jackknife empirical likelihood method for copulasyqi/mydownload/12test.pdfjackknife empirical...

Test (2012) 21:74–92DOI 10.1007/s11749-011-0236-4

O R I G I NA L PA P E R

Jackknife empirical likelihood method for copulas

Liang Peng · Yongcheng Qi · Ingrid Van Keilegom

Received: 5 May 2010 / Accepted: 28 January 2011 / Published online: 26 February 2011© Sociedad de Estadística e Investigación Operativa 2011

Abstract Copulas are used to depict dependence among several random variables.Both parametric and non-parametric estimation methods have been studied in theliterature. Moreover, profile empirical likelihood methods based on either empiri-cal copula estimation or smoothed copula estimation have been proposed to con-struct confidence intervals of a copula. In this paper, a jackknife empirical likelihoodmethod is proposed to reduce the computation with respect to the existing profileempirical likelihood methods.

Keywords Copulas · Empirical likelihood method · Jackknife

Mathematics Subject Classification (2000) 62E20 · 62F40 · 62G07 · 62G30 ·62H15

1 Introduction

Dependence among variables plays an important role in understanding and interpret-ing multivariate data series in economics, finance, insurance and other fields in social

Communicated by Domingo Morales.

L. PengSchool of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USAe-mail: [email protected]

Y. QiDepartment of Mathematics and Statistics, University of Minnesota Duluth, 1117 University Drive,Duluth, MN 55812, USAe-mail: [email protected]

I. Van Keilegom (�)Institute of Statistics, Université catholique de Louvain, Voie du Roman Pays 20,1348 Louvain-la-Neuve, Belgiume-mail: [email protected]

mailto:[email protected]



Jackknife empirical likelihood method for copulas 75

sciences. Although some commonly used dependence measures such as Pearson’scorrelation coefficient, Kendall’s tau and Spearman’s rho are useful in describing de-pendence, they cannot completely capture the dependence structure among variables.Instead, as a function independent of marginals, copulas become more or less a stan-dard tool in risk management (see McNeil et al. 2005).

Suppose (X1, Y1), . . . , (Xn,Yn) are independent and identically distributed ran-dom vectors with a joint distribution function F . The copula of F is defined asC(x, y) = F(F−

1 (x),F−2 (y)), where F1(x) = F(x,∞), F2(y) = F(∞, y) and (·)−

denotes the generalized inverse function of (·). We refer to Nelsen (1998) and Joe(1997) for an overview of copulas. A wide range of applications of copulas can befound in the literature of economics, econometrics and finance. For example, Zimmerand Trivedi (2006) used copulas to study self-selection and interdependence betweenhealth insurance and health care demand among married couples; Frees and Wang(2006) employed copula to insurance pricing; Van den Goorbergh et al. (2005) ap-plied dynamic copulas to option pricing; Cameron et al. (2004) modeled counted databy copulas; Hennessy and Lapan (2002) used copulas to study portfolio allocations;Junker and May (2005) proposed to use transformed copulas to study the aggregatefinancial risk on a portfolio level; Smith (2003) employed copulas to model data withselectivity bias; Chen and Fan (2006) used copulas to model errors of multivariatenon-linear time series.

For estimating a copula, both non-parametric and parametric methods have beenproposed in the literature. A simple non-parametric estimator is the so-called empir-ical copula:

Cn(x, y) = 1

n

n∑

i=1

I(Fn1(Xi) ≤ x,Fn2(Yi) ≤ y

), (1.1)

where

Fn1(x) = 1

n

n∑

i=1

I (Xi ≤ x) and Fn2(y) = 1

n

n∑

i=1

I (Yi ≤ y)

are the two marginal empirical distribution functions; see Deheuvels (1979). For adetailed study of the asymptotic limit of the empirical copula estimate, we refer toFermanian et al. (2004). Smoothing estimation of a copula was investigated by Fer-manian and Scaillet (2003) and Chen and Huang (2007).

Recently Chen et al. (2009) and Molanes Lopez et al. (2009) proposed empiricallikelihood methods, based on smoothed copula estimation or empirical copula esti-mation, to construct confidence intervals for a copula. The basic idea is to introducelinking variables s = F−

1 (x) and t = F−2 (y) and then apply profile empirical likeli-

hood methods to the following constraints:

F(s, t) = θ, F1(s) = x, F2(t) = y,

where θ = C(x, y) is the quantity which we like to construct a confidence intervalfor. The important reason for introducing the linking variables is to take the un-

76 L. Peng et al.

known marginals in (1.1) into account so as to obtain a chi-square limit. However,the constraints due to the linking variables cause computational burden in the empir-ical likelihood methods, especially for a high dimensional copula. Since we are onlyinterested in a confidence interval for θ , one may ask whether it is possible to havean empirical likelihood method with only one constraint and which has a chi-squarelimit.

Recently, Jing et al. (2009) proposed to apply empirical likelihood methods tojackknife pseudo samples for a U-statistic so as to avoid a non-linear minimizationproblem caused by direct application of the empirical likelihood method. In this pa-per, we show that the idea in Jing et al. (2009) can be employed to remove unneces-sary constraints in the setup of copulas. However, one has to work with smoothed cop-ula estimation. That is, a smoothed jackknife empirical likelihood method is needed.This new method has a great computational advantage over the existing profile empir-ical likelihood methods for constructing confidence intervals for a copula especiallyfor a high dimensional copula.

