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Nonparametric estimation of multivariate multiparameter conditional copulas
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Nonparametric estimation of multivariate multiparameter conditional copulas✩ Jin-Guan Lin a,∗ , Kong-Sheng Zhang b , Yan-Yong Zhao a a
Department of Statistics, Nanjing Audit University, Nanjing 211815, China
b
Department of Mathematics, Southeast University, Nanjing 211189, China
article
info
Article history: Received 1 December 2015 Accepted 31 August 2016 Available online xxxx AMS 2000 subject classifications: primary 62G08 secondary 62H12 Keywords: Conditional copula Calibration function Local linear smoothing Newton–Raphson method
abstract Nonparametric estimation of conditional copulas with one parameter has been investigated in Acar et al. (2011). The estimation for multivariate multiparameter conditional copulas, however, has not been considered so far. This paper adopts the local linear smoothing technique and Newton–Raphson method to estimate those copulas. Under some regularity conditions, the asymptotic normality of the estimators is obtained. Simulation work shows the efficiency of the proposed method. As an application, we analyze a life expectancies data set and show that the conditional t copula outperforms the conditional Clayton, Frank and Gumbel copulas. © 2016 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
1. Introduction Copulas have become an efficient tool in modeling the dependence structure between random variables. They have been widely used in many fields such as risk management (Embrechts, Lindskog, & McNeil, 2003), econometrics (Patton, 2012) and hydrology (Genest & Favre, 2007). Those broad applications mainly rely on the well known Sklar’s theorem (Nelsen, 2006) which enables us to model the marginal distributions and the copula C separately. Of importance is estimation of parameters in copulas. Two commonly used methods are full maximum likelihood and two-stage estimation approaches (Genest, Ghoudi, & Rivest, 1995; Nelsen, 2006). Also, Bayesian method is employed to tackle this issue (Min & Czado, 2010; Smith & Khaled, 2012; Smith, Min, Almeida, & Czado, 2010). Recently, Patton (2006) proposed the concept of conditional copula in which the correlation is affected by covariate(s). Based on Patton’s seminal work, many authors have payed considerable attention on the estimation and testing of conditional copulas. Gijbels, Veraverbeke, and Omelka (2011) presented two nonparametric estimators for conditional copulas. Abegaz, Gijbels, and Veraverbeke (2012) suggested estimating the conditional copula by using semiparametric method in which the marginal distributions are assumed to be unknown. Gijbels, Omelka, and Veraverbeke (2015) investigated the case where the covariate only affected the marginal distributions, but not the dependence structure. Acar, Craiu, and Yao (2011) proposed a nonparametric framework to estimate the calibration function. All the conditional copulas discussed in Acar et al. (2011) have two notable features: each copula contains only one parameter, and both the score
✩ The work is supported by National Natural Science Foundation of China (11571073, 11401094), Natural Science Foundation of Jiangsu Province of China (BK20141326, BK20140617), the Research Fund for the Doctoral Program of Higher Education of China under Grant (20120092110021). ∗ Corresponding author. E-mail address: [email protected] (J.-G. Lin).
http://dx.doi.org/10.1016/j.jkss.2016.08.003 1226-3192/© 2016 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
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function and Hessian matrix have closed forms which facilitate the calculation procedure for parameter estimation. In practice, however, we may encounter those copulas with the number of parameters being more than one, and the score function and Hessian matrix may have no simplified forms, for instance, Student t copula and mixture copula. These considerations motivate our present work. The remainder of the paper is organized as follows. In Section 2, we present nonparametric estimation methodology for conditional copula parameters and discuss the choice of optimal bandwidth. Section 3 is dedicated to investigate the asymptotic normality of the estimators. Section 4 carries out some simulation studies for illustrating the efficiency of the proposed method. A life expectancies data analysis is provided in Section 5. All proofs are deferred to Appendix. 2. Methodology Suppose that Y1 and Y2 are continuous variables, and X is a continuous covariate which may produce an effect on Y1 and Y2 . According to Sklar’s theorem (see, e.g., Nelsen, 2006), one has F (y1 , y2 |X = x) = C F1|X (y1 |x), F2|X (y2 |x); θ , α1 , α2 ,
where F (·, ·|X = x) denotes the conditional cumulative distribution of (Y1 , Y2 ) under X = x, and F1|X and F2|X are the marginal distributions of Y1 and Y2 with parameters α1 and α2 , respectively. C (·, ·) is a copula function with parameter θ , which describes the correlation dependence structure of (Y1 , Y2 ) under covariate X . Furthermore, we assume that the conditional marginal distributions are known since we are only interested in the dependence structure. Without loss of generality, we assume that the copula associated with (Y1 , Y2 ) under X has two parameters, however, the case where the number of parameters is more than two can be analyzed in a similar manner. Thus we only need to consider the model
(U , V |X = x) ∼ C u, v; θ = (θ1 (x), θ2 (x)) , where θ1 (x) = g1 (η1 (x)), θ2 (x) = g2 (η2 (x)), U = F1|X (Y1 ), and V = F2|X (Y2 ), the unknown functions η1 (x) and η2 (x) are called calibration functions (Acar et al., 2011), whereas the link functions g1 (x) and g2 (x) are known which guarantee that the copula parameters lie in the correct range. The main purpose of this paper is to estimate the calibration functions and then to estimate the conditional copula parameters. Assume that we have i.i.d. samples (Y1i , Y2i , Xi ), i = 1, 2, . . . , n, which are independent copies of (Y1 , Y2 , X ). By Taylor expansion with one order for η1 (Xi ) and η2 (Xi ), respectively, we have
η1 (Xi ) ≈ η1 (x) + η1′ (x)(Xi − x) ≡ xTix α and
η2 (Xi ) ≈ η2 (x) + η2′ (x)(Xi − x) ≡ xTix γ , (j)
η (x)
(j)
η (x)
dj η (x)
dj η (x)
where α = (α0 , α1 )T , γ = (γ0 , γ1 ), αj = 1 j! = dx1 j /j!, γj = 2 j! = dx2 j /j!, and xix = (1, Xi − x)T for j = 0, 1. The influence of the covariate Xi (i = 1, 2, . . . , n) on the relationship between Y1 and Y2 is incorporated into the conditional local log-likelihood
L(α, γ ; x, h) =
n
log c Ui , Vi ; g1 (xTix α), g2 (xTix γ ) Kh (Xi − x),
(1)
i =1
where Ui = F1|X (Y1i ), and Vi = F2|X (Y2i ), Kh (·) = K (·/h)/h with h denoting bandwidth, and K (x) is a kernel function and ∂ 2 C (u,v)
c (u, v) = ∂ u∂v is the density of the copula C . The Epanechnikov kernel K (t ) = 3/4(1 − t )2 I(|t | ≤ 1) is adopted in this paper where I indicates the indicator function. When the information on the conditional marginal distributions for Y1 and Y2 is not available, as in Abegaz et al. (2012), we can estimate them by n
Fˆj|X (y) =
wi I(Yji ≤ y)
i=1 Kh (Xi −x) j
where wi = n
k=1 Khj (Xk −x)
and hj denotes the bandwidth for j = 1, 2. For the sake of simplicity we set h1 = h2 and they can
be obtained by the method proposed by Hansen (2004) with Matlab software. A natural way to obtain the local maximum likelihood estimator βˆ = (α, ˆ γˆ ) = (αˆ 0 , αˆ 1 , γˆ0 , γˆ1 ) is to solve the estimating equation
∂ L(β; x, h) = 0, ∂β where the notation ∇ denotes the partial derivative of L with respect to β . ∇ L(β, x) =
We adopt the commonly used Newton–Raphson method to get the estimator βˆ with the recursive form
β t +1 = β t − {∇ 2 L(β t , x)}−1 ∇ L(β t , x),
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where ∇ L and ∇ 2 L stands for the score function and Hessian matrix. We are mainly interested in the estimates ηˆ 1 (x) = αˆ 0 and ηˆ 2 (x) = γˆ0 since the estimators of θ1 (x) and θ2 (x) can be obtained by θˆ1 (x) = g1 (ηˆ 1 (x)) = g1 (αˆ 0 ) and θˆ2 (x) = g2 (ηˆ 2 (x)) = g2 (γˆ0 ), respectively. The initial value of the parameter β in the case of conditional t copula can be set as βinitial = (g1−1 (ρ), ˆ 0, g2−1 (ˆν ), 0), where ρˆ and νˆ are the estimates of the t copula. Note that the copulas considered in Acar et al. (2011) are Clayton, Frank and Gumbel copulas whose first and second derivatives with respect to the copula parameter are available, however, for instance, the Student t copula does not have the property, therefore, we need to calculate the score function and Hessian matrix numerically. As we know, the bandwidth parameter h plays a crucial role in the estimation procedure for copula parameters. As in Acar et al. (2011), we also apply the leave-one-out cross-validation method to find the optimal bandwidth h∗ . The details on how to obtain h∗ are described as follows when we have the i.i.d. random sample (Ui , Vi , Xi )(i = 1, 2, . . . , n) with n being sample size.
• Step 1: Choose a suitable initial value βinitial for β . • Step 2: For i = 1, 2, . . . , n, delete ith individual to obtain a new sample (Uj , Vj , Xj ), j = 1, 2, . . . , n, j ̸= i. Based on the new sample and Newton–Raphson iteration approach, we can obtain the maximum likelihood estimate of the parameter θ , denote it by θˆh(−i) , then calculate log c (Ui , Vi ; θˆh(−i) ).
• Step 3: The optimal bandwidth h∗ can be achieved via maximizing the log-likelihood
n
i=1
(−i)
log c (Ui , Vi ; θˆh
).
