; ud : aÞ is continuous in ðu1; . . . ; ud; aÞ 2 ð0; 1Þd � A;

A5. For some g > 0 and any D1 2 (0,1/2), there exist b1 2 (0,1) and M1 > 0 such that

supka�akk�g

@2

@ai@ajVkðu1; . . . ; ud : aÞ

�� M1f^d

i¼1uig�b1 for ^di¼1 ui � D1

and

supka�akk�g

@2

@ai@ajVkðu1; . . . ; ud : aÞ

�� M1f1� _d

i¼1uig�b1 for _di¼1 ui 1� D1;

A6. For 1 � i � a and 1 � j � d, @2

@ai@ujVkðu1; . . . ; ud : aÞ is continuous in ðu1; . . . ; ud; aÞ 2 ð0; 1Þd � A;

A7. For any D2 2 (0,1/2), there exist b2 2 (0,1/2) and M2 > 0 such that

@2

@ai@ujVkðu1; . . . ; ud : akÞ

�� M2f^d

i¼1uig�b2 for ^di¼1 ui � D2

and

@2

@ai@ujVkðu1; . . . ; ud : akÞ

�� M2f1� _d

i¼1uig�b2 for _di¼1 ui 1� D2;

A8. Ck :¼ �E0fVkðaaÞðF01ð�0

11Þ; . . . ; F0dð�0

d1Þ : akÞg is finite and non-singular;A9. Rk :¼ Var0fVkðaÞðF0

1ð�011Þ; . . . ; F0

dð�0d1Þ : akÞ þ

Pdi¼1

RðIðF0

i ð�0itÞ � uiÞ � uiÞVkðaiÞðu1; . . . ; ud : akÞdC0ðu1; . . . ; ud : a0Þg is finite

and positive definite.

Further, we also assume that

E1. E0 @@h �itðh0Þ�� 2

< 1;

E2. E0 supkh� h0k�khn � h0k@2

@h@h0 �itðhÞ�� 2

< 1.

REMARK 2. (A1), (A8) and (A9) are standard conditions to obtain the asymptotic properties of the estimator ak;n. Meanwhile, (A3),(A5) and (A7) are similar to those imposed by Chan et al. (2009). Since we do not use the log-copula density in our study, theseconditions are easily met by commonly used copula densities.

REMARK 3. (E1) and (E2) imply that max1�t�n@@h �itðh0Þ�� ¼ opð

ffiffiffinpÞ and max1�t�n supkh�h0k�khn�h0k

@2

@h@h0 �itðhÞ�� ¼ opð

ffiffiffinpÞ, respec-

tively. See Lemma 1.

By Taylor’s theorem, we can get

�it � �0it ¼ �itðhnÞ � �itðh0Þ

¼ ðhn � h0Þ0@

@h�itðh0Þ þ

1

2ðhn � h0Þ0

@2

@h@h0�itð�hnÞ

� �ðhn � h0Þ

ð2Þ

B. KIM AND S. LEE

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for some �hn between hn and h0. Note that by eqn (2), (E1), (E2) and (D2), we can see that

max1�t�n

j�it � �0itj ¼ opð1Þ ð3Þ

and

Xn

t¼1

ð�it � �0itÞ

2 ¼ Opð1Þ: ð4Þ

Further, under (D1)–(D4), by virtue of Lemma A.1 of Chen and Fan (2006), it can be seen that uniformly over x 2 R,

FinðxÞ � F0i ðxÞ ¼ FinðxÞ � F0

i ðxÞ þ f 0i ðxÞ E0 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

hitðh0Þp @

@hlitðh01Þ

!þ xE0 1

2hitðh0Þ@

@hhitðh0Þ

� � !0ðhn � h0Þ þ opðn�1=2Þ; ð5Þ

where FinðxÞ ¼ 1n

Pnt¼1 Ið�0

it � xÞ. Based on these facts, we can obtain the following results (see Section 4 for the proofs).

Theorem 1.

Under assumptions (D1)–(D4), (A1)–(A3) and (E1)–(E2), we have kak;n � akk ¼ opð1Þ as n ! 1.

THEOREM 2. Under assumptions (D1)–(D4), (A1)–(A9) and (E1)–(E2), we have

ffiffiffinpðak;n � akÞ!

dNð0;C�1

k RkC�1k Þas n!1;

where Ck and Rk are the ones defined in (A8) and (A9), respectively.

REMARK 4. In general, there are no universal rules for the selection of k. Some guidelines are provided in Fujisawa and Eguchi(2006).