We organize this paper as follows. The new methodology is given in Sect. 2. Sec-tion 3 presents some simulation results. All proofs are put in Sect. 4.

2 Methodology

Define

Fn1,i (x) = 1

n − 1

∑

j �=i

I (Xj ≤ x), Fn2,i (y) = 1

n − 1

∑

j �=i

I (Yj ≤ y),

Cn,i(x, y) = 1

n − 1

∑

j �=i

I(Fn1,i (Xj ) ≤ x,Fn2,i (Yj ) ≤ y

).

As in Jing et al. (2009), the jackknife pseudo sample may be defined as

Vi(x, y) = nCn(x, y) − (n − 1)Cn,i(x, y), i = 1, . . . , n.

Based on the above pseudo sample, one can define the empirical likelihood functionas

Ln(x, y; θ) = sup

{n∏

i=1

pi : p1 > 0, . . . , pn > 0,

n∑

i=1

pi = 1,

n∑

i=1

piVi(x, y) = θ

},

where θ = C(x, y) is the quantity for which we like to construct a confidence interval.Unfortunately, the above procedure does not work although only one constraint isinvolved. The reason is as follows.

It is known that a key feature of the empirical likelihood method is the au-tomatic standardization, see Owen (2001) for an overview on empirical likeli-hood methods. In order to have the above procedure work, one needs to showthat

√n{ 1

n

∑ni=1 Vi(x, y) − C(x, y)} converges in distribution to N(0, σ 2) and


1n

∑ni=1{Vi(x, y) − C(x, y)}2 is a consistent estimator of σ 2, i.e.,

νn(x, y) = 1

n

n∑

i=1

{Vi(x, y) − 1

n

n∑

j=1

Vj (x, y)

}2

is a consistent estimator of σ 2. Note that νn(x, y) is the jackknife variance estimateof nVar(Cn(x, y)), see Shao and Tu (1995) for details on jackknife methods. Unfor-tunately, a brief simulation study, not reported here, shows that νn(x, y) is an incon-sistent estimator of nVar(Cn(x, y)). At first sight, one may argue that the definitionof copula involves quantiles and the jackknife variance estimate for a quantile is in-consistent, see Martin (1990). This is not the exact reason as we show that a smoothversion of νn(x, y) works with the non-smoothed marginal empirical distributionsFn1,i , Fn2,i , Fn1,Fn2 remained.

Let k be a density function and put K(x) = ∫ x

−∞ k(y) dy. Then we smoothCn(x, y) and Cn,i(x, y) by

⎧⎨

⎩Cn(x, y) = 1

n

∑nj=1 K(

x−Fn1(Xj )

h)K(

y−Fn2(Yj )

h),

Cn,i(x, y) = 1n−1

∑j �=i K(

x−Fn1,i (Xj )

h)K(

y−Fn2,i (Yj )

h),

(2.1)

where h = h(n) > 0 is a bandwidth, and define the smoothed jackknife pseudo sam-ple as

Vi(x, y) = nCn(x, y) − (n − 1)Cn,i(x, y).

Therefore, the jackknife variance estimate of nVar(Cn(x, y)), based on the pseudosample of Vi (x, y)’s is

νn(x, y) = 1

n

n∑

i=1

{Vi (x, y) − 1

n

n∑

j=1

Vj (x, y)

}2

.

Note that the asymptotic variance of√

n{Cn(x, y) − C(x, y)} is

σ 2(x, y) = C(x, y){1 − C(x, y)

} + x(1 − x)

{∂

∂xC(x, y)

}2

+ y(1 − y)

{∂

∂yC(x, y)

}2

− 2C(x, y)(1 − x)∂

∂xC(x, y)

− 2C(x, y)(1 − y)∂

∂yC(x, y)

+ 2

{C(x, y) − xy

}{∂

∂xC(x, y)

}{∂

∂yC(x, y)

}, (2.2)

which is the same as that of√

n{Cn(x, y) − C(x, y)}, see Fermanian et al. (2004).Our first result shows that νn(x, y) is consistent.

78 L. Peng et al.

Theorem 1 Assume that k is a symmetric density with support [−1,1] and that h =h(n) → 0, nh2 → ∞ and nh4 → 0 as n → ∞. Further assume that C(x, y) hascontinuous first derivatives. Then,

νn(x, y)/σ 2(x, y)p→ 1.

Based on Theorem 1, one can expect that an application of the empirical likelihoodmethod to the above smoothed jackknife pseudo sample of Vi(x, y)’s works. Now wedefine the empirical likelihood function as

Ln(x, y; θ) = sup

{n∏

i=1

pi : p1 > 0, . . . , pn > 0,

n∑

i=1

pi = 1,

n∑

i=1

piVi(x, y) = θ

}.

An application of the Lagrange multiplier method gives

pi = 1

n

1

1 + λ(Vi(x, y) − θ),

where λ = λ(x, y; θ) satisfies

1

n

n∑

i=1

Vi (x, y) − θ

1 + λ(Vi(x, y) − θ)= 0. (2.3)

Hence the log empirical likelihood ratio becomes

ln(x, y; θ) = −2 log{n−nLn(x, y; θ)

} = 2n∑

i=1

log{1 + λ

(Vi (x, y) − θ

)}.

Theorem 2 Under the conditions of Theorem 1, we have

ln(x, y;C(x, y)

) d→ χ2(1)

as n → ∞.

Based on Theorem 2, a confidence interval of level γ for C(x, y) is given by

Iγ (x, y;h) = {θ : ln(x, y; θ) ≤ χ2

1,γ

},

where χ21,γ is the γ quantile of χ2(1).