3. Asymptotic theory In this section, we investigate the theoretical results of the nonparametric estimate of copula θ. conditional j parameter For ease of presentation, let us first start by introducing some notation. Define µj = t j K (t )dt , vj = t K 2 (t )dt with K (t ) being a kernel function and zix = (1, c (u, v; θ ) = ∂ 2 log c (U ,V ; θ)
∂ C 2 (u,v; θ) ∂ u∂v
) for i = 1, . . . , n. Furthermore, let ℓ(θ; U , V ) = log c (U , V ; θ ) where
Xi −x T h
∂ log c (U ,V ; θ ) , ℓ′2 ∂θ1 ′′ ′′
, and ℓ1 ≡ ℓ1 (θ; U , V ) = ′
∂ 2 log c (U ,V ; θ)
≡ ℓ′2 (θ; U , V ) = ∂ 2 log c (U ,V ; θ )
∂ log c (U ,V ; θ ) , ℓ′′11 ∂θ2
≡ ℓ′′11 (θ; U , V ) =
, ℓ22 ≡ ℓ22 (θ; U , V ) = and X = {X1 , X2 , . . . , Xn }. ∂θ1 ∂θ2 ∂θ2 ∂θ2 Before stating the proofs, we list some assumptions which are necessary for Theorem 1. ∂θ1 ∂θ1
, ℓ′′12 ≡ ℓ′′12 (θ; U , V ) =
′
A1: The density function f (x) of covariate X is continuous and satisfies the Lipschitz condition, and X has a bounded support Ω . A2: The kernel function K (t ) is continuous, bounded and is symmetric on [−a, a] with a > 0. A3: gi (ηi (x)) and ηi (x) have continuous derivatives up to order two on a neighborhood of x N (x) ⊂ Ω , in addition, ℓ(θ (x), u, v) has three on N (x). continuous derivatives up to order A4: supx∈N (x) max |gi′ (ηi (x))|, |gi′′ (ηi (x))|, |ηi′ (x)|, |ηi′′ (x)| < ∞ for i = 1, 2.
A5: supx∈N (x)×(0,1)2 max |ℓ′i θ (x); u, v |, |ℓ′′ij θ (x); u, v |, |ℓ′′′ ijk (θ (x), u, v)| < ∞ for 1 ≤ i, j, k ≤ 2.
A6: supx∈N (x)×(0,1)2 Var{ℓ′′ij θ (x); u, v } < ∞ for 1 ≤ i, j ≤ 2.
A7: ∥∆∥ =
2 i =1
2
j =1
∆2ij < ∞ where ∆ is defined in (4) in the Appendix.
A8: E |X 3 | < ∞. Assumptions A1–A2 are commonly required in kernel estimation (Fan & Gijbels, 1996). Assumptions A3–A5 are used to obtain the Taylor expansion. Assumption A6 enables us to establish the relationship between the expectation of the variable and the variable itself. Assumption A7 guarantees the condition of Lindeberg–Feller central limit theorem in the proof of asymptotic normality. Assumption A8 is used to establish the order of the remainder in the second-order Taylor expansion. Theorem 1. Under the Assumptions A1–A8, h → 0 and nh → ∞ as n → ∞, we have
ηˆ 1 (x) − η1 (x) ′ ′ d h(ηˆ 1 (x) − η1 (x)) nh − Ψ 1 + op (1) − → N 0, Ξ , ηˆ 2 (x) − η2 (x) h(ηˆ 2′ (x) − η2′ (x)) α µ2 where Ψ = h2 f (x)Λ−1 D γ2 ⊗ µ3 , Ξ = Λ−1 ∆Λ−1 , 0 = (0, 0, 0, 0)T and the matrices Λ, D and ∆ are defined as 2
√
in (2), (3) and (4) in the Appendix, respectively. Remark 1. Theorem 1 establishes the asymptotically normal distribution theory for the calibration functions ηj (x) with convergence rate (nh)−1/2 as well as its corresponding derivative ηj′ (x) with convergence rate (nh3 )−1/2 , respectively, for j = 1, 2. The following corollary follows immediately from Theorem 1 by extracting the corresponding elements and the Delta method.
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Corollary 1. Suppose that all the conditions in Theorem 1 hold, then one has
√
′ d θˆ1 (x) − θ1 (x) Ψ (1) g1 (η1 (x)) Ξ (1, 1) nh − ⊙ ′ 1 + op (1) − → N 0, Ψ (3) Ξ (3, 1) g2 (η2 (x)) θˆ2 (x) − θ2 (x) g ′ (η (x))2 g1′ (η1 (x))g2′ (η2 (x)) where Υ = ′ 1 1 ′ . g (η (x))g (η (x)) g ′ (η (x))2 1
1
2
2
2
Ξ (1, 3) ⊙Υ Ξ (3, 3)
2
Remark 2. Corollary 1 only provides the asymptotically normal distribution theory for the calibration functions ηj (x) for j = 1, 2. 4. Monte Carlo experiments In this section, some simulations are carried out to indicate the finite sample performance of the proposed method. Specifically, we only investigate parameter estimation for the conditional Student t copula which admits the formula C (u, v; ρ, ν) = Tν (tν−1 (u), tν−1 (v); ρ, ν)
−1 tν (u)
−1 tν (v)
= −∞
−∞
ν+2 Γ ν+2 2 x2 + y2 − 2ρ xy − 2 dydx, 1+ ν ν(1 − ρ 2 ) Γ 2 π ν 1 − ρ2
where ρ is the correlation coefficient between U and V , and ν denotes the degrees of freedom (df for short). tv (·) (resp. tv−1 (·)) is the cumulative distribution (resp. quantile) of a univariate t distribution with mean zero and df = ν . Correspondingly, the density of t copula can be expressed as
Γ ν+2 2 Γ ν2 c (u, v; ρ, ν) = 2 Γ ν+2 1 1 − ρ2
1+
1+
2 −1 tν (u)
ν
1+
2 −1 tν (v)
ν+2 1
ν
2 2 −1 −1 −1 −1 tν (u) +tν (v) −2ρ tν (u)tν (v) ν(1−ρ 2 )
ν+2 2 .
We study the two different types of calibration functions. Model 1. η1 (X ) = 3 − X and η2 (X ) = 0.2X + 1, and X ∼ Uniform(1, 5). Model 2. η1 (X ) = 1 − 0.5X + 0.2X 2 and η2 (X ) = log(X ) + 0.1X , and X ∼ Uniform(1, 5). For each model, to ensure the two 1−exp(−x) parameters ρ and ν in their correct ranges, we use the link functions g1 (x) = 1+exp(−x) and g2 (x) = 1 + exp(x), respectively. In Model 1, it can be seen that the parameter ρ varies from −0.76 to 0.76 and ν covers the range (4.32, 8.39). Similarly, ρ ∈ (0.33, 0.94) and ν ∈ (2.10, 9.25) for Model 2. The random sample of the conditional t copula in Model 1 is distributed as
1 − exp(X − 3) , θ2 = exp(0.2X + 1) + 1 . (U , V )|X ∼ C u, v; θ1 = 1 + exp(X − 3) We first generate the sample Xi ∼ Uniform(1, 5)(i = 1, . . . , n) with sample size n, then for each Xi , we generate (Ui , Vi )(i = 1, . . . , n) from the corresponding conditional t copula, the tri-tuple sample (Ui , Vi , Xi )(i = 1, . . . , n) for (U , V , X ) is then obtained. We replicate this procedure 100 times, the precision of the proposed estimation method can be evaluated by using commonly used indexes such as integrated squared bias (IBIAS2 ) and integrated mean square error 11 (IMSE). The IBIAS2 and IMSE of the estimate of Kendall τ (x) = 4 0 0 C (u, v; θ (x))dC (u, v; θ (x)) − 1 are defined as 2 IBIAS2 (τˆ ) = E{τˆ (x)} − τ (x) dx X
and IMSE(τˆ ) =
2
E τˆ (x) − τ (x) dx,
X
respectively, where X is the possible set of x. We divide the interval [1.1, 4.9] into 50 equally space intervals and then obtain approximate IMSE and IBIAS2 for the estimates of ρ(x), ν(x) and τ (x). Note that Kendall τ for t copula has a simple form τ = 2/π · arcsin(ρ) (Embrechts et al., 2003). Table 1 reports the approximate integrated squared bias (IBIAS2 ) and integrated mean square error (IMSE) of the estimators of ρ, ν and τ . As expected, the IMSE of the estimators of ρ, ν and τ decreases when sample size n increases for the two models. It should be noticed that the range of the design point X is 4.9 − 1.1 = 3.8, thus, the average MSE (BIAS2 ) can be understood in the sense that the corresponding value in Table 1 divided by 3.8. Fig. 1 depicts the simulation results for conditional t copula as described in Models 1 and 2 with sample size 200. The solid line denotes the true value of Kendall τ , whereas the dashed line is the averaged local linear estimates and the dotted line represents the approximate 95% confidence intervals.
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Table 1 The integrated squared bias (IBIAS2 ) and integrated mean square error (IMSE) of parameter estimates of ρ, ν and τ for conditional t copula.
ρˆ
νˆ
τˆ
IBIAS2
IMSE
IBIAS2
IMSE
IBIAS2
IMSE
Model 1
n = 100 n = 200 n = 1000
0.0036 0.0022 0.0029
0.0691 0.0476 0.0235
3.1352 2.6483 1.8532
11.7050 9.1217 6.1046
0.0032 0.0011 0.0013
0.0359 0.0230 0.0176
Model 2
n = 100 n = 200 n = 1000
0.0012 0.0008 0.0006
0.0094 0.0075 0.0053
4.8365 3.0479 2.3941
16.3326 12.2282 10.3020
0.0012 0.0008 0.0007
0.0297 0.0128 0.0092
Fig. 1. (a) Model 1 and (b) Model 2. The simulation results for conditional t copula. 95% confidence intervals for Kendall tau: true value (solid lines), averaged local linear estimates (dashed line), approximate confidence intervals (dotted lines), sample size = 200.