3. SIMULATION RESULTS

In this section, we compare the performance of the MDPDE with k > 0 for the copula parameter with that of a pseudo maximumlikelihood estimator through a simulation study. To task this, we consider the semiparametric copula-based GARCH models definedin Chan et al. (2009). More precisely, we assume that the observations fYt ¼ ðY1;t; Y2;tÞ0 : t ¼ 1; . . . ; ng satisfy

Yi;t ¼ hi;t�i;t; h2i;t ¼ xi þ aiY

2i;t�1 þ bih

2i;t�1; i ¼ 1; 2; ð6Þ

where f�t ¼ ð�1;t; �2;tÞ0 : t ¼ 1; . . . ; ng is a sequence of i.i.d random vectors with marginal distribution N(0,1) andðx1; a1; b1Þ ¼ ð0:8; 0:4; 0:2Þ and ðx2; a2; b2Þ ¼ ð0:5; 0:2; 0:3Þ. We consider the Gaussian (true copula parameter a0 ¼ 1/3), Clayton(a0 ¼ 1), Gumbel (a0 ¼ 2) and Frank (a0 ¼ 2) copulas as the true copula for innovations. To illustrate the robust property of theMDPDE, we only consider the case that copula models are correctly specified. With regard to selecting a copula model problem, werefer to Chen and Fan (2006) and Chan et al. (2009).

We first consider the case that data are contaminated by additive outliers (AO). That is to say, we observe Xt ¼ ð1� PtÞYt þ Pt Nt ,where Pt are i.i.d Bernoulli random variables with success probability p and Nt are i.i.d two-dimensional multivariate normal

distribution with mean (0,0)0

and covariance matrix10 00 10

� �. It is assumed that Yt; Pt and Nt are all independent. In this study, we

consider the case of p ¼ 0,0.05 and 0.1. For the comparison, we investigate the sample mean, variance and mean squared error forestimators. The sample size under consideration is n ¼ 1000 and the repetition number in each simulation is 1000. Note thatðxi; ai; biÞ; i ¼ 1; 2, are estimated by the QMLE. The figures marked by the symbol

�stand for minimal MSEs and the dark area

represents the MDPDE with smaller MSEs than the pseudo maximum likelihood estimator. The cases of p ¼ 0 in Tables 1–4 show thatwhen the data are not contaminated by outliers, the pseudo maximum likelihood estimator outperforms the MDPDE with k > 0.However, we can see that the performance of the MDPDE with k close to 0 is similar to that of the pseudo maximum likelihoodestimator. This result confirms that the efficiency of the MDPDE decreases as k increases. Meanwhile, the cases of p ¼ 0.05 and 0.1 inTables 1–4 show that the MDPDE performs better when the data are contaminated by outliers. For example, when p ¼ 0.1 in Table 1,the relative MSE of the pseudo maximum likelihood estimator to the MDPDE with a ¼ 1 is about 5.1. Although the MDPDEs with highk in Tables 2 and 3 do not perform well, when k ¼ 0.3, the relative MSEs of the pseudo maximum likelihood estimator to the MDPDEare 1.9 and 3.2, respectively. Note that as p increases, the symbol

�moves downwards as seen in Table 1 and the dark area becomes

wider as seen in Table 3. This means that if data are severely contaminated by outliers, the MDPDE with high k performs better.



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Next, we consider the innovation outliers (IO) case. We assume that the data is observed from eqn (6) with contaminatedinnovations nt ¼ ð1� PtÞ�t þ Pt Nt , where Pt and Nt are defined identically in AO cases. We also consider the cases of p ¼ 0,0.05 and0.1. Tables 5–8 exhibit the results similar to that of the AO cases.

Finally, we consider the noise with fat tails instead of the Gaussian noise. We assume that the noise Nt is a bivariate t-

distributed random vector with location vector (0,0)0, scale matrix

1 00 1

� �and degrees of freedom 3. Here, we only illustrate

the result for the additive outlier case since the result for the innovation outlier case is similar. Tables 9–12 show results similarto those in Tables 1–4. Overall, our findings strongly support that the MDPDE method is a functional tool to yield a robustestimator.

4. PROOFS

Here, we provide the proofs for the theorems presented in Section 2. We denote F0t ¼ ðF0

1ð�01tÞ; . . . ; F0

dð�0dtÞÞ

0 andFt ¼ ðF1nð�1tÞ; . . . ; Fdnð�dtÞÞ0. The following is obvious and is stated without proof.Lemma 1.