3 Simulation study

In this section we investigate the finite sample behavior of the proposed jackknifeempirical likelihood method and compare it with the bootstrap confidence intervalbased on the empirical copula.


Using the ‘copula’ package in R, we draw 1,000 random samples of size n = 200and 1000 from the following mixture copula:

C(x, y; θ1, θ2, λ) = λ{x−θ1 + y−θ1 − 1

}−1/θ1

+ (1 − λ) exp{−(

(− logx)θ2 + (− logy)θ2)1/θ2

}

with marginals being a standard normal, where θ1 > 0, θ2 > 1 and λ ∈ [0,1]. Notethat the above mixture copula becomes the Clayton copula and the Gumbel copulawhen λ = 1 and 0, respectively. In particular, we consider

(θ1, θ2, λ) = (2,3,0.0), (2,3,0.5), (2,3,1.0),

(x, y) = (0.25,0.25), (0.5,0.5), (0.75,0.75), (0.25,0.5), (0.75,0.5)

and confidence levels γ = 0.9,0.95.The bootstrap confidence intervals are obtained by using 1,000 bootstrap sam-

ples. More specifically, we draw 1,000 bootstrap samples of size n from the orig-inal sample {(Xi, Yi)}ni=1 with replacement. Based on each bootstrap sample, wecalculate the empirical copula estimator. Hence we have 1,000 bootstrapped em-pirical copula estimators, say C∗1

n (x, y), . . . ,C∗1000n (x, y). Let a and b denote the

[1000(1 − γ )/2]th and [1000 − 1000(1 − γ )/2]th largest values of C∗1n (x, y) −

Cn(x, y), . . . ,C∗1000n (x, y)−Cn(x, y). Hence the bootstrap confidence interval is de-

fined as

I ∗γ (x, y) = [

Cn(x, y) − b, Cn(x, y) − a].

For computing the jackknife empirical likelihood-based confidence interval, weemploy the kernel k(x) = 3

4 (1 − x2)I (|x| ≤ 1) and the bandwidth

h = 0.2n−1/3, 0.5n−1/3, 0.8n−1/3,

since the optimal rate of the bandwidth for smoothing distributions is of the ordern−1/3. Also, the ‘emplik’ package in R is employed to calculate the jackknife empir-ical likelihood ratio.

Further, we consider the following bootstrap calibration for the proposed jack-knife empirical likelihood method. Put Cn(x, y) = 1

n

∑ni=1 Vi (x, y). As above, we

draw 1,000 bootstrap samples of size n from the original sample. For each bootstrapsample, we calculate the jackknife empirical likelihood ratio ln(x, y; Cn(x, y)). Thus,we obtain 1,000 bootstrapped jackknife empirical likelihood ratios and let d denotethe 1000γ th largest value. Therefore, the calibration interval is defined as

I cγ (x, y;h) = {

θ : ln(x, y; θ) ≤ d}.

In Tables 1 and 2 we report the empirical coverage probabilities for the jackknifeempirical likelihood confidence intervals Iγ (x, y;0.2n−1/3), Iγ (x, y;0.5n−1/3),Iγ (x, y;0.8n−1/3), the corresponding calibration intervals I c

γ (x, y;0.2n−1/3),

I cγ (x, y;0.5n−1/3), I c

γ (x, y;0.8n−1/3), and the bootstrap confidence interval I ∗γ (x, y)

80 L. Peng et al.

Table 1 Empirical coverage probabilities for the jackknife empirical likelihood-based confidence inter-val Iγ (x, y;h), the corresponding calibration interval I c

γ (x, y;h) and the bootstrap confidence interval

I∗γ (x, y) with sample size n = 200 and levels 0.90 and 0.95. Bandwidths are chosen as h1 = 0.2n−1/3,

h2 = 0.5n−1/3 and h3 = 0.8n−1/3

(λ, x, y) I∗0.90 I0.90(;h1) I0.90(;h2) I0.90(;h3) I c

0.90(;h1) I c0.90(;h2) I c

0.90(;h3)

I∗0.95 I0.95(;h1) I0.95(;h2) I0.95(;h3) I c

0.95(;h1) I c0.95(;h2) I c

0.95(;h3)

