Block Empirical Likelihood for Semiparametric Varying-Coefficient Partially Linear Errors-in-Variables Models with Longitudinal Data

Block empirical likelihood inference for semiparametric varying-coeffcient partially linear errors-in-variables models with longitudinal data is investigated. We apply the block empirical likelihood procedure to accommodate the within-group correlation of the longitudinal data. The block empirical log-likelihood ratio statistic for the parametric component is suggested. And the nonparametric version of Wilk’s theorem is derived under mild conditions. Simulations are carried out to access the performance of the proposed procedure.

In this paper, we consider models (1) and ( 2) with longitudinal data; one aim of this paper is to construct the confidence region for the parameter components.To achieve it, we apply the block empirical likelihood approach [20] to construct block empirical log-likelihood ratio statistic for parameter  and then prove nonparametric Wilk's phenomenon.Simulation studies assess the proposed method.The other aims are to prove that the maximum empirical likelihood estimator (MELE) for the parameter is asymptotically normal under some suitable conditions.
The rest of this paper is organized as follows.In Section 2, we construct the block empirical likelihood based confidence region for the parametric components.Assumption conditions and main results are given in Section 3. Simulation results are reported in Section 4. The proofs of the main results are stated in Section 5. Finally, some concluding remarks are given.

Methodology
In this section, we are to extend the result of Hu [21] to the semivarying coefficient errors-in-variables model with longitudinal data.
Suppose that  is known; then, model (3) can be reduced to a varying-coefficient regression model: Here, the local linear regression method is applied to estimate the coefficient function {  (⋅),  = 1, . . ., } in model (4).That is, for  in a small neighborhood of  0 , one can approximate   (⋅) locally by a linear function where    () =   ()/.This leads to the following weighted least-squares problem: find {(  ,   ),  = 1, . . ., } to minimize where  is a kernel function,  ℎ (⋅) = (⋅/ℎ)/ℎ, and ℎ is a bandwidth.Let Then, the solution to problem ( 6) is given by (â 1 () , . . ., â () , ℎ b1 () , . . ., ℎ b ()) Then, θ() can be given by where   is  ×  identity matrix and 0  is  ×  zero matrix.Denote Substituting ( 12) into (4), we can obtain the approximate residuals as the following: where Similar to Owen [10], {r  (),  = 1, . . ., ;  = 1, . . .,   } can be treated as a random sieve approximation of the random error sequence {  ,  = 1, . . ., ;  = 1, . . .,   }.In order to deal with the correlation within group, we use the block empirical likelihood method.The block empirical likelihood procedure takes the "data" r (),  = 1, . . .,   into account as a whole.Hence, similar to Xue and Zhu [13], we introduce the auxiliary random vector Following (13), if  is true, then {η  ()} =  (1).If one ignores the measurement error and replaces   by   in η (), one can show that the resulting estimator is inconsistent.As we all know, inconsistency caused by the measurement error can be overcome by applying the socalled correction for attenuation proposed by Fuller [22] in linear regression.With a similar way as in Zhao and Xue [14], the corrected-attenuation auxiliary vector is introduced and defined as where aims to avoid the underestimating for the parameter caused by the measurement error.Therefore, the empirical likelihood ratio function for  is defined as A unique value for R() exists, provided that 0 is inside the convex hull of the point ( η 1 (), . . ., η  ()).Using the Lagrange multiplier technique, the optimal value for   is where  = ( 1 , . . .,   )  is the solution of the equation Then, the block empirical log-likelihood ratio function is In addition, by maximizing LR(), we can obtain the maximum empirical likelihood estimator (MELE) β .Let If the matrix Γ is invertible, then the MELE of  can be given by According to β , we can define the estimator {  (⋅),  = 1, . . ., } as

Theorem 1. Assume that the Assumptions 1-6 hold; if 𝛽 is the true value of the parameter, then
where D  → denotes the convergence in distribution and  2  is a chi-square distribution with  degrees of freedom.
Then, we can construct the confidence regions for the parameter .More precisely, for any 0 <  < 1, let   be such that ( 2  >   ) ≤ 1 − .Then, constitute a confidence region for  with asymptotic coverage 1 − .
Theorem 2. Assume that the Assumptions 1-6 hold.Then, one has where
In the simulation studies, for each combination of   , and , we draw 1,000 random samples of sizes 100 or 200 from the above model, respectively.For each sample, a 95% confidence interval for  = 1.5 is computed using our block empirical likelihood method.The kernel function is taken as the Gauss kernel  ℎ () = (1/ √ 2ℎ) exp(−() 2 /2ℎ 2 ).The "leave-onesample-out" method is used to select the bandwidth ℎ.We define the score of ℎ as follows: Then cross-validation smoothing parameter ℎ CV is the minimizer of CV(ℎ).Some representative coverage probabilities are reported in Table 1.

Proof of the Main Results
In order to prove the main results, we first introduce several lemmas.Let ) . (30) Proof.This lemma can be found in Mack and Silverman [23].
Proof of Theorem 1. From (36), using the same arguments as were used in the proof of Owen [10], we have where  is defined in (19).Then, we have size