If fXtg is a strictly stationary process with EkXtk2 < 1, then

max1�t�n

kXtk ¼ opðn1=2Þ as n!1:

Table 1. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is normal and AOs exist

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 0.325 (0.080/0.087)�

0.273 (0.123/0.484) 0.232 (0.140/1.172)

M k ¼ 0.1 0.325 (0.081/0.088) 0.284 (0.127/0.372) 0.244 (0.152/0.942)D 0.3 0.324 (0.084/0.092) 0.302 (0.134/0.229) 0.270 (0.176/0.573)P 0.5 0.322 (0.089/0.101) 0.316 (0.142/0.173) 0.293 (0.193/0.356)D 0.7 0.319 (0.096/0.117) 0.323 (0.151/0.161)

�0.308 (0.198/0.260)

E 1 0.312 (0.115/0.162) 0.325 (0.178/0.185) 0.317 (0.203/0.230)�

Table 2. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Clayton and AOs exist

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 0.912 (0.554/1.323)�

0.776 (0.548/5.578) 0.654 (0.543/12.48)

M k ¼ 0.1 0.911 (0.615/1.407) 0.816 (0.621/4.023) 0.701 (0.645/9.562)D 0.3 0.894 (0.803/1.933) 0.857 (0.883/2.936)

�0.762 (0.972/6.619)

�

P 0.5 0.845 (1.554/3.970) 0.818 (1.737/5.061) 0.729 (1.704/9.046)D 0.7 0.765 (3.268/8.778) 0.699 (3.122/12.17) 0.596 (2.270/18.61)E 1 0.650 (10.70/22.95) 0.489 (5.066/31.21) 0.391 (2.006/39.13)

Table 3. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Gumbel and AOs exist

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 1.941 (0.357/0.709)�

1.779 (0.375/5.280) 1.649 (0.359/12.66)

M k ¼ 0.1 1.940 (0.371/0.735) 1.835 (0.410/3.135) 1.718 (0.429/8.376)D 0.3 1.925 (0.547/1.109) 1.896 (0.627/1.704)

�1.820 (0.699/3.920)

�

P 0.5 1.887 (1.496/2.763) 1.872 (1.204/2.836) 1.813 (1.274/4.771)D 0.7 1.818 (3.618/6.939) 1.761 (2.473/8.192) 1.683 (2.424/12.46)E 1 1.612 (145.8/160.8) 1.454 (70.00/99.76) 1.371 (98.18/137.6)

Table 4. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Frank and AOs exist

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 1.991 (4.267/4.271) 1.816 (4.450/7.821) 1.658 (4.638/16.33)

M k ¼ 0.1 1.991 (4.265/4.270)�

1.825 (4.530/7.571) 1.679 (4.799/15.13)D 0.3 1.990 (4.305/4.310) 1.853 (4.662/6.806) 1.724 (5.046/12.67)P 0.5 1.989 (4.361/4.368) 1.875 (4.822/6.374) 1.765 (5.335/10.86)D 0.7 1.988 (4.440/4.450) 1.894 (4.997/6.115) 1.802 (5.629/9.536)E 1 1.986 (4.593/4.608) 1.917 (5.293/5.980)

�1.850 (6.074/8.313)

�

B. KIM AND S. LEE


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Table 6. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Clayton and IOs exist

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 0.910 (0.579/1.382)�

0.750 (0.539/6.796) 0.615 (0.553/15.40)

M k ¼ 0.1 0.910 (0.610/1.411) 0.790 (0.621/5.019) 0.654 (0.663/12.65)D 0.3 0.896 (0.769/1.847) 0.834 (0.858/3.628)

�0.703 (0.959/9.766)

�

P 0.5 0.850 (1.453/3.687) 0.793 (1.573/5.851) 0.669 (1.585/12.57)D 0.7 0.774 (3.223/8.338) 0.671 (2.732/13.58) 0.554 (2.288/22.14)E 1 0.661 (10.56/22.02) 0.477 (8.346/35.64) 0.389 (3.423/40.72)

Table 5. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is normal and IOs exist

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 0.326 (0.080/0.086)�

0.266 (0.130/0.577) 0.225 (0.145/1.319)

M k ¼ 0.1 0.326 (0.081/0.087) 0.279 (0.136/0.435) 0.238 (0.159/1.063)D 0.3 0.325 (0.085/0.092) 0.302 (0.150/0.249) 0.267 (0.192/0.636)P 0.5 0.323 (0.090/0.100) 0.320 (0.163/0.181) 0.294 (0.223/0.379)D 0.7 0.320 (0.097/0.115) 0.332 (0.178/0.178)

�0.314 (0.246/0.281)

�

E 1 0.312 (0.113/0.157) 0.339 (0.224/0.227) 0.329 (0.285/0.287)

Table 7. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Gumbel and IOs exist

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 1.943 (0.381/0.706)�

1.750 (0.366/6.623) 1.608 (0.350/15.70)

M k ¼ 0.1 1.942 (0.398/0.730) 1.805 (0.414/4.199) 1.667 (0.434/11.55)D 0.3 1.929 (0.539/1.041) 1.869 (0.616/2.337)

�1.754 (0.709/6.753)

�

P 0.5 1.891 (0.980/2.162) 1.846 (1.116/3.501) 1.743 (1.239/7.821)D 0.7 1.820 (2.497/5.750) 1.733 (2.351/9.498) 1.619 (2.270/16.79)E 1 1.687 (17.93/27.68) 1.435 (61.87/93.78) 1.327 (62.00/107.2)