(0,0.25,0.25) 0.858 0.857 0.884 0.806 0.934 0.895 0.818

0.909 0.924 0.935 0.885 0.974 0.943 0.887

(0,0.25,0.5) 0.732 0.847 0.870 0.857 0.958 0.902 0.871

0.848 0.895 0.917 0.917 0.977 0.950 0.931

(0,0.5,0.5) 0.848 0.878 0.881 0.841 0.959 0.898 0.848

0.913 0.931 0.935 0.914 0.989 0.947 0.918

(0,0.75,0.5) 0.775 0.770 0.826 0.837 0.923 0.896 0.869

0.790 0.807 0.881 0.903 0.941 0.950 0.926

(0,0.75,0.75) 0.863 0.868 0.871 0.590 0.957 0.883 0.598

0.922 0.925 0.931 0.719 0.982 0.939 0.739

(0.5,0.25,0.25) 0.863 0.860 0.886 0.805 0.939 0.889 0.813

0.916 0.917 0.931 0.888 0.980 0.937 0.891

(0.5,0.25,0.5) 0.840 0.873 0.890 0.871 0.976 0.919 0.879

0.863 0.928 0.936 0.935 0.994 0.969 0.942

(0.5,0.5,0.5) 0.852 0.862 0.878 0.869 0.937 0.889 0.875

0.913 0.911 0.937 0.932 0.973 0.943 0.936

(0.5,0.75,0.5) 0.807 0.886 0.910 0.888 0.970 0.922 0.896

0.878 0.940 0.950 0.948 0.989 0.959 0.949

(0.5,0.75,0.75) 0.868 0.865 0.888 0.840 0.941 0.894 0.847

0.928 0.935 0.939 0.908 0.974 0.943 0.911

(1,0.25,0.25) 0.831 0.877 0.892 0.811 0.938 0.906 0.827

0.901 0.932 0.945 0.898 0.971 0.945 0.905

(1,0.25,0.5) 0.770 0.869 0.903 0.886 0.976 0.932 0.898

0.795 0.915 0.944 0.948 0.986 0.975 0.957

(1,0.5,0.5) 0.865 0.868 0.872 0.880 0.944 0.887 0.881

0.927 0.930 0.943 0.933 0.981 0.951 0.935

(1,0.75,0.5) 0.865 0.884 0.900 0.901 0.951 0.917 0.904

0.913 0.942 0.948 0.948 0.987 0.959 0.952

(1,0.75,0.75) 0.865 0.884 0.892 0.895 0.935 0.894 0.894

0.928 0.941 0.945 0.946 0.972 0.947 0.944


Table 2 Empirical coverage probabilities for the jackknife empirical likelihood-based confidence inter-val Iγ (x, y;h), the corresponding calibration interval I c



h2 = 0.5n−1/3 and h3 = 0.8n−1/3

(λ, x, y) I∗0.90 I0.90(;h1) I0.90(;h2) I0.90(;h3) I c

0.90(;h1) I c0.90(;h2) I c

0.90(;h3)

I∗0.95 I0.95(;h1) I0.95(;h2) I0.95(;h3) I c

0.95(;h1) I c0.95(;h2) I c

0.95(;h3)

(0,0.25,0.25) 0.875 0.889 0.895 0.831 0.929 0.899 0.831

0.936 0.938 0.952 0.911 0.976 0.952 0.918

(0,0.25,0.5) 0.833 0.880 0.887 0.876 0.959 0.905 0.878

0.901 0.935 0.937 0.931 0.979 0.952 0.935

(0,0.5,0.5) 0.879 0.879 0.896 0.851 0.938 0.904 0.860

0.929 0.937 0.944 0.911 0.974 0.946 0.917

(0,0.75,0.5) 0.812 0.857 0.878 0.868 0.958 0.908 0.876

0.827 0.907 0.927 0.929 0.983 0.952 0.936

(0,0.75,0.75) 0.870 0.879 0.873 0.697 0.933 0.883 0.701

0.929 0.933 0.941 0.807 0.971 0.946 0.813

(0.5,0.25,0.25) 0.878 0.890 0.887 0.821 0.923 0.892 0.823

0.930 0.941 0.945 0.906 0.970 0.950 0.906

(0.5,0.25,0.5) 0.849 0.873 0.886 0.863 0.940 0.897 0.873

0.903 0.932 0.933 0.927 0.979 0.951 0.936

(0.5,0.5,0.5) 0.886 0.887 0.882 0.868 0.932 0.888 0.871

0.943 0.944 0.947 0.932 0.974 0.954 0.929

(0.5,0.75,0.5) 0.833 0.869 0.900 0.892 0.934 0.903 0.887

0.886 0.940 0.946 0.947 0.975 0.948 0.945

(0.5,0.75,0.75) 0.877 0.888 0.902 0.851 0.923 0.900 0.853

0.941 0.936 0.950 0.918 0.963 0.949 0.915

(1,0.25,0.25) 0.867 0.886 0.883 0.816 0.926 0.887 0.818

0.925 0.937 0.940 0.889 0.970 0.943 0.893

(1,0.25,0.5) 0.845 0.883 0.887 0.879 0.938 0.895 0.882

0.893 0.945 0.947 0.937 0.980 0.957 0.940

(1,0.5,0.5) 0.873 0.871 0.894 0.902 0.921 0.902 0.901

0.934 0.940 0.950 0.948 0.973 0.956 0.948

(1,0.75,0.5) 0.846 0.886 0.902 0.904 0.930 0.905 0.903

0.917 0.944 0.951 0.954 0.973 0.953 0.954

(1,0.75,0.75) 0.873 0.876 0.890 0.887 0.908 0.891 0.889

0.923 0.939 0.941 0.938 0.958 0.945 0.941

82 L. Peng et al.

with levels γ = 0.9 and 0.95. The corresponding average lengths of those intervalsare reported in Tables 3 and 4. Some QQ-plots are given in Figs. 1 and 2.

Tables 1 and 2 show that the jackknife empirical likelihood method has bettercoverage accuracy than the bootstrap method based on the empirical copula estima-tor. Also, the choice of h = 0.5n−1/3 gives good results in general and the proposedcalibration method based on h = 0.5n−1/3 works well. Coverage probabilities withn = 400 in Table 2 are more accurate than those with n = 200 in Table 1 for almostall cases; see Figs. 1 and 2 as well. Tables 3 and 4 show that the intervals basedon the proposed jackknife empirical likelihood method are shorter than those basedon the bootstrap method. Comparing with Tables 1 and 2 in Chen et al. (2009) andTable 1 in Molanes Lopez et al. (2009), we find that the jackknife empirical likeli-hood method is comparable to the profile empirical likelihood methods in the abovetwo papers. However, as explained in the introduction, the computation of the pro-posed jackknife empirical likelihood method is much less intensive than the profileempirical likelihood method.