Fig. 2. Scatter plots: the life expectancies at birth for males and females (left), the conditional marginal distribution transformation for males and females (right).
5. Real data analysis In this section, we apply the proposed estimation method and conditional t copula to model conditional dependence between males and females for life expectancies of at birth in terms of gross domestic product(GDP). For the purpose of comparison, we also employed the Clayton, Frank and Gumbel copulas. Abegaz et al. (2012) and Gijbels et al. (2011) analyzed similar data with sample size 222 which is collected in 2009. Here, the data set with sample size 216 used in this paper is collected in 2013. The data of life expectancies at birth of males and females was collected by the world Factbook of the Central Intelligence Agency (CIA) while the data of GDP in USD per capita is available at http://data.worldbank.org/indicator/SP.DYN.LE00.MA.IN?page=2. Fig. 2 presents the scatter plot of the life expectancies of males and females, also gives the scatter plot of the conditional marginal distribution transformation for males and females. Fig. 3 provides the scatter plots of log10 (GDP) and life expectancies of males and females, where log10 (GDP) denotes the logarithm transformation of GDP with base 10. Kendall τ rank correlation coefficient 0.856 between males and females
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Fig. 3. Scatter plot of log10 (GDP) and life expectancies at birth for males and females.
indicates that there exists a relatively strong positive relationship between males and females. Fig. 3 also gives some evidence that the covariate GDP affects the life expectancies both for males and females. In addition, we model this data with constant t copula, which results in ρˆ = 0.89 and νˆ = 3.60 with standard errors 0.014 and 1.726, respectively. It is necessary to test the null hypothesis whether the unconditional t copula fits the data well before using conditional t copula. In other words, we tend to test the null hypothesis H0 : the copula belongs to t copula family. Based on the parametric bootstrap method in Genest, Rémillard, and Beaudoin (2009), we obtain the p-value 0.28 which indicates that we should not reject the null hypothesis at the significance level 5%. Therefore, it is reasonable to consider the usage of conditional t copula and the effect of the covariate GDP. To illustrate the superiority of the conditional t copula over the conditional Clayton, Frank and Gumbel copulas, we evaluate the cross-validated prediction error (Acar et al., 2011) for each copula. The cross-validated prediction error can be calculated as CVPE(C ) =
n
Ui − Eˆ (−i) (Ui |Vi , Xi )
2
2 + Vi − Eˆ (−i) (Vi |Ui , Xi ) ,
i=1
where Eˆ (−i) (Ui |Vi , Xi ) =
1
(−i)
uc u, Vi ; θˆh∗ (Xi ) du.
0
We evaluate the integration by Monte Carlo method. Specifically, we generate N = 10000 standard uniformly distributed random samples {U1 , . . . , UN } and then the integration can be approximated by the mean N 1
N t =1
(−i)
c (Ut , Vi ; θˆh∗ (Xi )).
In addition, as one referee suggested, we also calculate the logarithmic score which is approximated by n 1
n i =1
(−i)
logc Ui , Vi ; θˆh∗ (Xi ) .
For each conditional copula, we split the interval [0.08, range(X )/2] into 20 subintervals with equal space, then calculate these conditional local log-likelihoods at those interval endpoints, and then choose the one as optimal bandwidth which n (−i) maximizes the log-likelihood i=1 log c (Ui , Vi ; θˆh ). Table 2 summarizes the cross-validated prediction errors and optimal bandwidths. According to the CVPE and logarithmic score, we find that the conditional t copula outperforms other copulas. Fig. 4 reveals that the relationship between males and females for life expectancies at birth is positive for those poor or rich countries. The dashed line with symbol ‘*’ provides the conditional Kendall τ estimates at the observations of covariate and the dotted lines denote the corresponding pointwise confidence interval bounds at the confidence level 95%, and the
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Table 2 CVPE and optimal bandwidth for different copulas. Copula
CVPE
Conditional t Conditional Gaussian Conditional Clayton Conditional Frank Conditional Gumbel
8.59 9.07 9.45 10.16 23.87
Logarithmic score
h∗
184.07
0.31 0.08 0.08 1.18 0.08
−505.62 −5168 107.26 −14301
Fig. 4. Kendall τ estimation of the conditional t copula.
Fig. 5. The estimation of degree of freedom ν of the conditional t copula.
solid line describes the unconditional Kendall τ estimate. It can be seen that the confidence interval is wide when the covariate takes small values. The conditional Kendall tau varies from about 0.593 to 0.936. It can be found in Fig. 4 that more poor or rich country admits more strong relationship between males and females. Fig. 5 gives the plot of the estimation of degree of freedom ν for conditional t copula. It can be observed that the value of ν is large when the covariate takes value around its mean, hence, in medium income level countries and regions, the conditional t copula will reduce to the Gaussian copula. This gives us some evidence that the conditional t copula will produce some different effect on the estimation process compared with the conditional Gaussian copula.
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6. Concluding remarks In this article, we study the nonparametric estimation of conditional copula with multiple parameters and its application. The estimates can be obtained by utilizing local linear fitting and Newton–Raphson iterative algorithms. Simulation results show that the proposed method has a good performance in finite sample. In real data analysis, we consider the use of conditional t copula to describe the influence of GDP on the relationship between males and females. This estimation method can be directly extended to those copulas with more than two parameters such as geometric t copula (Zhang, Lin, & Xu, 2016) or mixture of copulas, however, the structure of the variance will become more difficult since the covariance should be taken into account. Testing whether the calibration function is linear is of importance. We are in favor of the parametric estimation instead of the nonparametric estimation when the calibration is indeed linear. Our future work will consist of the testing process. Acknowledgments The authors would like to thank the two anonymous referees for their valuable suggestions. Appendix Proof of Theorem 1. By Assumptions A2–A4 and Taylor’s theorem, we have
η1 (Xi ) = η1 (x) + η1′ (x)(Xi − x) + op (Xi − x) = η1 (x) + op (1) if |
X i −x h
| ≤ a. Similarly,
η2 (Xi ) = η2 (x) + η2′ (x)(Xi − x) + op (Xi − x) = η2 (x) + op (1). More precisely, η1 (Xi ) = η1 (x) + η1′ (x)(Xi − x) + r1i ≡ η¯ 1i + r1i and η2 (Xi ) = η2 (x) + η2′ (x)(Xi − x) + r2i ≡
∞ η2(t ) (x) − x)t and r2i = (Xi − x)t . For ease of presentation, let r1 = t =2 t! 2 2 = −E ℓ′′11 θ1 (x), θ2 (x); U , V , σ12 = (r11 , r12 , . . . , r1n )T , r2 = (r21 , r22 , . . . , r2n )T , g1′ = g1′ η1 (x) and g2′ = g2′ η2 (x) , σ11 2 −E ℓ′′12 θ1 (x), θ2 (x); U , V and σ22 = −E ℓ′′12 θ1 (x), θ2 (x); U , V . T T X −x as well as b2 = ξn−1 γ0 −η2 (x), h γ1 −η2′ (x) . Let ξn = (nh)−1/2 , zix = (1, ih )T , b1 = ξn−1 α0 −η1 (x), h α1 −η1′ (x)
η¯ 2i + r2i where r1i =
∞
t =2
η1 (t ) (x) (Xi t!
If (α0 , α1 , γ0 , γ1 ) is equal to the local log-likelihood maximum estimate (αˆ 0 , αˆ 1 , γˆ0 , γˆ1 ) in (1), the corresponding value
bˆ ≡ (bˆ T1 , bˆ T2 )T = ξn−1 αˆ 0 − η1 (x), h αˆ 1 − η1′ (x) , γˆ0 − η2 (x), h γˆ1 − η2′ (x)
L(b) =
n
T
maximizes the equivalent form of (1), i.e.,
log c Ui , Vi ; g1 (η¯ 1i + ξn bT1 zix ), g2 (η¯ 2i + ξn bT2 zix ) Kh (Xi − x)
i =1 n = ℓ g1 (η¯ 1i + ξn bT1 zix ), g2 (η¯ 2i + ξn bT2 zix ); Ui , Vi Kh (Xi − x). i =1
It is clear that bˆ also maximizes the following function n T T L (b) = ℓ g1 (η¯ 1i + ξn b1 zix ), g2 (η¯ 2i + ξn b2 zix ); Ui , Vi − ℓ g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi K (Xi − x).