Table 8. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Frank and IOs exist

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 1.993 (4.132/4.133)�

1.806 (4.363/8.125) 1.643 (5.111/17.86)

M k ¼ 0.1 1.993 (4.142/4.143) 1.821 (4.478/7.672) 1.672 (5.351/16.12)D 0.3 1.992 (4.199/4.202) 1.859 (4.678/6.669) 1.732 (5.807/12.98)P 0.5 1.990 (4.275/4.281) 1.891 (4.920/6.101) 1.791 (6.351/10.73)D 0.7 1.989 (4.380/4.389) 1.920 (5.183/5.812) 1.846 (6.941/9.291)E 1 1.986 (4.580/4.596) 1.958 (5.617/5.786)

�1.922 (7.878/8.472)

�

Table 9. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is normal and AOs exist (t-distributed noise case)

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 0.324 (0.081/0.089)�

0.303 (0.090/0.182) 0.282 (0.100/0.360)

M k ¼ 0.1 0.324 (0.083/0.090) 0.306 (0.090/0.165) 0.287 (0.100/0.313)D 0.3 0.324 (0.087/0.096) 0.310 (0.092/0.148) 0.294 (0.102/0.253)P 0.5 0.322 (0.092/0.106) 0.311 (0.097/0.146)

�0.299 (0.108/0.226)

D 0.7 0.319 (0.100/0.122) 0.311 (0.105/0.156) 0.301 (0.118/0.221)�

E 1 0.311 (0.117/0.165) 0.307 (0.131/0.198) 0.301 (0.151/0.253)

Table 10. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Clayton and AOs exist (t-distributed noise case)

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 0.917 (0.568/1.251)�

0.844 (0.582/3.024) 0.775 (0.558/5.612)

M k ¼ 0.1 0.918 (0.601/1.278) 0.855 (0.603/2.704)�

0.794 (0.592/4.851)D 0.3 0.904 (1.078/1.991) 0.856 (0.757/2.822) 0.807 (0.768/4.505)

�

P 0.5 0.856 (1.627/3.686) 0.817 (1.432/4.790) 0.773 (1.425/6.583)D 0.7 0.786 (4.563/9.130) 0.737 (2.962/9.885) 0.693 (2.779/12.23)E 1 0.685 (14.20/24.08) 0.606 (9.139/24.63) 0.560 (8.107/27.46)



30

7

LEMMA 2. Suppose that (D1)–(D4) and (E1)–(E2) hold and h(u:a) is any continuous function in u ¼ ðu1; . . . ; udÞ0 2 ð0; 1Þd anda 2 A that satisfies the following conditions:

(1) A � Ra is a compact parameter space;

(2) h(u:a) is continuous in ðu; aÞ 2 ð0; 1Þd � A;

(3) For any D 2 (0,1/2), there exist b 2 (0,1) and M > 0 such that

supa2Ajhðu1; . . . ; ud : aÞj � Mf^d

i¼1uig�b for ^di¼1 ui � D

and

supa2Ajhðu1; . . . ; ud : aÞj � Mf1� _d

i¼1uig�b for _di¼1 ui 1� D:

Then, we have

supa2A

1

n

Xn

t¼1

hðF1nð�1tÞ; . . . ; Fdnð�dtÞ : aÞ � E0ðhðF01ð�0

11Þ; . . . ; F0dð�0

d1Þ : aÞÞ��

�� ¼ opð1Þ:

PROOF. Note that

supa2A

1

n

Xn

t¼1

hðFt : aÞ � E0ðhðF01 : aÞÞ

��

� supa2A

1

n

Xn

t¼1

hðF0t : aÞ � E0ðhðF0

1 : aÞÞ��

��þ supa2A

1

n

Xn

t¼1

hðFt : aÞ � 1

n

Xn

t¼1

hðF0t : aÞ

��

¼ In þ IIn:

First, we show that In ¼ opð1Þ. By the conditions in (1)–(3) and the compactness of ½D; 1 � D�d � A, we can have

E0 supa2AjhðF0

1 : aÞj� �

¼Zð0;1Þd

supa2Ajhðu : aÞjdC0ðu : a0Þ

¼Zð0;1Þd

supa2Ajhðu : aÞj Ið^d

i¼1ui < DÞ þ Ið_di¼1ui > 1� DÞ

þ IðD � ^d

i¼1ui � _di¼1ui � 1� DÞ

dC0ðu : a0Þ

�Zð0;1Þd

MXd

i¼1

u�bi Iðui < DÞ þ ð1� uiÞ�bIðui > 1� DÞ

n odC0ðu : a0Þ þ sup

a2A;u2½D;1�D�djhðu : aÞj

¼ dM

Z D

0

u�bduþZ 1

1�Dð1� uÞ�bdu

� �þ sup

a2A;u2½D;1�D�djhðu : aÞj ¼ 2dM

1

1� bD1�b þ sup

a2A;u2½D;1�D�djhðu : aÞj <1:

Table 11. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Gumbel and AOs exist (t-distributed noise case)

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 1.941 (0.374/0.717)�

1.845 (0.344/2.740) 1.759 (0.330/6.114)

M k ¼ 0.1 1.941 (0.383/0.728) 1.861 (0.354/2.272) 1.785 (0.343/4.956)D 0.3 1.928 (0.505/1.022) 1.870 (0.517/2.211)

�1.809 (0.499/4.128)

�

P 0.5 1.890 (0.922/2.128) 1.841 (1.014/3.542) 1.789 (0.960/5.393)D 0.7 1.817 (2.412/5.764) 1.768 (2.335/7.727) 1.716 (2.238/10.28)E 1 1.656 (28.33/40.14) 1.551 (103.7/123.8) 1.484 (105.2/131.7)

Table 12. Sample Mean(Variance·102/MSE·102) of copula parameter estimators when true copula is Frank and AOs exist (t-distributed noise case)

p ¼ 0 p ¼ 0.05 p ¼ 0.1

Pseudo MLE(k ¼ 0) 1.979 (4.007/4.049)�

1.883 (4.131/5.501) 1.775 (4.162/9.234)

M k ¼ 0.1 1.979 (4.017/4.059) 1.884 (4.158/5.497) 1.777 (4.196/9.152)D 0.3 1.978 (4.063/4.108) 1.890 (4.205/5.414) 1.789 (4.264/8.707)P 0.5 1.977 (4.121/4.170) 1.894 (4.279/5.406)

�1.798 (4.350/8.435)

D 0.7 1.976 (4.204/4.258) 1.897 (4.370/5.437) 1.805 (4.453/8.249)E 1 1.973 (4.363/4.431) 1.899 (4.544/5.554) 1.814 (4.643/8.098)

�

B. KIM AND S. LEE


30

8

Next, we deal with IIn. Let K ¼ fmax1�t�n _di¼1jFinð�itÞ � F0

i ð�0itÞj � D

2g. By eqn (3), eqn (5), (D2), (D4), and the Glivenko–Cantellitheorem, we have

max1�t�n

Finð�itÞ � F0i ð�0

itÞ�� sup

x2RFinðxÞ � F0

i ðxÞ�� þ sup

x2Rf 0

i ðxÞ max1�t�n

�it � �0it

�� ¼ opð1Þ:

ð7Þ

Hence, P(Kc) ! 0 as n ! 1.On the set K, we can express IIn � II1n þ II2n þ II3n þ II4n þ II5n; where

II1n ¼ supa2A

1

n

Xn

t¼1

hðFt : aÞ�� I ^d

i¼1Finð�itÞ <D2

or _di¼1 Finð�itÞ > 1� D

2

� �;

II2n ¼ supa2A

1

n

Xn

t¼1

hðF0t : aÞ

�� I ^di¼1F0

i ð�0itÞ <

D2

or _di¼1 F0

i ð�0itÞ > 1� D

2

� �;

II3n ¼ supa2A

1

n

Xn

t¼1

hðFt : aÞ�� I D

2� ^d

i¼1Finð�itÞ � _di¼1Finð�itÞ � 1� D

2

� �

� I ^di¼1F0

i ð�0itÞ <

D2

or _di¼1 F0


2

� �;

II4n ¼ supa2A

1

n

Xn

t¼1

hðF0t : aÞ

�� I D2� ^d

i¼1F0i ð�0

itÞ � _di¼1F0

i ð�0itÞ � 1� D

2

� �

� I ^di¼1Finð�itÞ <

D2

or _di¼1 Finð�itÞ > 1� D

2

� �;

II5n ¼ supa2A

1

n

Xn

t¼1

hðFt : aÞ � hðF0t : aÞ

�� I D2� ^d

i¼1Finð�itÞ � _di¼1Finð�itÞ � 1� D

2

� �

� ID2� ^d

i¼1F0i ð�0

itÞ � _di¼1F0

i ð�0itÞ � 1� D

2

� �:

First, note that

II1n �M

n

Xn

t¼1

ð^di¼1Finð�itÞÞ�bI ^d

i¼1Finð�itÞ <D2

� ��þð1� _d

i¼1Finð�itÞÞ�bI _di¼1Finð�itÞ > 1� D

2

� ��

� M

n

Xn

t¼1

Xd

i¼1

t

nþ 1

� ��b

It

nþ 1<

D2

� �þ 1� t

nþ 1

� ��b

It

nþ 1> 1� D

2

� �( )

� M

ndðnþ 1Þ

Z D=2

0

u�bduþZ 1

1�D=2

ð1� uÞ�bdu

( )

¼ 2dM 1þ 1

n

� �1

1� bD2

� �1�b

! 0 as n!1;D! 0;

which implies II1n ¼ opð1Þ as n ! 1, D ! 0.Second, by the strong law of large numbers, as n ! 1,