4 Proofs

Lemma 1 Under the conditions of Theorem 1, we have

√n{Cn(x, y) − C(x, y)

}

D→ W(x,y) := B(x, y) − ∂

∂xC(x, y)B(x,1) − ∂

∂yC(x, y)B(1, y),

where B(x, y) is a Gaussian process with zero mean and covariance

E{B(x1, y1)B(x2, y2)

} = C(x1 ∧ x2, y1 ∧ y2) − C(x1, y1)C(x2, y2).

Further, for any fixed x, y ∈ (0,1),

√n{Cn(x, y) − C(x, y)

} d→ N(0, σ 2(x, y)

),

where σ 2(x, y) is defined in (2.2), and

∂Cn(x, y)

∂x

p→ ∂C(x, y)

∂x,

∂Cn(x, y)

∂y

p→ ∂C(x, y)

∂y. (4.1)

Proof See Fermanian et al. (2004). �


√n

{1

n

n∑

i=1

Vi(x, y) − C(x, y)

}d→ N

(0, σ 2(x, y)

)

for any fixed x, y ∈ (0,1).


Table 3 Average interval lengths for the jackknife empirical likelihood-based confidence intervalIγ (x, y;h), the corresponding calibration interval I c



h2 = 0.5n−1/3 and h3 = 0.8n−1/3

(λ, x, y) I∗0.90 I0.90(;h1) I0.90(;h2) I0.90(;h3) I c

0.90(;h1) I c0.90(;h2) I c

0.90(;h3)

I∗0.95 I0.95(;h1) I0.95(;h2) I0.95(;h3) I c

0.95(;h1) I c0.95(;h2) I c

0.95(;h3)

(0,0.25,0.25) 0.046 0.041 0.033 0.028 0.050 0.035 0.028

0.054 0.049 0.040 0.034 0.061 0.042 0.034

(0,0.25,0.5) 0.029 0.026 0.022 0.019 0.057 0.027 0.021

0.035 0.032 0.027 0.024 0.089 0.038 0.027

(0,0.5,0.5) 0.050 0.045 0.037 0.032 0.060 0.039 0.033

0.059 0.054 0.045 0.039 0.072 0.047 0.039

(0,0.75,0.5) 0.024 0.021 0.017 0.016 0.076 0.026 0.018

0.029 0.025 0.021 0.019 0.116 0.043 0.024

(0,0.75,0.75) 0.043 0.037 0.029 0.023 0.048 0.030 0.023

0.051 0.044 0.034 0.027 0.059 0.036 0.028

(0.5,0.25,0.25) 0.045 0.040 0.033 0.028 0.050 0.034 0.028

0.054 0.048 0.040 0.034 0.061 0.042 0.034

(0.5,0.25,0.5) 0.032 0.029 0.025 0.022 0.048 0.027 0.023

0.038 0.035 0.030 0.027 0.073 0.036 0.029

(0.5,0.5,0.5) 0.053 0.048 0.042 0.037 0.062 0.043 0.037

0.063 0.058 0.050 0.044 0.074 0.052 0.045

(0.5,0.75,0.5) 0.038 0.034 0.030 0.027 0.045 0.031 0.027

0.044 0.042 0.036 0.032 0.058 0.038 0.034

(0.5,0.75,0.75) 0.048 0.043 0.036 0.032 0.052 0.037 0.032

0.057 0.052 0.044 0.038 0.062 0.045 0.038

(1,0.25,0.25) 0.046 0.041 0.034 0.029 0.051 0.036 0.029

0.055 0.049 0.041 0.034 0.062 0.043 0.036

(1,0.25,0.5) 0.035 0.032 0.028 0.025 0.049 0.030 0.026

0.042 0.039 0.034 0.030 0.068 0.039 0.032

(1,0.5,0.5) 0.057 0.053 0.047 0.042 0.065 0.048 0.043

0.067 0.063 0.056 0.050 0.079 0.058 0.051

(1,0.75,0.5) 0.045 0.042 0.037 0.033 0.051 0.038 0.033

0.054 0.050 0.044 0.040 0.062 0.046 0.040

(1,0.75,0.75) 0.052 0.047 0.041 0.036 0.055 0.042 0.037

0.062 0.056 0.049 0.044 0.066 0.050 0.044

84 L. Peng et al.

Table 4 Average interval lengths for the jackknife empirical likelihood-based confidence intervalIγ (x, y;h), the corresponding calibration interval I c



h2 = 0.5n−1/3 and h3 = 0.8n−1/3

(λ, x, y) I∗0.90 I0.90(;h1) I0.90(;h2) I0.90(;h3) I c

0.90(;h1) I c0.90(;h2) I c

0.90(;h3)

I∗0.95 I0.95(;h1) I0.95(;h2) I0.95(;h3) I c

0.95(;h1) I c0.95(;h2) I c

0.95(;h3)