∗
i=1
By Taylor’s theorem, under Assumptions A3–A5, we have n
L∗ (b) = bT1 ξn
ℓ′1 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g1′ (η¯ 1i )zix K (Xi − x) 1 + op (1)
i =1
G1
n + bT2 ξn ℓ′2 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g2′ (η¯ 2i )zix K (Xi − x) 1 + op (1) i=1
1
+ bT1 2
G2
n 1
n i =1
ℓ′′11 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi (g1′ (η¯ 1i ))2 zix zTix Kh (Xi − x) 1 + op (1) b1
M11
J.-G. Lin et al. / Journal of the Korean Statistical Society (
1
+ bT2
n 1
2
n i=1
n 1
2
9
M22
1
–
ℓ′′22 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi (g2′ (η¯ 2i ))2 zix zTix Kh (Xi − x) 1 + op (1) b2
+ 2 × bT1
)
n i=1
ℓ′′12 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g1′ (η¯ 1i )g2′ (η¯ 2i )zix zTix Kh (Xi − x) 1 + op (1) b2
M12
+R
1
≡ bT G + bT 2
M11 M12
M12 M22
b + R,
where R=
1 6
ξn3
n 2
c c η2i ) 2 (bTj zix )(bTk zix )(bTl zix ) K (Xi − x), ℓ′′′ η1i ), g2 ( η2i ); Ui , Vi g1′ ( η1i ) 1 g2′ ( jkl g1 (
j,k,l=1
i =1
G = (GT1 , GT2 )T and cs = I(j = s)+ I(k = s)+ I(l = s) for s = 1, 2. ηji is between η¯ ji and η¯ ji +ξn bTj zix for i = 1, . . . , n, j = 1, 2. It is worth noting that the method of Acar (2011) cannot be applied directly here since she ignored the effect of the link functions. Our proof is similar to Cai, Fan, and Li (2000). First, under the Assumptions A4, A5 and A8, we assert that the expectation of the absolute value of R is of order O(γn ). Consequently, we obtain that R = op (γn ) = op (1). Next we will show that
M =
M11 M12
M12 M22
≡ −Λ 1 + op (1) ,
(2)
where Λ = D ⊗ µf ,
D=
2 ′2 σ11 g1 2 ′ ′ σ12 g1 g2
2 ′ ′ σ12 g1 g2 , 2 ′2 σ22 g2
(3)
and
µ0 µf = f (x) µ1
µ1 . µ2
Note that under Assumptions A1–A2, the (k1 , k2 )th entry of the matrix E z1x zT1x Kh (X1 − x) can be written as
t −x
k1 +k2
K
h
t −x h
f (t )dt = f (x)µk1 +k2 1 + o(1)
for 0 ≤ k1 , k2 ≤ 1. Hence, E z1x zT1x Kh (X1 − x) = µf 1 + o(1) .
The expectation of M11 can be obtained as follows E (M11 ) = E E (M11 |X)
=E
n 1 n i=1
= E E ℓ11 ′′
ℓ′′11 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g1′2 (η¯ 1i )zix zTix Kh (Xi − x)|X
′2 T g1 (η¯ 1i ), g2 (η¯ 2i ); U , V |X g1 (η¯ 1i )z1x z1x Kh (X1 − x)
2 ′2 = −σ11 g1 E z1x zT1x Kh (X1 − x) 1 + o(1) 2 ′2 = −σ11 g1 µf 1 + o(1) .
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2 ′2 Therefore, under Assumption A6, M11 = E (M11 ) 1 + op (1) = −σ11 g1 µf 1 + op (1) . 2 ′2 2 ′ ′ g2 µf 1 + op (1) . Thus, we have Similarly, we have M12 = −σ12 g1 g2 µf 1 + op (1) and M22 = −σ22 L∗ (b) = bT G −
1 2
bT Λb 1 + op (1) .
Utilizing the quadratic approximation lemma (Fan & Gijbels, 1996), we have bˆ = Λ−1 G 1 + op (1) ,
under the situation where G is a random vector with stochastic bound. Next, we will calculate the mean and variance of G, respectively. Note that
ℓ′1 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi = ℓ′1 g1 (η1 (Xi ) − r1i ), g2 (η2 (Xi ) − r2i ); Ui , Vi = ℓ′1 g1 (η1 (Xi )) − r1i g1′ (η1 (Xi )) 1 + op (1) , g2 (η2 (Xi )) − r2i g2′ (η2 (Xi )) 1 + op (1) ; Ui , Vi = ℓ′1 g1 (η1 (Xi )), g2 (η2 (Xi )); Ui , Vi − ℓ′′11 g1 (η1 (Xi )), g2 (η2 (Xi )); Ui , Vi r1i g1′ (η1 (Xi )) × 1 + op (1) − ℓ′′12 g1 (η1 (Xi )), g2 (η2 (Xi )); Ui , Vi r2i g2′ (η2 (Xi )) 1 + op (1) and the fact E ℓ′1 (g1 (η1 (Xi )), g2 (η2 (Xi )); Ui , Vi ) = 0, together with Assumption A3, we have E (G1 ) = E E (G1 |X) n ′ ′ = ξn E E ℓ1 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g1 (η¯ 1i )zix K (Xi − x) 1 + op (1) X i =1
= ξn g1 σ ′2
2 11 nE
2 z1x K (X1 − x)r11 1 + o(1) + ξn g1′ g2′ σ12 nE z1x K (X1 − x)r21 1 + o(1) .
Note that the jth entry of the vector E z1x K (X1 − x)r11 can be written as
( s − x) j K
∞ s − x
h
αt (s − x)t f (s)ds = α2 h3 f (x)µ2+j 1 + o(1)
t =2
for j = 0, 1.
µ
µ
Thus, E z1x K (X1 − x)r11 = α2 h3 f (x) µ2 1 + o(1) . Similarly, E z1x K (X1 − x)r21 = γ2 h3 f (x) µ2 3 3 Based on the previous arguments, we have
α 2 √ 2 µ2 ⊗ nhh f (x) 1 + o(1) . γ2 µ3
E (G) = D
Next, the variance of the vector G is derived as Var(G) =
L111 + L112 L121 + L122
L121 + L122 L221 + L222
,
where
L111 = E Var(G1 X)
= ξn2 E
n
Var ℓ′1 (g1 (η¯ 1i ), g2 (η¯ 2i )); Ui , Vi X g1′ (η¯ 1i )2 zix zTix K 2 (Xi − x) 1 + op (1)
i =1
1 = g1′2 Var ℓ′1 (θ (x); U , V ) E z1x zT1x K 2 (X1 − x) 1 + o(1) , h
L112 = Var E (G1 |X) = o(1),
L121 = E Cov G1 |X, G2 |X
=
1 ′ ′ g1 g2 Cov ℓ′1 (θ (x); U , V ), ℓ′2 (θ (x); U , V ) E z1x zT1x K 2 (X1 − x) 1 + o(1) , h
L122 = Cov
E (G1 |X), E (G2 |X)
= o(1),
1 + o(1) .
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1 L221 = g2′2 Var ℓ′2 (θ (x); U , V ) E z1x zT1x K 2 (X1 − x) 1 + o(1) , h L222 = Var E (G2 |X) = o(1). υ υ In a similar argument we have 1h E z1x zT1x K 2 (X1 − x) = υf 1 + o(1) , where υf = h2 f (x) υ0 υ1 . 1 2 Let ℓ′1 ≡ ℓ′1 (θ; U , V ), ℓ′2 ≡ ℓ′2 (θ; U , V ), ℓ′′11 ≡ ℓ′′11 (θ; U , V ), ℓ′′12 ≡ ℓ′′12 (θ; U , V ) and ℓ′′22 ≡ ℓ′′22 (θ; U , V ). By using first and second order Bartlett’s identity E (ℓ′1 ) = E (ℓ′2 ) = 0 and E (ℓ′1 ℓ′2 ) + E (ℓ′′12 ) = 0, we have Var(ℓ′1 ) = E (ℓ′1 ℓ′1 ) = −E (ℓ′′11 ) = 2 2 2 σ11 , Var(ℓ′2 ) = σ22 and Cov(ℓ′1 , ℓ′2 ) = −E (ℓ′′12 ) = σ12 . Hence, the variance of vector G has the following expression 2 ′2 2 ′ ′ σ11 g1 σ12 g1 g2 ⊗ vf 1 + o(1) Var(G) = 2 ′ ′ 2 ′2 σ12 g1 g2 σ22 g2 = D ⊗ vf 1 + o(1) ≡ ∆ 1 + o(1) . (4)
In order to prove the asymptotic normality of random vector G, we only need to show that for any unit vector a,
aT Var(G)a
−1/2
aT G − aT E (G) → N (0, 1).
ℓ′ g (η¯ , g η¯ ); U , V
(5)
Let ϕi = ξn aT ℓ′1 g1 (η¯ 1i , g2 η¯ 2i ); Ui , Vi ⊗ zix K ( ih ), then aT G = i=1 ϕi . Note that ϕi is independent and identically i i 2 2i 2 1 1i 1 T distributed for each i and the fact Var(ϕi ) = n a Var(G)a. By using Assumption A7, it follows from Lindeberg–Feller central limit theorem that the asymptotic distribution of ϕi is normal which then verifies (5). The proof of Corollary 1 is completed via Delta method. X −x
n
References Abegaz, F., Gijbels, I., & Veraverbeke, N. (2012). Semiparametric estimation of conditional copulas. Journal of Multivariate Analysis, 110, 43–73. Acar, E. F. (2011). Nonparametric estimation and inference for the copula parameter in conditional copulas (Ph.D. thesis). University of Toronto. Acar, E. F., Craiu, R. V., & Yao, F. (2011). Dependence calibration in conditional copulas: A nonparametric approach. Biometrics, 67, 445–453. Cai, Z. W., Fan, J. Q., & Li, R. Z. (2000). Efficient estimation and inferences for varying coefficient models. Journal of the American Statistical Association, 95, 888–902. Embrechts, P., Lindskog, F., & McNeil, A. (2003). Modelling dependence with copulas and applications to risk management. In S. Rachev (Ed.), Handbook of heavy tailed distributions in finance (pp. 329–384). Elsevier, Chapter 8. Fan, J., & Gijbels, I. (1996). Local polynomial modelling and its applications, vol. 66. London: Chapman & Hall. Genest, C., & Favre, A. C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12, 347–368. Genest, C., Ghoudi, K., & Rivest, L. P. (1995). Semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82, 543–552. Genest, C., Rémillard, B., & Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics & Economics, 44, 199–213. Gijbels, I., Omelka, M., & Veraverbeke, N. (2015). Estimation of a Copula when a covariate affects only marginal distributions. Scandinavian Journal of Statistics, http://dx.doi.org/10.1111/sjos.12154. Gijbels, I., Veraverbeke, N., & Omelka, M. (2011). Conditional copulas, association measures and their application. Computational Statistics and Data Analysis, 55, 1919–1932. Hansen, BE. (2004). Bandwidth selection for nonparametric distribution estimation. University of Wisconsin. Working Paper. Min, A., & Czado, C. (2010). Bayesian inference for multivariate copulas using pair-copula constructions. Journal of Financial Econometrics, 8, 511–546. Nelsen, R. B. (2006). An introduction to copulas. Springer. Patton, A. J. (2006). Modeling asymmetric exchange rate dependence. International Economic Review, 47, 527–556. Patton, A. J. (2012). A review of copula models for economic time series. Journal of Multivariate Analysis, 110, 4–18. Smith, M., & Khaled, M. (2012). Estimation of copula models with discrete margins via Bayesian data augmentation. Journal of the American Statistical Association, 107, 290–303. Smith, M., Min, A., Almeida, C., & Czado, C. (2010). Modeling longitudinal data using a pair-copula decomposition of serial dependence. Journal of the American Statistical Association, 105, 1467–1479. Zhang, K. S., Lin, J. G., & Xu, P. R. (2016). A new class of copulas involved geometric distribution: estimation and applications. Insurance: Mathematics & Economics, 66, 1–10.