II2n �M

n

Xn

t¼1

Xd

i¼1

ðF0i ð�0

itÞÞ�bI F0

i ð�0itÞ <

D2

� �þ ð1� F0

i ð�0itÞÞ�bI F0


2

� ��

!a:s: 2dM

Z D=2

0

u�bdu ¼ 2dM1

1� bD2

� �1�b

:

Hence, II2n ¼ opð1Þ as n ! 1, D ! 0.Third, on K,

II3n � supa2A

1

n

Xn

t¼1

hðFt : aÞ�� I ^d

i¼1Finð�itÞ < D or _di¼1 Finð�itÞ > 1� D

�and

II4n � supa2A

1

n

Xn

t¼1

hðF0t : aÞ

�� I ^di¼1F0

i ð�0itÞ < D or _d

i¼1 F0i ð�0

itÞ > 1� D �

:

Thus, we can show II3n ¼ opð1Þ and II4n ¼ opð1Þ similarly to II1n and II2n, respectively.



30

9

Finally, on K, by the continuity of h and the compactness of ½D=2; 1 � D�d �A, it can be easily seen that II5n ¼ opð1Þ as n ! 1.Therefore, we have II1n þ II2n þ II3n þ II4n þ II5n ¼ opð1Þ as n ! 1 and D ! 0. This validates the lemma.

PROOF OF THEOREM 1. It is sufficient to show that

supa2A

1

n

Xn

t¼1

cðFt : aÞk � E0 cðF01 : aÞk

� �� ¼ opð1Þ:

However, since the above is directly yielded by Lemma 2 with h(u:a) ¼ c(u:a)k for 0 < k � 1, the theorem is established.By Theorem 2.1 and (A1), we have

limn!1

Pfak;n is an interior point of Ag ¼ 1:

In case that ak;n is in the interior of A, by Taylor’s theorem, we have

0 ¼ @Hk;nðak;nÞ@a

¼ @Hk;nðakÞ@a

þ @2Hk;nð�ak;nÞ@a@a0

ðak;n � akÞ;

where �ak;n is an appropriate intermediate point between ak;n and ak. Hence,

ffiffiffinpðak;n � akÞ ¼ C�1

k

ffiffiffinp @Hk;nðakÞ

@aþ

ffiffiffinp

C�1k

@2Hk;nð�ak;nÞ@a@a0

þ Ck

� �ðak;n � akÞ :

¼ C�1k


@aþ

ffiffiffinp

Dk;n:

Further, by applying eqn (5) and Taylor’s theorem, we have following expression:


@a¼ A0n þ

Xd

i¼1

Bin þXd

i¼1

X3

j¼1

ðCjÞin þXd

i¼1

X3

j¼1

ðDjÞin þXd

i¼1

X3

j¼1

ðEjÞin þXd

i¼1

Fin þXd

i¼1

Gin;

where

A0n ¼1ffiffiffi

npXn

t¼1

VkðaÞðF0t : akÞ;

Bin ¼1ffiffiffi

npXn

t¼1

Finð�itÞ � F0i ð�0

itÞ �

VkðaiÞð�ut : akÞ � VkðaiÞðF0t : akÞ

�;

ðC1Þin ¼1

n

Xn

t¼1

Uinð�0itÞVkðaiÞðF0

t : akÞ � E0 Uinð�0i1ÞVkðaiÞðF0

1 : akÞ �

;

ðC2Þin ¼1

n

Xn

t¼1

f 0i ð�0

itÞVkðaiÞðF0t : akÞgið�0

itÞ0 ffiffiffinp ðhn � h0Þ � E0 f 0

i ð�0i1ÞVkðaiÞðF0

1 : akÞgið�0i1Þ0 � ffiffiffi

npðhn � h0Þ;

ðC3Þin ¼1

n

Xn

t¼1

f 0i ð�0

itÞVkðaiÞðF0t : akÞ

@

@h�itðh0Þ

� �0 ffiffiffinpðhn � h0Þ � E0 f 0


1 : akÞ@

@h�i1ðh0Þ

� �0� � ffiffiffinpðhn � h0Þ;

ðD1Þin ¼1

n

Xn

t¼1

ðUinð�itÞ � Uinð�0itÞÞVkðaiÞðF0

t : akÞ;

ðD2Þin ¼1

n

Xn

t¼1

f 0i ð�itÞgið�itÞ0 � f 0

i ð�0itÞgið�0

itÞ0 � ffiffiffi

npðhn � h0ÞVkðaiÞðF0

t : akÞ;

ðD3Þin ¼1

n

Xn

t¼1

f 0i ð��itÞ � f 0

i ð�0itÞ

�VkðaiÞðF0

t : akÞ@

@h�itðh0Þ

� �0 ffiffiffinpðhn � h0Þ;