(0,0.25,0.25) 0.031 0.028 0.024 0.021 0.032 0.024 0.021

0.037 0.033 0.029 0.025 0.038 0.029 0.025

(0,0.25,0.5) 0.020 0.018 0.016 0.014 0.022 0.016 0.014

0.024 0.022 0.019 0.017 0.030 0.020 0.018

(0,0.5,0.5) 0.034 0.031 0.027 0.024 0.037 0.028 0.024

0.040 0.037 0.032 0.029 0.044 0.033 0.029

(0,0.75,0.5) 0.016 0.014 0.012 0.011 0.029 0.014 0.012

0.019 0.017 0.015 0.014 0.051 0.020 0.015

(0,0.75,0.75) 0.029 0.025 0.021 0.017 0.030 0.021 0.018

0.034 0.030 0.025 0.021 0.036 0.026 0.021

(0.5,0.25,0.25) 0.031 0.028 0.024 0.021 0.032 0.024 0.021

0.037 0.034 0.029 0.025 0.038 0.029 0.025

(0.5,0.25,0.5) 0.022 0.020 0.018 0.016 0.024 0.018 0.016

0.026 0.024 0.021 0.019 0.030 0.022 0.020

(0.5,0.5,0.5) 0.036 0.033 0.030 0.027 0.038 0.030 0.027

0.043 0.040 0.035 0.032 0.046 0.036 0.032

(0.5,0.75,0.5) 0.025 0.023 0.021 0.019 0.027 0.021 0.019

0.030 0.028 0.025 0.023 0.032 0.025 0.023

(0.5,0.75,0.75) 0.033 0.029 0.026 0.023 0.033 0.026 0.023

0.039 0.035 0.030 0.028 0.039 0.031 0.028

(1,0.25,0.25) 0.032 0.028 0.025 0.022 0.032 0.025 0.022

0.038 0.034 0.030 0.026 0.039 0.030 0.026

(1,0.25,0.5) 0.024 0.022 0.019 0.018 0.025 0.020 0.018

0.029 0.026 0.023 0.022 0.031 0.024 0.022

(1,0.5,0.5) 0.039 0.037 0.033 0.031 0.042 0.034 0.031

0.047 0.044 0.040 0.037 0.050 0.041 0.037

(1,0.75,0.5) 0.031 0.029 0.026 0.024 0.032 0.026 0.024

0.037 0.034 0.031 0.029 0.039 0.032 0.029

(1,0.75,0.75) 0.035 0.032 0.029 0.027 0.035 0.029 0.027

0.042 0.039 0.035 0.032 0.043 0.035 0.032


Fig. 1 A QQ plot with the theoretical quantiles of the χ21 distribution plotted against the correspond-

ing sample quantiles obtained from 1000 values of ln(x, y;C(x, y)) for different settings of n = 200,λ = 0,0.5,1.0, (x, y) = (0.25,0.25), (0.25,0.5), and (0.5,0.5)

Proof Write Vi (x, y) = Vi,1(x, y) + Vi,2(x, y), where

{Vi,1(x, y) = ∑n

j=1{K(x−Fn1(Xj )

h)K(

y−Fn2(Yj )

h) − K(

x−Fn1,i (Xj )

h)K(

y−Fn2,i (Yj )

h)},

Vi,2(x, y) = K(x−Fn1,i (Xi)

h)K(

y−Fn2,i (Yi )

h).

(4.2)

86 L. Peng et al.

Fig. 2 A QQ plot with the theoretical quantiles of the χ21 distribution plotted against the correspond-

ing sample quantiles obtained from 1000 values of ln(x, y;C(x, y)) for different settings of n = 400,λ = 0,0.5,1.0, (x, y) = (0.25,0.25), (0.25,0.5), and (0.5,0.5)

Since

⎧⎪⎨

⎪⎩

sup1≤i≤n |Fn1,i (x) − Fn1(x)| = sup1≤i≤n | 1n−1Fn1(x) − 1

n−1I (Xi ≤ x)| ≤ n−1,

sup1≤i≤n |Fn2,i (y) − Fn2(y)| = sup1≤i≤n | 1n−1Fn2(y) − 1

n−1I (Yi ≤ y)| ≤ n−1,∑n

i=1{Fn1,i (x) − Fn1(x)} = ∑ni=1{Fn2,i (y) − Fn2(y)} = 0,

(4.3)


it follows from the mean-value theorem that for fixed x, y ∈ (0,1)

1

n

n∑

i=1

Vi,1(x, y)

= 1

n

n∑

i=1

n∑

j=1

Fn1,i (Xj ) − Fn1(Xj )

hk

(x − Fn1(Xj )

h

)K

(y − Fn2(Yj )

h

)

+ 1

n

n∑

i=1

n∑

j=1

Fn2,i (Yj ) − Fn2(Yj )

hK

(x − Fn1(Xj )

h

)k

(y − Fn2(Yj )

h

)

+ 1

n

n∑

i=1

n∑

j=1

{1

2

(Fn1,i (Xj ) − Fn1(Xj )

h

)2

k′(

x − ξn1,i,j

h

)K

(y − ξn2,i,j

h

)

+ 1

2

(Fn2,i (Yj ) − Fn2(Yj )

h

)2

K

(x − ξn1,i,j

h

)k′

(y − ξn2,i,j

h

)

+ Fn1,i (Xj ) − Fn1(Xj )

h


hk

(x − ξn1,i,j

h

)k

(y − ξn2,i,j

h

)}

= 1

n

n∑

i=1

n∑

j=1

{1

2

(Fn1,i (Xj ) − Fn1(Xj )

h

)2

k′(

x − ξn1,i,j

h

)K

(y − ξn2,i,j

h

)

+ 1

2

(Fn2,i (Yj ) − Fn2(Yj )

h

)2

K

(x − ξn1,i,j

h

)k′

(y − ξn2,i,j

h

)

+ Fn1,i (Xj ) − Fn1(Xj )

h


hk

(x − ξn1,i,j

h

)k

(y − ξn2,i,j

h

)},

where ξn1,i,j is between Fn1,i (Xj ) and Fn1(Xj ) and ξn2,i,j is between Fn2,i (Yj ) andFn2(Yj ). By (4.3), we have

{sup1≤i≤n I (| x−ξn1,i,j

h| ≤ 1) ≤ I (x − h − n−1 ≤ Fn1(Xj ) ≤ x + h + n−1),

sup1≤i≤n I (| y−ξn2,i,j

h| ≤ 1) ≤ I (x − h − n−1 ≤ Fn2(Yj ) ≤ x + h + n−1).

