Imputed Empirical Likelihood for Varying Coefficient Models with Missing Covariates

doi:10.4236/ojapps.2013.31B1009

Paper Menu >>

Journal Menu >>

Open Journal of Applied Sciences, 2013, 3, 44-48

doi:10.4236/ojapps.2013.31B1009 Published Online April 2013 (http://www.scirp.org/journal/ojapps)

Imputed Empirical Likelihood for Varying Coefficient

Models with Missing Covariates

Peixin Zhao

Department of Mathematics, Hechi University, Guangxi Yizhou, China

Email: [email protected]

Received 2013

ABSTRACT

The empirical likelihood-based inference for varying coefficient models with missing covariates is investigated. An

imputed empirical likelihood ratio function for the coefficient functions is proposed, and it is shown that iis limiting

distribution is standard chi-squared. Then the corresponding confidence intervals for the regression coefficients are

constructed. Some simulations show that the proposed procedure can attenuate the effect of the missing data, and

performs well for the finite sample.

Keywords: Empirical Likelihood; Varying Coefficient Model; Missing Covariate

1. Introduction

In practice, missing data frequ ently occur in many appli-

cation literatures, and the literatures on statistical analy-

sis of data with missing values have been flourished in

the past decade. Parametric regression models with miss-

ing data have been widely discussed (see [1,2]). In many

practical situations, however, the parametric regression

models are not flexible enough to capture the underlying

relation between the response and the associate covari-

ates. Hence, Wang [3] and Liang et al. [4] considered the

statistical inferences for the partially linear model with

missing covariates, which is a useful extension of the

parametric regression model. In addition, the following

varying coefficient model is another useful extension of

the parametric regression model, which has more imple-

ments and stronger explanations than the parametric re-

gression model. This paper aims to present an imputed

empirical likelihood method for analyzing the varying

coefficient model with covariate data missing at random.

Consider the following varying co efficient model



YX U





 (1)

where Y is the response variable, X is the covariate

vector, U is the scalar covariate, and 1p

 







,, T





is a vector of unknown smooth functions. The error



has mean zero conditional on X and U. In this paper, we

focus mainly on the case that the covariate X may be

missing at random. That is, the available incomplete d ata

with the sample size of n are denoted as





,,,, 1,2,,

iiii

YU in





where 0





if i

is missing, otherwise 1





, and

it satisfies that









1,, 1,

iiii iiii

PXYUPYUZ



,



(2)

where





iii

YU. The supposition (2) is commonly

used in the literature of missing data (see [2-5]). It is well

known that, in the presence of missing data, the complete

case analysis often generate a considerable bias and lose

efficiency. Then, it is important to develop some new

methods which can take the partially incomplete data

into account.

In this paper, an imputed empirical likelihood procedure

is proposed to study model (1) under missing covariates.

The proposed method can use the information of the in-

complete data efficiently, and the limiting distribution of

the proposed empirical log-likelihood ratio function is

shown to be standard chi-squared. Then the correspond-

ing confiden ce intervals of the regression coefficients are

constructed. Some simulations show that the proposed

procedure can attenuate the effect of missing data, and

performs well for finite sample.

Compared with the Wald-type confidence intervals,

the empirical likelihood based confidence intervals pos-

sess several attractive features such as the circumvention

of asymptotic variance estimation and the flexible shapes

of the confidence intervals determined by data (see [6]).

This paper provides an additional positive result of the

empirical likelihood inferences varying coefficient models

with missing data, which extends the application litera-

ture of the empirical likelihood method.

*This research was supported by the National Natural Science Founda-

tion of China (Grant No. 11101119).

P. X. ZHAO 45

2. Methodology and Main Results

Let

 



 

uiiii

zEXYX uZz

Yg zgzu



 

 ,

where



zEXZz and









zEXXZz.

Then, by a simple calculation, we have that



iiiiu ii

EXYXU ZUufu



























where i





, and



u is the density function

of i. Hence, using this information, an auxiliary ran-

dom vector can be defined as



iiii

uXYXu Z

KU u



 























where

 

uKuh, is a kernel function, and h

is the band w i d th. For a n y given u, note that



K



1,,

 



are independent each other, and satisfy















if and only if is the true parameter. Hence using

the empirical likelihood method proposed by [6], an em-





pirical log-likelihood ratio function for can be defined

based on . However, contains the

unknown functions ,























and



z, then it

can not be used directly for the statistical inference for

. A natural idea to solve this problem is to replace











z and



z with the following kernel

estimators respectively.



ˆ,

ˆ.

ih i

ii hi

KZ z

zKZ z

XK Zz

gz KZz

XK Zz

gz KZ z





















Then, we obtain the following estimated auxiliary ran-

dom vector



ˆ1

ˆˆ

iiii

uXYXu Z

KU u



 

























(3)

where





ˆˆ



 and

















ˆˆˆ

ui iii

Yg ZgZu



 .