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Contents lists available at ScienceDirect
Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss
Nonparametric estimation of multivariate multiparameter conditional copulas✩ Jin-Guan Lin a,∗ , Kong-Sheng Zhang b , Yan-Yong Zhao a a
Department of Statistics, Nanjing Audit University, Nanjing 211815, China
b
Department of Mathematics, Southeast University, Nanjing 211189, China
article
info
Article history: Received 1 December 2015 Accepted 31 August 2016 Available online xxxx AMS 2000 subject classifications: primary 62G08 secondary 62H12 Keywords: Conditional copula Calibration function Local linear smoothing Newton–Raphson method
abstract Nonparametric estimation of conditional copulas with one parameter has been investigated in Acar et al. (2011). The estimation for multivariate multiparameter conditional copulas, however, has not been considered so far. This paper adopts the local linear smoothing technique and Newton–Raphson method to estimate those copulas. Under some regularity conditions, the asymptotic normality of the estimators is obtained. Simulation work shows the efficiency of the proposed method. As an application, we analyze a life expectancies data set and show that the conditional t copula outperforms the conditional Clayton, Frank and Gumbel copulas. © 2016 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
1. Introduction Copulas have become an efficient tool in modeling the dependence structure between random variables. They have been widely used in many fields such as risk management (Embrechts, Lindskog, & McNeil, 2003), econometrics (Patton, 2012) and hydrology (Genest & Favre, 2007). Those broad applications mainly rely on the well known Sklar’s theorem (Nelsen, 2006) which enables us to model the marginal distributions and the copula C separately. Of importance is estimation of parameters in copulas. Two commonly used methods are full maximum likelihood and two-stage estimation approaches (Genest, Ghoudi, & Rivest, 1995; Nelsen, 2006). Also, Bayesian method is employed to tackle this issue (Min & Czado, 2010; Smith & Khaled, 2012; Smith, Min, Almeida, & Czado, 2010). Recently, Patton (2006) proposed the concept of conditional copula in which the correlation is affected by covariate(s). Based on Patton’s seminal work, many authors have payed considerable attention on the estimation and testing of conditional copulas. Gijbels, Veraverbeke, and Omelka (2011) presented two nonparametric estimators for conditional copulas. Abegaz, Gijbels, and Veraverbeke (2012) suggested estimating the conditional copula by using semiparametric method in which the marginal distributions are assumed to be unknown. Gijbels, Omelka, and Veraverbeke (2015) investigated the case where the covariate only affected the marginal distributions, but not the dependence structure. Acar, Craiu, and Yao (2011) proposed a nonparametric framework to estimate the calibration function. All the conditional copulas discussed in Acar et al. (2011) have two notable features: each copula contains only one parameter, and both the score
✩ The work is supported by National Natural Science Foundation of China (11571073, 11401094), Natural Science Foundation of Jiangsu Province of China (BK20141326, BK20140617), the Research Fund for the Doctoral Program of Higher Education of China under Grant (20120092110021). ∗ Corresponding author. E-mail address: [email protected] (J.-G. Lin).
http://dx.doi.org/10.1016/j.jkss.2016.08.003 1226-3192/© 2016 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
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function and Hessian matrix have closed forms which facilitate the calculation procedure for parameter estimation. In practice, however, we may encounter those copulas with the number of parameters being more than one, and the score function and Hessian matrix may have no simplified forms, for instance, Student t copula and mixture copula. These considerations motivate our present work. The remainder of the paper is organized as follows. In Section 2, we present nonparametric estimation methodology for conditional copula parameters and discuss the choice of optimal bandwidth. Section 3 is dedicated to investigate the asymptotic normality of the estimators. Section 4 carries out some simulation studies for illustrating the efficiency of the proposed method. A life expectancies data analysis is provided in Section 5. All proofs are deferred to Appendix. 2. Methodology Suppose that Y1 and Y2 are continuous variables, and X is a continuous covariate which may produce an effect on Y1 and Y2 . According to Sklar’s theorem (see, e.g., Nelsen, 2006), one has F (y1 , y2 |X = x) = C F1|X (y1 |x), F2|X (y2 |x); θ , α1 , α2 ,
where F (·, ·|X = x) denotes the conditional cumulative distribution of (Y1 , Y2 ) under X = x, and F1|X and F2|X are the marginal distributions of Y1 and Y2 with parameters α1 and α2 , respectively. C (·, ·) is a copula function with parameter θ , which describes the correlation dependence structure of (Y1 , Y2 ) under covariate X . Furthermore, we assume that the conditional marginal distributions are known since we are only interested in the dependence structure. Without loss of generality, we assume that the copula associated with (Y1 , Y2 ) under X has two parameters, however, the case where the number of parameters is more than two can be analyzed in a similar manner. Thus we only need to consider the model
(U , V |X = x) ∼ C u, v; θ = (θ1 (x), θ2 (x)) , where θ1 (x) = g1 (η1 (x)), θ2 (x) = g2 (η2 (x)), U = F1|X (Y1 ), and V = F2|X (Y2 ), the unknown functions η1 (x) and η2 (x) are called calibration functions (Acar et al., 2011), whereas the link functions g1 (x) and g2 (x) are known which guarantee that the copula parameters lie in the correct range. The main purpose of this paper is to estimate the calibration functions and then to estimate the conditional copula parameters. Assume that we have i.i.d. samples (Y1i , Y2i , Xi ), i = 1, 2, . . . , n, which are independent copies of (Y1 , Y2 , X ). By Taylor expansion with one order for η1 (Xi ) and η2 (Xi ), respectively, we have
η1 (Xi ) ≈ η1 (x) + η1′ (x)(Xi − x) ≡ xTix α and
η2 (Xi ) ≈ η2 (x) + η2′ (x)(Xi − x) ≡ xTix γ , (j)
η (x)
(j)
η (x)
dj η (x)
dj η (x)
where α = (α0 , α1 )T , γ = (γ0 , γ1 ), αj = 1 j! = dx1 j /j!, γj = 2 j! = dx2 j /j!, and xix = (1, Xi − x)T for j = 0, 1. The influence of the covariate Xi (i = 1, 2, . . . , n) on the relationship between Y1 and Y2 is incorporated into the conditional local log-likelihood
L(α, γ ; x, h) =
n
log c Ui , Vi ; g1 (xTix α), g2 (xTix γ ) Kh (Xi − x),
(1)
i =1
where Ui = F1|X (Y1i ), and Vi = F2|X (Y2i ), Kh (·) = K (·/h)/h with h denoting bandwidth, and K (x) is a kernel function and ∂ 2 C (u,v)
c (u, v) = ∂ u∂v is the density of the copula C . The Epanechnikov kernel K (t ) = 3/4(1 − t )2 I(|t | ≤ 1) is adopted in this paper where I indicates the indicator function. When the information on the conditional marginal distributions for Y1 and Y2 is not available, as in Abegaz et al. (2012), we can estimate them by n
Fˆj|X (y) =
wi I(Yji ≤ y)
i=1 Kh (Xi −x) j
where wi = n
k=1 Khj (Xk −x)
and hj denotes the bandwidth for j = 1, 2. For the sake of simplicity we set h1 = h2 and they can
be obtained by the method proposed by Hansen (2004) with Matlab software. A natural way to obtain the local maximum likelihood estimator βˆ = (α, ˆ γˆ ) = (αˆ 0 , αˆ 1 , γˆ0 , γˆ1 ) is to solve the estimating equation
∂ L(β; x, h) = 0, ∂β where the notation ∇ denotes the partial derivative of L with respect to β . ∇ L(β, x) =
We adopt the commonly used Newton–Raphson method to get the estimator βˆ with the recursive form
β t +1 = β t − {∇ 2 L(β t , x)}−1 ∇ L(β t , x),
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3
where ∇ L and ∇ 2 L stands for the score function and Hessian matrix. We are mainly interested in the estimates ηˆ 1 (x) = αˆ 0 and ηˆ 2 (x) = γˆ0 since the estimators of θ1 (x) and θ2 (x) can be obtained by θˆ1 (x) = g1 (ηˆ 1 (x)) = g1 (αˆ 0 ) and θˆ2 (x) = g2 (ηˆ 2 (x)) = g2 (γˆ0 ), respectively. The initial value of the parameter β in the case of conditional t copula can be set as βinitial = (g1−1 (ρ), ˆ 0, g2−1 (ˆν ), 0), where ρˆ and νˆ are the estimates of the t copula. Note that the copulas considered in Acar et al. (2011) are Clayton, Frank and Gumbel copulas whose first and second derivatives with respect to the copula parameter are available, however, for instance, the Student t copula does not have the property, therefore, we need to calculate the score function and Hessian matrix numerically. As we know, the bandwidth parameter h plays a crucial role in the estimation procedure for copula parameters. As in Acar et al. (2011), we also apply the leave-one-out cross-validation method to find the optimal bandwidth h∗ . The details on how to obtain h∗ are described as follows when we have the i.i.d. random sample (Ui , Vi , Xi )(i = 1, 2, . . . , n) with n being sample size.