(4.4)

Hence, (4.3) and (4.4) imply that for some M1 > 0

∣∣∣∣∣1

n

n∑

i=1

Vi,1(x, y)

∣∣∣∣∣ ≤ M1

n3h2

n∑

i=1

n∑

j=1

{I

(∣∣∣∣x − ξn1,i,j

h

∣∣∣∣ ≤ 1

)+ I

(∣∣∣∣y − ξn2,i,j

h

∣∣∣∣ ≤ 1

)}

≤ 2M1

nh, (4.5)

88 L. Peng et al.

which is of order o(1/√

n). Since

K

(x − Fn1,i (Xi)

h

)K

(y − Fn2,i (Yi)

h

)

= K

( n−1n

x + 1n

− Fn1(Xi)

(n − 1)h/n

)K

( n−1n

y + 1n

− Fn2(Yi)

(n − 1)h/n

),

Lemma 1 implies that

√n

{1

n

n∑

i=1

Vi,2(x, y) − C(x, y)

}d→ N

(0, σ 2(x, y)

). (4.6)

Hence, the lemma follows from (4.5) and (4.6). �


1

n

n∑

i=1

{Vi (x, y) − C(x, y)

}2 p→ σ 2(x, y)

for fixed x, y ∈ (0,1).

Proof It is straightforward to check that

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

1n

∑ni=1{Fnj,i(x1) − Fnj (x1)}{Fnj,i(x2) − Fnj (x2)}

= 1(n−1)2 {Fnj (x1 ∧ x2) − Fnj (x1)Fnj (x2)},

1n

∑ni=1{Fn1,i (x1) − Fn1(x1)}{Fn2,i (x2) − Fn2(x2)}

= 1(n−1)2 {Fn(x1, x2) − Fn1(x1)Fn2(x2)}

(4.7)

for j = 1,2 and x1, x2 ∈ �, where Fn(x1, x2) = 1n

∑ni=1 I (Xi ≤ x1, Yi ≤ x2). Let

Vi,1(x, y) and Vi,2(x, y) be defined as in decomposition (4.2). Then, as in the proofof (4.5), we can show that

1

n

n∑

i=1

V 2i,1(x, y)

= 1

n

n∑

i=1

{n∑

j=1

Fn1,i (Xj ) − Fn1(Xj )

hk

(x − Fn1(Xj )

h

)K

(y − Fn2(Yj )

h

)

+n∑

j=1


hK

(x − Fn1(Xj )

h

)k

(y − Fn2(Yj )

h

)}2

+ op(1),


where by (4.7), we have

1

n

n∑

i=1

V 2i,1(x, y)

=n∑

j=1

n∑

k=1

1

(n − 1)2h2

{Fn1(Xj ∧ Xk) − Fn1(Xj )Fn1(Xk)

}k

(x − Fn1(Xj )

h

)

× K

(y − Fn2(Yj )

h

)k

(x − Fn1(Xk)

h

)K

(y − Fn2(Yk)

h

)

+n∑

j=1

n∑

k=1

1

(n − 1)2h2

{Fn2(Yj ∧ Yk) − Fn2(Yj )Fn2(Yk)

}K

(x − Fn1(Xj )

h

)

× k

(y − Fn2(Yj )

h

)K

(x − Fn1(Xk)

h

)k

(y − Fn2(Yk)

h

)

+ 2n∑

j=1

n∑

k=1

1

(n − 1)2h2

{Fn(Xj ,Yk) − Fn1(Xj )Fn2(Yk)

}k

(x − Fn1(Xj )

h

)

× K

(y − Fn2(Yj )

h

)K

(x − Fn1(Xk)

h

)k

(y − Fn2(Yk)

h

)+ op(1)

=∫ 1

0

∫ 1

0

∫ 1

0

∫ 1

0h−2{x1 ∧ x2 − x1x2}k

(x − x1

h

)K

(y − y1

h

)k

(x − x2

h

)

× K

(y − y2

h

)dC(x1, y1) dC(x2, y2)

+∫ 1

0

∫ 1

0

∫ 1

0

∫ 1

0h−2{y1 ∧ y2 − y1y2}

× K

(x − x1

h

)k

(y − y1

h

)K

(x − x2

h

)k

(y − y2

h

)dC(x1, y1) dC(x2, y2)

+ 2∫ 1

0

∫ 1

0

∫ 1

0

∫ 1

0h−2{C(x1, y2) − x1y2

}k

(x − x1

h

)K

(y − y1

h

)

× K

(x − x2

h

)k

(y − y2

h

)dC(x1, y1) dC(x2, y2) + op(1)

= {x − x2}

{∂

∂xC(x, y)

}2

+ {y − y2}

{∂

∂yC(x, y)