ðE1Þin ¼ E0 Uinð�0i1ÞVkðaiÞðF0

1 : akÞ �

;

ðE2Þin ¼ E0 f 0i ð�0

i1ÞVkðaiÞðF01; akÞgið�0

i1Þ0 � ffiffiffi

npðhn � h0Þ;

ðE3Þin ¼ E0 f 0i ð�0

i1ÞVkðaiÞðF01 : akÞ

@

@h�i1ðh0ÞÞ0

� � ffiffiffinpðhn � h0Þ

� �;

Fin ¼1

n

Xn

t¼1

ffiffiffinpðFinð�itÞ � F0

i ð�itÞÞ � Uinð�itÞ � f 0i ð�itÞgið�itÞ0

ffiffiffinpðhn � h0Þ

� �VkðaiÞðF0

t : akÞ;

Gin ¼1

2

1ffiffiffinpXn

t¼1

f 0i ð��itÞðhn � h0Þ0

@2

@h@h0�itð�hÞ

� �ðhn � h0ÞVkðaiÞðF0

t : akÞ;

B. KIM AND S. LEE


31

0

and finally,

�ut ¼ ð�u1t; . . . ; �udtÞ0 where �uit is an appropriate intermidiate point betweenFinð�itÞ and F0i ð�0

itÞ:

Additionally, Uin(x) and gi(x) are defined by

UinðxÞ ¼ffiffiffinp 1

n

Xn

t¼1

Ið�0it � xÞ � F0

i ðxÞ !

and

giðxÞ ¼ E0 1ffiffiffiffiffiffiffiffiffiffiffiffiffihitðh0Þ

p @

@hlitðh01Þ

!þ xE0 1

2hitðh0Þ@

@hhitðh0Þ

� �:

LEMMA 3. Suppose that the conditions in Theorem 2 hold. Then, we have

A0n þXd

i¼1

X3

j¼1

ðEjÞin!d

Nð0;RkÞ;

where Rk is defined in (A9).

PROOF. Since ltðh01Þ and Htðh0Þ are independent of �tðh0Þ, ðE2Þin þ ðE3Þin ¼ 0. Thus,

A0n þXd

i¼1

X3

j¼1

ðEjÞin ¼ A0n þXd

i¼1

ðE1Þin

¼ 1ffiffiffinpXn

t¼1

VkðaÞðF0t : akÞ þ

Xd

i¼1

ZfIðF0

i ð�0itÞ � uiÞ � uigVkðaiÞðu : akÞdC0ðu : a0Þ

" #:

Since ak ¼ argmina2AE0ðVkðF0t : aÞÞ and E0½

RfIðF0

i ð�0itÞ � uiÞ � uigVkðaiÞðu : akÞdC0ðu : a0Þ� ¼ 0,

fVkðaÞðF0t : akÞ þ

Pdi¼1

RfIðF0

i ð�0itÞ � uiÞ � uigVkðaiÞðu : akÞdC0ðu : a0Þ : t ¼ 1; . . . ; ng are iid random vectors with mean 0. Therefore,

A0n þXd

i¼1

X3

j¼1

ðEjÞin!d

Nð0;RkÞ:

This completes the proof.


kBink ¼ opð1Þ as n!1; i ¼ 1; . . . ; d:

PROOF. By the Cauchy–Schwarz inequality,

kBink2 � 1

n

Xn

t¼1


�� 2Xn

t¼1

ðFinð�itÞ � F0i ð�0

itÞÞ2:

Since VkðaiÞðu1; . . . ; udÞ is continuous function, due to (A7) and eqn (7), we have

1

n

Xn

t¼1


�� 2¼ opð1Þ:

Further, in view of eqn (5), by (D2), (D4) and (2.4),



31

1

Xn

t¼1

ðFinð�itÞ � F0i ð�0

itÞÞ2 � 2n sup

x2RðFinðxÞ � F0

i ðxÞÞ2 þ 2 sup

x2Rf 0

i ðxÞ� �2Xn

t¼1

ð�it � �0itÞ

2 ¼ Opð1Þ:

Therefore, kBink ¼ opð1Þ for i¼1, . . . ,d. This validates the lemma.


kðCjÞink ¼ opð1Þ as n!1; i ¼ 1; . . . ; d; j ¼ 1; 2; 3:

PROOF. By using (A6) and (A7), we can verify kðC1Þink ¼ opð1Þ for i ¼ 1, . . . ,d in a similar fashion as in Corollary 5.5 of Ruymgaartet al. (1972).