Hence, an empirical log-likelihood ratio can be given by











111

2maxlog0,1,1 .

nnn

iii ii

iii

np pppu











 









For any given u, provided that zero is inside the convex

hull of the points













1,, ,

 

u then a unique

value for











exists. By using the Lagrange mul-

tiplier method to find the optimal , then











can be represented as





2log1 ,

nTi

Ru u











(4)

where



is a 1p



vector given as the solution to









ˆ0.



 





 (5)

Next we will show that is asymptotically chi-

square distributed when















is the true parameter for

given u. To derive a theory for , the following

assumptions will be required.









Assumption 1. The bandwidth h satisfies that

and .

 50nh 

Assumptio n 2. The k ernel function



u is a bounded

and symmetric probability density function, and satisfies





uKudu





.

Assumption 3. The density function





u is

bounded away from zero, and has continuous first de-

rivatives. The function







has bounded partial de-

rivatives up to the order 2 with .



inf 0





Assumption 4.











u and 2



u are twice

continuously differentiable. Furthermore, we assume that











, 1, ,kp



, and



u is a positive defi-

nite matrix for any given u.

Assumption5. The error



and covariate X satisfy





supuEUu





 and



supuEX U u



 ,

respectively, where



denotes the Euclidean distance.

Under these assumptions, the following theorem gives

the asymptotic distribution of .





Theorem 1. Suppose that Assumptions 1-5 hold. For

any given u, if







is the true value of the parameter,

then













where “D





2” denotes the convergence in distribution

and “



” denotes the chi-square distribution with p de-

grees of freedom.

By Theorem 1, the 1





confidence interval for







can be defined as







 







,Cu uRu









where





satisfies













 . In addition, to

P. X. ZHAO

implement this estimation procedure, we need to choose

the bandwidth h. One can select h by optimizing some

data driven criteria, such as the classical criteria CV,

GCV and BIC. For the facilitation of calculation, we

suggest to choose the bandwidth based on the CV criteria.

More specifically, we can estimate h by minimizing the

following cross-validation score





[]

CV ,

iiiii

hYXU









where is the estimator of after deleting

the ith subject. From our simulation experience, we

found that such a choice of the bandwidth is workable.



[]

ˆiu







Next we give the proof Theorem 1. The proof the

Theorem 1 relies on the following lemma.

Lemma 1. Under the assumptions 1-5, we have



 



1ˆ0, ,

uNvuu









where and

 

vufuKsds

 



uE XEXZUu















 







Proof. From the definition of in (3), it is

easy to show that





ˆiu







 

1ˆ

iiiih i

niui hi

YX uKUu

KU u

















 











(6)

Then, similar to the proof of Theorem 4 in Wang (2009),

we can prove that

 



11,

niiihip

niii ihi

AXKUuo

EXZ K Uu





























Hence, using the central limit theorem, we have







0, ,



Nvu u (7)







0, ,



Nvu u (8)

where













vufuKsds,

 



uEX Uu











 







and

 



uE EXZUu















 









. Finally,

this lemma follows immediately by (6) - (8).

Proof of Theorem 1. Together with the proof of

Lemma 1 an d using the s ame argu ment as ar e used in the

proof of Lemma 1 in [7], we can show that





1ˆ

max .

in uonh



 



(9)

Similar to the proof of (2.14) in [6], we can prove that





12 .

Onh









(10)

Then, invoking (9) and (10), and applying the Taylor

expansion to (4), it is easy to show that















ˆˆ

nTT



  









1.



(11)

Furthermore, from (5) and invoking (9) and (10), we

can prove that



ˆˆ ˆ

ii i

onh

 

















(12)



ˆˆ

  









 (13)

Using (11)-(13), we obtain that



1ˆ

Ruu u

























(14)

where

 





ˆˆˆ

 







u. Invoking the

proof of the Lemma 1 and using the law of large numbers,

we obtain that





ˆ.

uvu



This together with (14) and Lemma 1 yields Theorem 1.

3. Simulation Studies

In this section, some Monte Carlo simulations are con-

ducted to evaluate the finite sample performance of the

proposed empirical likelihood method. The data are gen-

erated from the following model



,YXu









where









sin 2uu





, the covariates U and X are

generated according to and



~0,1UU







~0,1XN ,

P. X. ZHAO 47

respectively. The response Y is generated according the

model with . In the following simulation

procedure, we choose the following two missing data

mechanism:



~0,0.5N







Case1:



exp 10.50.450.5exp 10.50.45



 u

Case 2:











exp1 0.50.451exp1 0.50.45

yu yu





The average missing rates of these two cases are 0.15

and 0.25 respectively. For each case, we take 1000 simu-

lation runs. In addition, the sample size is taken as n =

200.