• Step 1: Choose a suitable initial value βinitial for β . • Step 2: For i = 1, 2, . . . , n, delete ith individual to obtain a new sample (Uj , Vj , Xj ), j = 1, 2, . . . , n, j ̸= i. Based on the new sample and Newton–Raphson iteration approach, we can obtain the maximum likelihood estimate of the parameter θ , denote it by θˆh(−i) , then calculate log c (Ui , Vi ; θˆh(−i) ).
• Step 3: The optimal bandwidth h∗ can be achieved via maximizing the log-likelihood
n
i=1
(−i)
log c (Ui , Vi ; θˆh
).
3. Asymptotic theory In this section, we investigate the theoretical results of the nonparametric estimate of copula θ. conditional j parameter For ease of presentation, let us first start by introducing some notation. Define µj = t j K (t )dt , vj = t K 2 (t )dt with K (t ) being a kernel function and zix = (1, c (u, v; θ ) = ∂ 2 log c (U ,V ; θ)
∂ C 2 (u,v; θ) ∂ u∂v
) for i = 1, . . . , n. Furthermore, let ℓ(θ; U , V ) = log c (U , V ; θ ) where
Xi −x T h
∂ log c (U ,V ; θ ) , ℓ′2 ∂θ1 ′′ ′′
, and ℓ1 ≡ ℓ1 (θ; U , V ) = ′
∂ 2 log c (U ,V ; θ)
≡ ℓ′2 (θ; U , V ) = ∂ 2 log c (U ,V ; θ )
∂ log c (U ,V ; θ ) , ℓ′′11 ∂θ2
≡ ℓ′′11 (θ; U , V ) =
, ℓ22 ≡ ℓ22 (θ; U , V ) = and X = {X1 , X2 , . . . , Xn }. ∂θ1 ∂θ2 ∂θ2 ∂θ2 Before stating the proofs, we list some assumptions which are necessary for Theorem 1. ∂θ1 ∂θ1
, ℓ′′12 ≡ ℓ′′12 (θ; U , V ) =
′
A1: The density function f (x) of covariate X is continuous and satisfies the Lipschitz condition, and X has a bounded support Ω . A2: The kernel function K (t ) is continuous, bounded and is symmetric on [−a, a] with a > 0. A3: gi (ηi (x)) and ηi (x) have continuous derivatives up to order two on a neighborhood of x N (x) ⊂ Ω , in addition, ℓ(θ (x), u, v) has three on N (x). continuous derivatives up to order A4: supx∈N (x) max |gi′ (ηi (x))|, |gi′′ (ηi (x))|, |ηi′ (x)|, |ηi′′ (x)| < ∞ for i = 1, 2.
A5: supx∈N (x)×(0,1)2 max |ℓ′i θ (x); u, v |, |ℓ′′ij θ (x); u, v |, |ℓ′′′ ijk (θ (x), u, v)| < ∞ for 1 ≤ i, j, k ≤ 2.
A6: supx∈N (x)×(0,1)2 Var{ℓ′′ij θ (x); u, v } < ∞ for 1 ≤ i, j ≤ 2.
A7: ∥∆∥ =
2 i =1
2
j =1
∆2ij < ∞ where ∆ is defined in (4) in the Appendix.
A8: E |X 3 | < ∞. Assumptions A1–A2 are commonly required in kernel estimation (Fan & Gijbels, 1996). Assumptions A3–A5 are used to obtain the Taylor expansion. Assumption A6 enables us to establish the relationship between the expectation of the variable and the variable itself. Assumption A7 guarantees the condition of Lindeberg–Feller central limit theorem in the proof of asymptotic normality. Assumption A8 is used to establish the order of the remainder in the second-order Taylor expansion. Theorem 1. Under the Assumptions A1–A8, h → 0 and nh → ∞ as n → ∞, we have
ηˆ 1 (x) − η1 (x) ′ ′ d h(ηˆ 1 (x) − η1 (x)) nh − Ψ 1 + op (1) − → N 0, Ξ , ηˆ 2 (x) − η2 (x) h(ηˆ 2′ (x) − η2′ (x)) α µ2 where Ψ = h2 f (x)Λ−1 D γ2 ⊗ µ3 , Ξ = Λ−1 ∆Λ−1 , 0 = (0, 0, 0, 0)T and the matrices Λ, D and ∆ are defined as 2
√
in (2), (3) and (4) in the Appendix, respectively. Remark 1. Theorem 1 establishes the asymptotically normal distribution theory for the calibration functions ηj (x) with convergence rate (nh)−1/2 as well as its corresponding derivative ηj′ (x) with convergence rate (nh3 )−1/2 , respectively, for j = 1, 2. The following corollary follows immediately from Theorem 1 by extracting the corresponding elements and the Delta method.
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Corollary 1. Suppose that all the conditions in Theorem 1 hold, then one has
√
′ d θˆ1 (x) − θ1 (x) Ψ (1) g1 (η1 (x)) Ξ (1, 1) nh − ⊙ ′ 1 + op (1) − → N 0, Ψ (3) Ξ (3, 1) g2 (η2 (x)) θˆ2 (x) − θ2 (x) g ′ (η (x))2 g1′ (η1 (x))g2′ (η2 (x)) where Υ = ′ 1 1 ′ . g (η (x))g (η (x)) g ′ (η (x))2 1
1
2
2
2
Ξ (1, 3) ⊙Υ Ξ (3, 3)
2
Remark 2. Corollary 1 only provides the asymptotically normal distribution theory for the calibration functions ηj (x) for j = 1, 2. 4. Monte Carlo experiments In this section, some simulations are carried out to indicate the finite sample performance of the proposed method. Specifically, we only investigate parameter estimation for the conditional Student t copula which admits the formula C (u, v; ρ, ν) = Tν (tν−1 (u), tν−1 (v); ρ, ν)
−1 tν (u)
−1 tν (v)
= −∞
−∞
ν+2 Γ ν+2 2 x2 + y2 − 2ρ xy − 2 dydx, 1+ ν ν(1 − ρ 2 ) Γ 2 π ν 1 − ρ2
where ρ is the correlation coefficient between U and V , and ν denotes the degrees of freedom (df for short). tv (·) (resp. tv−1 (·)) is the cumulative distribution (resp. quantile) of a univariate t distribution with mean zero and df = ν . Correspondingly, the density of t copula can be expressed as
Γ ν+2 2 Γ ν2 c (u, v; ρ, ν) = 2 Γ ν+2 1 1 − ρ2
1+
1+
2 −1 tν (u)
ν
1+
2 −1 tν (v)
ν+2 1
ν
2 2 −1 −1 −1 −1 tν (u) +tν (v) −2ρ tν (u)tν (v) ν(1−ρ 2 )
ν+2 2 .
We study the two different types of calibration functions. Model 1. η1 (X ) = 3 − X and η2 (X ) = 0.2X + 1, and X ∼ Uniform(1, 5). Model 2. η1 (X ) = 1 − 0.5X + 0.2X 2 and η2 (X ) = log(X ) + 0.1X , and X ∼ Uniform(1, 5). For each model, to ensure the two 1−exp(−x) parameters ρ and ν in their correct ranges, we use the link functions g1 (x) = 1+exp(−x) and g2 (x) = 1 + exp(x), respectively. In Model 1, it can be seen that the parameter ρ varies from −0.76 to 0.76 and ν covers the range (4.32, 8.39). Similarly, ρ ∈ (0.33, 0.94) and ν ∈ (2.10, 9.25) for Model 2. The random sample of the conditional t copula in Model 1 is distributed as
1 − exp(X − 3) , θ2 = exp(0.2X + 1) + 1 . (U , V )|X ∼ C u, v; θ1 = 1 + exp(X − 3) We first generate the sample Xi ∼ Uniform(1, 5)(i = 1, . . . , n) with sample size n, then for each Xi , we generate (Ui , Vi )(i = 1, . . . , n) from the corresponding conditional t copula, the tri-tuple sample (Ui , Vi , Xi )(i = 1, . . . , n) for (U , V , X ) is then obtained. We replicate this procedure 100 times, the precision of the proposed estimation method can be evaluated by using commonly used indexes such as integrated squared bias (IBIAS2 ) and integrated mean square error 11 (IMSE). The IBIAS2 and IMSE of the estimate of Kendall τ (x) = 4 0 0 C (u, v; θ (x))dC (u, v; θ (x)) − 1 are defined as 2 IBIAS2 (τˆ ) = E{τˆ (x)} − τ (x) dx X
and IMSE(τˆ ) =
2
E τˆ (x) − τ (x) dx,
X
respectively, where X is the possible set of x. We divide the interval [1.1, 4.9] into 50 equally space intervals and then obtain approximate IMSE and IBIAS2 for the estimates of ρ(x), ν(x) and τ (x). Note that Kendall τ for t copula has a simple form τ = 2/π · arcsin(ρ) (Embrechts et al., 2003). Table 1 reports the approximate integrated squared bias (IBIAS2 ) and integrated mean square error (IMSE) of the estimators of ρ, ν and τ . As expected, the IMSE of the estimators of ρ, ν and τ decreases when sample size n increases for the two models. It should be noticed that the range of the design point X is 4.9 − 1.1 = 3.8, thus, the average MSE (BIAS2 ) can be understood in the sense that the corresponding value in Table 1 divided by 3.8. Fig. 1 depicts the simulation results for conditional t copula as described in Models 1 and 2 with sample size 200. The solid line denotes the true value of Kendall τ , whereas the dashed line is the averaged local linear estimates and the dotted line represents the approximate 95% confidence intervals.
