Adjusted Empirical Likelihood for Varying Coefficient Partially Linear Models with Censored Data

By constructing an adjusted auxiliary vector ingeniously, we propose an adjusted empirical likelihood ratio function for the parametric components of varying coefficient partially linear models with censored data. It is shown that its limiting distribution is standard central chi-squared. en the con�dence intervals for the parametric components are constructed. A simulation study and a real data analysis are undertaken to assess the �nite sample performance of the proposed method.

e con�dence intervals obtained by the empirical likelihood method possess several attractive features compared to the conventional Wald-type con�dence intervals, such as circumvention of asymptotic variance estimation to compute the standard error, �exible shapes of the con�dence intervals determined by data and range-preserving property (see [9]).Wang and Li [2] considered the adjusted empirical likelihood inferences for a class of partially linear models with right censored data.Yang et al. [3] considered the adjusted empirical likelihood inferences for a partially linear single-index model with right censored data.is paper also contributes to the rapidly growing literature on the adjustment-based empirical likelihood method and provides additional positive results for the Wilks phenomena in the parametric components of varying coefficient partially linear models with censored data, which extends the application literature of the adjusted empirical likelihood method.e more works on empirical likelihood methods can be found in [10,11].

Methodology and Results
As in [3] where ( *  |   ,   ,   ) = 0.When () is known as well, model ( 2) is a standard semiparametric varying coefficient partially linear model.For given , using the same arguments as Fan and Huang [7], we can get the weighted local leastsquares estimator of () by minimizing where  ℎ () = ℎ −1 (/ℎ), () is a kernel function, ℎ is a bandwidth, and   denotes the th component of   .Let  = ( 1 , … ,   )  , let  *  = ( * 1 , … ,  *  )  , let   be    identity matrix, let 0  be    zero matrix, let Ω  = diag( ℎ (− 1 ), … , let  ℎ (−  )) be  diagonal matrix.en, the solution to minimize (3) can be given by where By (5) we have that  is true if and only if  η () = (1).�sing such information, we can de�ne a pro�le empirical log-likelihood ratio function for .However, () is usually unknown, and η () cannot be used directly to make inference for .To solve this problem, we replace () in η () by its estimator.In this paper, we employ the Kaplan-Meier estimator where  + () =   =1 ( *  > ).Hence, an estimator of η () can be de�ned as where Y *   =  *   −  (  )    and  () =   =1   () *   .en, we can de�ne the pro�le empirical log-likelihood ratio function for  as where   =   (),  = 1, … , .A unique value for  () exists, provided that 0 is inside the convex hull of the points (  1 (), … ,    ()).Based on the method of Lagrange multipliers to �nd the optimal   , then the empirical log-likelihood ratio function can be represented as where  = () is a   1 vector, which satis�es Let e following theorem gives the asymptotic distribution of  ().
From eorem 1, we can see that the asymptotic distribution of  () is a mixture of central chi-square distributions.Hence, the con�dence regions for the parametric components can be constructed if the unknown weights are estimated.Next, we give an adjusted empirical log-likelihood ratio function that has an asymptotic standard chi-square distribution.en the con�dence regions for the parametric components are constructed, and the estimation for the unknown weights is avoided.Let   (⋅) be the Kapaln-Meier estimator for the distribution function of , and en, as in [3], an adjusted empirical log-likelihood ratio function can be de�ned by where   () =   Σ −1 ()  ()   Σ −1 0 ()  ().e following theorem shows that the asymptotic distribution of   ad () can be approximated by standard chi-square distribution.eorem 2. Suppose that conditions (C1)-(C5) in the Appendix hold.If  0 is the true parameter, then Let  2  (1 − ) be the 1 −  quantile of  2  , (0    1).eorem 2 implies that an approximate 1 −  con�dence region for  can be de�ned by

Numerical Results
In this section, we conduct several simulation experiments to illustrate the �nite sample performances of the proposed method and consider a real data set analysis for further illustration.

Simulation Studies.
To evaluate the performance of the proposed adjustment-based empirical likelihood (AEL) method, we present some simulation results.In the simulation, we simulated data from the following model: where  = 1, () = 2 For comparison, we consider two approaches for constructing the con�dence intervals: the AEL method proposed in this paper and normality approximation (NA) method proposed by [12].To construct con�dence intervals based on the NA method, we need to estimate the asymptotic variance.However, note that the asymptotic variance of the estimator has a complicated structure; then we estimate the asymptotic variance by the bootstrap method.e average lengths and coverage probabilities of the con�dence intervals are shown in Table 1.Here the nominal level is taken as 1 −  = , and the simulation is computed with 1000 simulation runs.From Table 1, we can see the following observations.(i) For any given level of CR and the size of sample, although the coverage probabilities of the con�dence intervals obtained by the AEL method and the NA method are similar, the lengths of the con-�dence intervals obtained by the AEL method are shorter than those obtained by the NA method.is implies that the con�dence intervals obtained by the AEL method outperform those obtained by the NA method.
(ii) For the given size of sample , the performances of the con�dence intervals, obtained by the AEL method under the moderate censoring rate, are all close to the nominal level 95.is implies that the adjustment scheme is workable for the moderate censoring rate.However, the simulation results also suggest that larger samples would be needed when the censoring rate is relatively high such as CR = 0.45.

Application to CGD Data.
We illustrate the applicability of our proposed method using the chronic granulomatous disease (CGD) data set from the International CGD Cooperative Study Group.is data set was designed to have a single interim analysis when the follow-up data as of July 15, 1989 were complete.e monitoring committee for the trial terminated the trial at a meeting on September 22, 1989.e treatment given each patient was unblinded at the �rst scheduled visit for the patient following the decision of the monitoring committee.More details for this data description and analysis can be seen in [13].e variables contained here are : treatment code, 1 = rIFN, 2 = placebo;  1 : pattern of inheritance, 1 = linked, 2 = autosomal recessive;  2 : age, in years;  3 : height, in cm;  4 : weight, in kg;  5 : using corticosteroids at time of study entry, 1 = yes, 2 = no;  6 : using prophylactic antibiotics at time of study entry, 1 = yes, 2 = no;  7 : 1 = male, 2 = female;  8 : hospital category, 1 = US-NIH, 2 = US-other, 3 = Europe, Amsterdam, 4 = Europe, other; : elapsed time (in days) from randomization to diagnosis of a serious infection or if a censored observation: elapsed time from randomization to censoring date; : censoring indicator, 1 = Noncensored observation, 2 = censored observation.: sequence number.For each patient, the infection records are in sequence number order.
Based on the generalized likelihood ratio testing, Jiang and Qian [14] show that this data set can be �tted by the following model: where  = (),  = 20, and  0 = 1 as the intercept term.e estimators and 95% con�dence intervals are shown in Table 2. From Table 2, we see that the covariates  1 ,  5 , and  7 have somewhat stronger associations with the response.
In addition, we also can see that the con�dence intervals obtained by the AEL method are workable.