Next, we deal with kðC2Þink. By the strong law of large numbers,

1

n

Xn

t¼1

f 0i ð�0


itÞ0 !a:s: E0 f 0


1 : akÞgið�0i1Þ0 �

as n!1:

Hence, together with (D2), we obtain

kðC2Þink �1

n

Xn

t¼1

f 0i ð�0


itÞ0 � E0 f 0


1 : akÞgið�0i1Þ0 �� ffiffiffi

npðhn � h0Þ

�� ¼ opð1Þ as n!1; i ¼ 1; . . . ; d:

Finally, by the strong law of large numbers, as n ! 1,

1

n

Xn

t¼1

f 0i ð�0

itÞVkðaiÞðF0t : akÞ

@

@h�itðh0Þ

� �0!a:s: E0 f 0


1 : akÞ@

@h�i1ðh0Þ

� �0� �:

By using this, we can show that kðC3Þink ¼ opð1Þ for i ¼ 1, . . . ,d similarly to kðC2Þink. This validates the lemma.


kðDjÞink ¼ kFink ¼ kGink ¼ opð1Þ as n!1; i ¼ 1; . . . ; d; j ¼ 1; 2; 3:

PROOF. By the strong law of large numbers and central limit theorem with (A7), we have

1

n

Xn

t¼1

kVkðaiÞðF0t : akÞk ¼

1ffiffiffinpXn

t¼1

kVkðaiÞðF0t : akÞk ¼ Opð1Þ:

Further, by the Glivenko–Cantelli theorem,

1ffiffiffinp max

1�t�nkUinð�itÞ � Uinð�0

itÞk � 2 supx2R

1

n

Xn

t¼1

Ið�0it � xÞ � F0

i ðxÞ��

��!a:s: 0:By using the uniform continuity of f 0

i ðxÞ and xf 0i ðxÞ, (D2), (E1), (E2), eqn (3) and eqn (5), we obtain

B. KIM AND S. LEE


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2

kðD1Þink �1ffiffiffi

np max

1�t�nkUinð�itÞ � Uinð�0

itÞk1ffiffiffi

npXn

t¼1

kVkðaiÞðF0t : akÞk ¼ opð1Þ;

kðD2Þink � max1�t�n

kf 0i ð�itÞgið�itÞ0 � f 0

i ð�0itÞgið�0

itÞ0kk

ffiffiffinpðhn � h0Þk

1

n

Xn

t¼1

kVkðaiÞðF0t : akÞk

¼ opð1Þ;

kðD3Þink � max1�t�n

jf 0i ð��itÞ � f 0

i ð�0itÞj

1ffiffiffinp max

1�t�n

@

@h�itðh0Þ

��

� �kffiffiffinpðhn � h0Þk

� 1ffiffiffinpXn

t¼1


kFink � supx2R

ffiffiffinpðFinðxÞ � F0

i ðxÞÞ � UinðxÞ � f 0i ðxÞgiðxÞ0

ffiffiffinpðhn � h0Þ

�� 1

n

Xn

t¼1


kGink � supx2R

f 0i ðxÞ

1ffiffiffinp max

1�t�nsup

kh�h0k�khn�h0k

@2

@h@h0�itðhÞ

��

!kffiffiffinpðhn � h0Þk2

� 1

n

Xn

t¼1

kVkðaiÞðF0t : akÞk ¼ opð1Þ:

Hence, the lemma is established.


ffiffiffinpkDk;nk ¼ opð1Þ as n!1:

PROOF. By Lemma 2, Theorem 1 and the fact that E0ðVkðaaÞðF01 : aÞÞ is continuous in a, we have


þ Ck

�� sup

ka�akk�g

1

n

Xn

t¼1

VkðaaÞðFt : aÞ � E0 VkðaaÞðF01 : aÞ

��þ E0ðVkðaaÞðF0

1 : �ak;nÞÞ � E0ðVkðaaÞðF01 : akÞÞ

�� ¼ opð1Þ: ð8Þ

Further, If@2Hk;nð�ak;nÞ@a@a0 is invertible,

ffiffiffinp

Dk;n can be written as

ffiffiffinp

Dk;n ¼ �C�1k


þ Ck

� �@2Hk;nð�ak;nÞ@a@a0

� ��1 ffiffiffinp @Hk;nðakÞ

@a

� �:

Since Ck is finite and nonsingular, there exists a positive constant n such that kM�1k � 2kC�1k k for any a · a matrix M with

||M + Ck|| � n. Therefore, by Lemma 3–6 and eqn (8), we have

ffiffiffinpkDk;nk ¼ C�1

k

�� @2Hk;nð�ak;nÞ@a@a0

þ Ck

�� @2Hk;nð�ak;nÞ

@a@a0

� ��1��

�� ffiffiffinp @Hk;nðakÞ

@a

��

¼ opð1Þ as n!1:

Hence, the lemma is asserted.

PROOF OF THEOREM 2. The theorem is a direct result of Lemmas 3–7.

ACKNOWLEDGEMENTS

We like to thank the referee for his/her helpful comments. This work was supported by Mid-career Researcher Program throughNRF grant funded by the MEST (No. 2010-0000374).



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robust estimation for copula parameter in scomdy models

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