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Table 1 The integrated squared bias (IBIAS2 ) and integrated mean square error (IMSE) of parameter estimates of ρ, ν and τ for conditional t copula.
ρˆ
νˆ
τˆ
IBIAS2
IMSE
IBIAS2
IMSE
IBIAS2
IMSE
Model 1
n = 100 n = 200 n = 1000
0.0036 0.0022 0.0029
0.0691 0.0476 0.0235
3.1352 2.6483 1.8532
11.7050 9.1217 6.1046
0.0032 0.0011 0.0013
0.0359 0.0230 0.0176
Model 2
n = 100 n = 200 n = 1000
0.0012 0.0008 0.0006
0.0094 0.0075 0.0053
4.8365 3.0479 2.3941
16.3326 12.2282 10.3020
0.0012 0.0008 0.0007
0.0297 0.0128 0.0092
Fig. 1. (a) Model 1 and (b) Model 2. The simulation results for conditional t copula. 95% confidence intervals for Kendall tau: true value (solid lines), averaged local linear estimates (dashed line), approximate confidence intervals (dotted lines), sample size = 200.
Fig. 2. Scatter plots: the life expectancies at birth for males and females (left), the conditional marginal distribution transformation for males and females (right).
5. Real data analysis In this section, we apply the proposed estimation method and conditional t copula to model conditional dependence between males and females for life expectancies of at birth in terms of gross domestic product(GDP). For the purpose of comparison, we also employed the Clayton, Frank and Gumbel copulas. Abegaz et al. (2012) and Gijbels et al. (2011) analyzed similar data with sample size 222 which is collected in 2009. Here, the data set with sample size 216 used in this paper is collected in 2013. The data of life expectancies at birth of males and females was collected by the world Factbook of the Central Intelligence Agency (CIA) while the data of GDP in USD per capita is available at http://data.worldbank.org/indicator/SP.DYN.LE00.MA.IN?page=2. Fig. 2 presents the scatter plot of the life expectancies of males and females, also gives the scatter plot of the conditional marginal distribution transformation for males and females. Fig. 3 provides the scatter plots of log10 (GDP) and life expectancies of males and females, where log10 (GDP) denotes the logarithm transformation of GDP with base 10. Kendall τ rank correlation coefficient 0.856 between males and females
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Fig. 3. Scatter plot of log10 (GDP) and life expectancies at birth for males and females.
indicates that there exists a relatively strong positive relationship between males and females. Fig. 3 also gives some evidence that the covariate GDP affects the life expectancies both for males and females. In addition, we model this data with constant t copula, which results in ρˆ = 0.89 and νˆ = 3.60 with standard errors 0.014 and 1.726, respectively. It is necessary to test the null hypothesis whether the unconditional t copula fits the data well before using conditional t copula. In other words, we tend to test the null hypothesis H0 : the copula belongs to t copula family. Based on the parametric bootstrap method in Genest, Rémillard, and Beaudoin (2009), we obtain the p-value 0.28 which indicates that we should not reject the null hypothesis at the significance level 5%. Therefore, it is reasonable to consider the usage of conditional t copula and the effect of the covariate GDP. To illustrate the superiority of the conditional t copula over the conditional Clayton, Frank and Gumbel copulas, we evaluate the cross-validated prediction error (Acar et al., 2011) for each copula. The cross-validated prediction error can be calculated as CVPE(C ) =
n
Ui − Eˆ (−i) (Ui |Vi , Xi )
2
2 + Vi − Eˆ (−i) (Vi |Ui , Xi ) ,
i=1
where Eˆ (−i) (Ui |Vi , Xi ) =
1
(−i)
uc u, Vi ; θˆh∗ (Xi ) du.
0
We evaluate the integration by Monte Carlo method. Specifically, we generate N = 10000 standard uniformly distributed random samples {U1 , . . . , UN } and then the integration can be approximated by the mean N 1
N t =1
(−i)
c (Ut , Vi ; θˆh∗ (Xi )).
In addition, as one referee suggested, we also calculate the logarithmic score which is approximated by n 1
n i =1
(−i)
logc Ui , Vi ; θˆh∗ (Xi ) .
For each conditional copula, we split the interval [0.08, range(X )/2] into 20 subintervals with equal space, then calculate these conditional local log-likelihoods at those interval endpoints, and then choose the one as optimal bandwidth which n (−i) maximizes the log-likelihood i=1 log c (Ui , Vi ; θˆh ). Table 2 summarizes the cross-validated prediction errors and optimal bandwidths. According to the CVPE and logarithmic score, we find that the conditional t copula outperforms other copulas. Fig. 4 reveals that the relationship between males and females for life expectancies at birth is positive for those poor or rich countries. The dashed line with symbol ‘*’ provides the conditional Kendall τ estimates at the observations of covariate and the dotted lines denote the corresponding pointwise confidence interval bounds at the confidence level 95%, and the
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Table 2 CVPE and optimal bandwidth for different copulas. Copula
CVPE
Conditional t Conditional Gaussian Conditional Clayton Conditional Frank Conditional Gumbel
8.59 9.07 9.45 10.16 23.87
Logarithmic score
h∗
184.07
0.31 0.08 0.08 1.18 0.08
−505.62 −5168 107.26 −14301
Fig. 4. Kendall τ estimation of the conditional t copula.
Fig. 5. The estimation of degree of freedom ν of the conditional t copula.
solid line describes the unconditional Kendall τ estimate. It can be seen that the confidence interval is wide when the covariate takes small values. The conditional Kendall tau varies from about 0.593 to 0.936. It can be found in Fig. 4 that more poor or rich country admits more strong relationship between males and females. Fig. 5 gives the plot of the estimation of degree of freedom ν for conditional t copula. It can be observed that the value of ν is large when the covariate takes value around its mean, hence, in medium income level countries and regions, the conditional t copula will reduce to the Gaussian copula. This gives us some evidence that the conditional t copula will produce some different effect on the estimation process compared with the conditional Gaussian copula.
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6. Concluding remarks In this article, we study the nonparametric estimation of conditional copula with multiple parameters and its application. The estimates can be obtained by utilizing local linear fitting and Newton–Raphson iterative algorithms. Simulation results show that the proposed method has a good performance in finite sample. In real data analysis, we consider the use of conditional t copula to describe the influence of GDP on the relationship between males and females. This estimation method can be directly extended to those copulas with more than two parameters such as geometric t copula (Zhang, Lin, & Xu, 2016) or mixture of copulas, however, the structure of the variance will become more difficult since the covariance should be taken into account. Testing whether the calibration function is linear is of importance. We are in favor of the parametric estimation instead of the nonparametric estimation when the calibration is indeed linear. Our future work will consist of the testing process. Acknowledgments The authors would like to thank the two anonymous referees for their valuable suggestions. Appendix Proof of Theorem 1. By Assumptions A2–A4 and Taylor’s theorem, we have
η1 (Xi ) = η1 (x) + η1′ (x)(Xi − x) + op (Xi − x) = η1 (x) + op (1) if |
X i −x h
| ≤ a. Similarly,
η2 (Xi ) = η2 (x) + η2′ (x)(Xi − x) + op (Xi − x) = η2 (x) + op (1). More precisely, η1 (Xi ) = η1 (x) + η1′ (x)(Xi − x) + r1i ≡ η¯ 1i + r1i and η2 (Xi ) = η2 (x) + η2′ (x)(Xi − x) + r2i ≡
∞ η2(t ) (x) − x)t and r2i = (Xi − x)t . For ease of presentation, let r1 = t =2 t! 2 2 = −E ℓ′′11 θ1 (x), θ2 (x); U , V , σ12 = (r11 , r12 , . . . , r1n )T , r2 = (r21 , r22 , . . . , r2n )T , g1′ = g1′ η1 (x) and g2′ = g2′ η2 (x) , σ11 2 −E ℓ′′12 θ1 (x), θ2 (x); U , V and σ22 = −E ℓ′′12 θ1 (x), θ2 (x); U , V . T T X −x as well as b2 = ξn−1 γ0 −η2 (x), h γ1 −η2′ (x) . Let ξn = (nh)−1/2 , zix = (1, ih )T , b1 = ξn−1 α0 −η1 (x), h α1 −η1′ (x)
η¯ 2i + r2i where r1i =
∞
t =2
η1 (t ) (x) (Xi t!
If (α0 , α1 , γ0 , γ1 ) is equal to the local log-likelihood maximum estimate (αˆ 0 , αˆ 1 , γˆ0 , γˆ1 ) in (1), the corresponding value
bˆ ≡ (bˆ T1 , bˆ T2 )T = ξn−1 αˆ 0 − η1 (x), h αˆ 1 − η1′ (x) , γˆ0 − η2 (x), h γˆ1 − η2′ (x)
L(b) =
n
T
maximizes the equivalent form of (1), i.e.,
log c Ui , Vi ; g1 (η¯ 1i + ξn bT1 zix ), g2 (η¯ 2i + ξn bT2 zix ) Kh (Xi − x)
i =1 n = ℓ g1 (η¯ 1i + ξn bT1 zix ), g2 (η¯ 2i + ξn bT2 zix ); Ui , Vi Kh (Xi − x). i =1
It is clear that bˆ also maximizes the following function n T T L (b) = ℓ g1 (η¯ 1i + ξn b1 zix ), g2 (η¯ 2i + ξn b2 zix ); Ui , Vi − ℓ g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi K (Xi − x).