For comparison, we consider two methods for construct-

ing the confiden ce intervals: the imputed estimation method

(IEL) proposed by this paper, and the naïve empirical

likelihood method (NEL). The latter is neglecting the

incomplete data information, and constructing the confi-

dence intervals for the regression coefficients only based

on the complete data. The averages of the confid ence in t er -

vals with the nominal level

1 95%,





 computed with

1000 simulation runs, are summarized in Figures 1 and 2.

Figure 1 is the simulation results under the missing

mechanism Case 1, and Figure 2 is the simulation results

under the missing mechanism Case 2, where the dashed

curves mean the results obtained by IEL method, the

dotted curves mean the results obtained by NEL method,

and the solid curve represents the real curve of







From Figures 1 and 2, we can make the following ob-

servations:

(i) The confidence intervals based on the I EL method

outperform those based on the NEL method, because

lengths of the confidence intervals obtained by the IEL

method are shorter than those obtained by the NEL method.

(ii) The performances of the confidence intervals based

on the IEL method are similar for all levels of missing

mechanisms. This implies that the imputed empirical

likelihood procedure can attenuate the effect of missing

Figure 1. The 95% confidence intervals of θ(u) under the

missing mechanism Case 1 based on IEL method (dashed

curve) and NEL method ( dotted curve).

Figure 2. The 95% confidence intervals of θ(u) under the

missing mechanism Case 2 based on IEL method (dashed

curve) and NEL method ( dotted curve).

data.

4. Conclusions and Discussions

We have proposed an imputed empirical likelihood pro-

cedure for varying coefficient models when some covari-

ates are missing. The proposed method can attenuate the

effect of missing data efficiently, and extends the impu-

tation-based estimation method to the varying coefficient

models with missing covariates. Simulation studies indi-

cated that the proposed method was very effective in at-

tenuating the effect of missing data and constructing the

confidence intervals for the coefficient functions.

In this paper, although we assume that all components

of the covariate are subject to missing, it is not essential.

The proposed estimation method can easily extend the

case that only some components of the covariate are

measured with missing. In addition, one useful extension

of the varying coefficient model is the varying coeffi-

cient partially linear model. For such model, Zhao and

Xue [8] considered the statistical inferences for regres-

sion coefficients when the response with missing. Then,

another interesting topic of further research is investigat-

ing the inferences for such varying coefficient partially

linear models with missing covariates.

REFERENCES

[1] Q. H. Wang and J. N. K. Rao, “Empirical Likelihood for

Linear Models under Imputation for Missing Responses,”

The Canadian Journal of Statistics, Vol. 29, No. 4，2001,

pp. 597-608 . doi：10.2307/3316009

[2] L. G. Xue, “Empirical Likelihood for Linear Models with

Missing Responses,” Journal of Multivariate Analysis,

Vol. 100, No. 7,2009, pp. 1353-1366.

doi：10.1016/j.jmva.2008.12.009

[3] Q. H. Wang, “Statistical Estimation in Partial Linear

Models with Covariate Data Missing at Random,” Annals

of the Institute of Statistical Mathematics, Vol. 61, No. 1,

2009, pp. 47-84.

doi:10.1007/s10463-007-0137-1

[4] H. Liang, S. J. Wang, J. M. Robbins and R. J. Carroll,

“Estimation in Partially Linear Models with Missing

P. X. ZHAO

Covariates,” Journal of the American Statistical

Association, Vol. 99, No. 466, 2004, pp. 357-367.

doi：10.1198/016214504000000421

[5] H. Wong, S. J. Guo, M. Chen and W.C. IP, “On Locally

Weighted Estimation and Hypothesis Testing of Varying

Coefficient Models with Missing Covariates,” Journal of

Statistical Planning and Inference, Vol. 139, No. 9, No. 1,

2009, pp. 2933-2951. doi：10.1016/j.jspi.2009.01.016

[6] A. B. Owen, “Empirical Likelihood Ratio Confidence

Regions,” The Annals of Statistics, Vol. 18, No. 1, 1990,

pp. 90-120. doi：10.1214/aos/1176347494

[7] L. G. Xue and L. X. Zhu, “Empirical Likelihood for a

Varying Coefficient Model with Longitudinal Data,”

Journal of the American Statistical Association, Vol. 102,

No. 478, 2007, pp. 642-654.

doi：10.1198/016214507000000293

[8] P. X. Zhao and L. G. Xue, “Variable Selection for

Semiparametric Varying Coefficient Partially Lin-

ear Models with Missing Response at Random,”

Acta Mathematica Sinica, English Series, Vol. 27,

No. 11, 2011, pp. 2 205-2216.

doi：10.1007/s10114-011-9200-1