}2

+ 2{C(x, y) − xy

}{ ∂

∂xC(x, y)

}{∂

∂yC(x, y)

}+ op(1). (4.8)

Note that in the last step we have used (4.1). Similarly, we can show that

1

n

n∑

i=1

Vi,1(x, y)Vi,2(x, y)

= 1

n

n∑

i=1

n∑

j=1

K

(x − Fn1(Xi)

h

)K

(y − Fn2(Yi)

h

)Fn1,i (Xj ) − Fn1(Xj )

h

90 L. Peng et al.

× k

(x − Fn1(Xj )

h

)K

(y − Fn2(Yj )

h

)

+ 1

n

n∑

i=1

n∑

j=1

K

(x − Fn1(Xi)

h

)K

(y − Fn2(Yi)

h

)Fn2,i (Yj ) − Fn2(Yj )

h

× K

(x − Fn1(Xj )

h

)k

(y − Fn2(Yj )

h

)+ op(1)

=n∑

j=1

1

n

n∑

i=1

K

(x − Fn1(Xi)

h

)K

(y − Fn2(Yi)

h

)1

(n − 1)h

× {Fn1(Xj ) − I (Xi ≤ Xj)

}k

(x − Fn1(Xj )

h

)K

(y − Fn2(Yj )

h

)

+n∑

j=1

1

n

n∑

i=1

K

(x − Fn1(Xi)

h

)K

(y − Fn2(Yi)

h

)1

(n − 1)h

× {Fn2(Yj ) − I (Yi ≤ Yj )

}K

(x − Fn1(Xj )

h

)k

(y − Fn2(Yj )

h

)+ op(1)

= 1

(n − 1)h

n∑

j=1

{1

n

n∑

i=1

K

(x − Fn1(Xi)

h

)K

(y − Fn2(Yi)

h

)}

× Fn1(Xj )k

(x − Fn1(Xj )

h

)K

(y − Fn2(Yj )

h

)

− 1

(n − 1)h

n∑

j=1

{1

n

n∑

i=1

K

(x − Fn1(Xi)

h

)K

(y − Fn2(Yi)

h

)

× I(Fn1(Xi) ≤ Fn1(Xj )

)}

k

(x − Fn1(Xj )

h

)K

(y − Fn2(Yj )

h

)

+ 1

(n − 1)h

n∑

j=1

{1

n

n∑

i=1

K

(x − Fn1(Xi)

h

)K

(y − Fn2(Yi)

h

)}

× Fn2(Yj )K

(x − Fn1(Xj )

h

)k

(y − Fn2(Yj )

h

)

− 1

(n − 1)h

n∑

j=1

{1

n

n∑

i=1

K

(x − Fn1(Xi)

h

)K

(y − Fn2(Yi)

h

)

× I(Fn2(Yi) ≤ Fn2(Yj )

)}

K

(x − Fn1(Xj )

h

)k

(y − Fn2(Yj )

h

)+ op(1)


= C(x, y)x∂

∂xC(x, y) − C(x, y)

∂

∂xC(x, y)

+ C(x, y)y∂

∂yC(x, y) − C(x, y)

∂

∂yC(x, y) + op(1). (4.9)

It follows from (4.3) that

max1≤i≤n

supx

∣∣Fnj,i(x) − Fj (x)∣∣ p→ 0 (4.10)

for j = 1,2. By Lemma 1 and (4.10), we have

1

n

n∑

i=1

V 2i,2(x, y)

p→ C(x, y). (4.11)

Hence, the lemma follows from (4.8), (4.9), (4.11) and the fact that

1

n

n∑

i=1

Vi(x, y)p→ C(x, y),

which is implied by Lemma 2. �

Proof of Theorem 1 This follows from Lemmas 2 and 3. �

Proof of Theorem 2 It follows from the mean-value theorem that

Vi,1(x, y) =n∑

j=1

{Fn1,i (Xj ) − Fn1(Xj )

hk

(x − ξn1,i,j

h

)K

(y − ξn2,i,j

h

)

+ Fn2,i (Yj ) − Fn2(Yj )

hK

(x − ξn1,i,j

h

)k

(y − ξn2,i,j

h

)},

where ξn1,i,j lies between Fn1,i (Xj ) and Fn1(Xj ), ξn2,i,j lies between Fn2,i (Yj ) andFn2(Yj ), and Vi,1(x,Y ) is defined in the proof of Lemma 2. By (4.3) and (4.4), wecan show that max1≤i≤n |Vi,1(x, y)| is bounded, and so is max1≤i≤n |Vi (x, y)|. Bystandard arguments of the empirical likelihood method (see Owen 1988), we canshow that

ln(x, y;C(x, y)

) ={

n∑

i=1

Vi(x, y) − C(x, y)

}2/ n∑

i=1

{Vi(x, y) − C(x, y)

}2+ op(1)

d→ χ2(1)

as n → ∞. The details are omitted here. �

92 L. Peng et al.

Acknowledgements We thank two reviewers for their helpful comments. Peng’s research was sup-ported by NSA grant H98230-10-1-0170 and NSF grant DMS1005336. Qi’s research was supported byNSA grant H98230-10-1-0161 and NSF grant DMS1005345. Ingrid Van Keilegom acknowledges finan-cial support from IAP research network P6/03 of the Belgian Government (Belgian Science Policy), andfrom the European Research Council under the European Community’s Seventh Framework Programme(FP7/2007-2013)/ERC Grant agreement No. 203650.

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