∗
i=1
By Taylor’s theorem, under Assumptions A3–A5, we have n
L∗ (b) = bT1 ξn
ℓ′1 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g1′ (η¯ 1i )zix K (Xi − x) 1 + op (1)
i =1
G1
n + bT2 ξn ℓ′2 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g2′ (η¯ 2i )zix K (Xi − x) 1 + op (1) i=1
1
+ bT1 2
G2
n 1
n i =1
ℓ′′11 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi (g1′ (η¯ 1i ))2 zix zTix Kh (Xi − x) 1 + op (1) b1
M11
J.-G. Lin et al. / Journal of the Korean Statistical Society (
1
+ bT2
n 1
2
n i=1
n 1
2
9
M22
1
–
ℓ′′22 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi (g2′ (η¯ 2i ))2 zix zTix Kh (Xi − x) 1 + op (1) b2
+ 2 × bT1
)
n i=1
ℓ′′12 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g1′ (η¯ 1i )g2′ (η¯ 2i )zix zTix Kh (Xi − x) 1 + op (1) b2
M12
+R
1
≡ bT G + bT 2
M11 M12
M12 M22
b + R,
where R=
1 6
ξn3
n 2
c c η2i ) 2 (bTj zix )(bTk zix )(bTl zix ) K (Xi − x), ℓ′′′ η1i ), g2 ( η2i ); Ui , Vi g1′ ( η1i ) 1 g2′ ( jkl g1 (
j,k,l=1
i =1
G = (GT1 , GT2 )T and cs = I(j = s)+ I(k = s)+ I(l = s) for s = 1, 2. ηji is between η¯ ji and η¯ ji +ξn bTj zix for i = 1, . . . , n, j = 1, 2. It is worth noting that the method of Acar (2011) cannot be applied directly here since she ignored the effect of the link functions. Our proof is similar to Cai, Fan, and Li (2000). First, under the Assumptions A4, A5 and A8, we assert that the expectation of the absolute value of R is of order O(γn ). Consequently, we obtain that R = op (γn ) = op (1). Next we will show that
M =
M11 M12
M12 M22
≡ −Λ 1 + op (1) ,
(2)
where Λ = D ⊗ µf ,
D=
2 ′2 σ11 g1 2 ′ ′ σ12 g1 g2
2 ′ ′ σ12 g1 g2 , 2 ′2 σ22 g2
(3)
and
µ0 µf = f (x) µ1
µ1 . µ2
Note that under Assumptions A1–A2, the (k1 , k2 )th entry of the matrix E z1x zT1x Kh (X1 − x) can be written as
t −x
k1 +k2
K
h
t −x h
f (t )dt = f (x)µk1 +k2 1 + o(1)
for 0 ≤ k1 , k2 ≤ 1. Hence, E z1x zT1x Kh (X1 − x) = µf 1 + o(1) .
The expectation of M11 can be obtained as follows E (M11 ) = E E (M11 |X)
=E
n 1 n i=1
= E E ℓ11 ′′
ℓ′′11 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g1′2 (η¯ 1i )zix zTix Kh (Xi − x)|X
′2 T g1 (η¯ 1i ), g2 (η¯ 2i ); U , V |X g1 (η¯ 1i )z1x z1x Kh (X1 − x)
2 ′2 = −σ11 g1 E z1x zT1x Kh (X1 − x) 1 + o(1) 2 ′2 = −σ11 g1 µf 1 + o(1) .
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2 ′2 Therefore, under Assumption A6, M11 = E (M11 ) 1 + op (1) = −σ11 g1 µf 1 + op (1) . 2 ′2 2 ′ ′ g2 µf 1 + op (1) . Thus, we have Similarly, we have M12 = −σ12 g1 g2 µf 1 + op (1) and M22 = −σ22 L∗ (b) = bT G −
1 2
bT Λb 1 + op (1) .
Utilizing the quadratic approximation lemma (Fan & Gijbels, 1996), we have bˆ = Λ−1 G 1 + op (1) ,
under the situation where G is a random vector with stochastic bound. Next, we will calculate the mean and variance of G, respectively. Note that
ℓ′1 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi = ℓ′1 g1 (η1 (Xi ) − r1i ), g2 (η2 (Xi ) − r2i ); Ui , Vi = ℓ′1 g1 (η1 (Xi )) − r1i g1′ (η1 (Xi )) 1 + op (1) , g2 (η2 (Xi )) − r2i g2′ (η2 (Xi )) 1 + op (1) ; Ui , Vi = ℓ′1 g1 (η1 (Xi )), g2 (η2 (Xi )); Ui , Vi − ℓ′′11 g1 (η1 (Xi )), g2 (η2 (Xi )); Ui , Vi r1i g1′ (η1 (Xi )) × 1 + op (1) − ℓ′′12 g1 (η1 (Xi )), g2 (η2 (Xi )); Ui , Vi r2i g2′ (η2 (Xi )) 1 + op (1) and the fact E ℓ′1 (g1 (η1 (Xi )), g2 (η2 (Xi )); Ui , Vi ) = 0, together with Assumption A3, we have E (G1 ) = E E (G1 |X) n ′ ′ = ξn E E ℓ1 g1 (η¯ 1i ), g2 (η¯ 2i ); Ui , Vi g1 (η¯ 1i )zix K (Xi − x) 1 + op (1) X i =1
= ξn g1 σ ′2
2 11 nE
2 z1x K (X1 − x)r11 1 + o(1) + ξn g1′ g2′ σ12 nE z1x K (X1 − x)r21 1 + o(1) .
Note that the jth entry of the vector E z1x K (X1 − x)r11 can be written as
( s − x) j K
∞ s − x
h
αt (s − x)t f (s)ds = α2 h3 f (x)µ2+j 1 + o(1)
t =2
for j = 0, 1.
µ
µ
Thus, E z1x K (X1 − x)r11 = α2 h3 f (x) µ2 1 + o(1) . Similarly, E z1x K (X1 − x)r21 = γ2 h3 f (x) µ2 3 3 Based on the previous arguments, we have
α 2 √ 2 µ2 ⊗ nhh f (x) 1 + o(1) . γ2 µ3
E (G) = D
Next, the variance of the vector G is derived as Var(G) =
L111 + L112 L121 + L122
L121 + L122 L221 + L222
,
where
L111 = E Var(G1 X)
= ξn2 E
n
Var ℓ′1 (g1 (η¯ 1i ), g2 (η¯ 2i )); Ui , Vi X g1′ (η¯ 1i )2 zix zTix K 2 (Xi − x) 1 + op (1)
i =1
1 = g1′2 Var ℓ′1 (θ (x); U , V ) E z1x zT1x K 2 (X1 − x) 1 + o(1) , h
L112 = Var E (G1 |X) = o(1),
L121 = E Cov G1 |X, G2 |X
=
1 ′ ′ g1 g2 Cov ℓ′1 (θ (x); U , V ), ℓ′2 (θ (x); U , V ) E z1x zT1x K 2 (X1 − x) 1 + o(1) , h
L122 = Cov
E (G1 |X), E (G2 |X)
= o(1),
1 + o(1) .
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1 L221 = g2′2 Var ℓ′2 (θ (x); U , V ) E z1x zT1x K 2 (X1 − x) 1 + o(1) , h L222 = Var E (G2 |X) = o(1). υ υ In a similar argument we have 1h E z1x zT1x K 2 (X1 − x) = υf 1 + o(1) , where υf = h2 f (x) υ0 υ1 . 1 2 Let ℓ′1 ≡ ℓ′1 (θ; U , V ), ℓ′2 ≡ ℓ′2 (θ; U , V ), ℓ′′11 ≡ ℓ′′11 (θ; U , V ), ℓ′′12 ≡ ℓ′′12 (θ; U , V ) and ℓ′′22 ≡ ℓ′′22 (θ; U , V ). By using first and second order Bartlett’s identity E (ℓ′1 ) = E (ℓ′2 ) = 0 and E (ℓ′1 ℓ′2 ) + E (ℓ′′12 ) = 0, we have Var(ℓ′1 ) = E (ℓ′1 ℓ′1 ) = −E (ℓ′′11 ) = 2 2 2 σ11 , Var(ℓ′2 ) = σ22 and Cov(ℓ′1 , ℓ′2 ) = −E (ℓ′′12 ) = σ12 . Hence, the variance of vector G has the following expression 2 ′2 2 ′ ′ σ11 g1 σ12 g1 g2 ⊗ vf 1 + o(1) Var(G) = 2 ′ ′ 2 ′2 σ12 g1 g2 σ22 g2 = D ⊗ vf 1 + o(1) ≡ ∆ 1 + o(1) . (4)
In order to prove the asymptotic normality of random vector G, we only need to show that for any unit vector a,
aT Var(G)a
−1/2
aT G − aT E (G) → N (0, 1).
ℓ′ g (η¯ , g η¯ ); U , V
(5)
Let ϕi = ξn aT ℓ′1 g1 (η¯ 1i , g2 η¯ 2i ); Ui , Vi ⊗ zix K ( ih ), then aT G = i=1 ϕi . Note that ϕi is independent and identically i i 2 2i 2 1 1i 1 T distributed for each i and the fact Var(ϕi ) = n a Var(G)a. By using Assumption A7, it follows from Lindeberg–Feller central limit theorem that the asymptotic distribution of ϕi is normal which then verifies (5). The proof of Corollary 1 is completed via Delta method. X −x
n
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