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

Abstract

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.

1. Introduction

For longitudinal data, we consider semiparametric varying-coefficient partially linear model which has the following form: 𝑌(𝑡)=𝑋(𝑡)^𝜏𝛽+𝑍(𝑡)^𝜏𝜃(𝑡)+𝜖(𝑡), (1) where 𝑌(𝑡) is the response variable, 𝑋, 𝑍, and 𝑡 are regressors, 𝛽=(𝛽₁,…,𝛽_𝑝)^𝜏 is a 𝑝-dimensional vector of unknown parameters, 𝜃(𝑡)=(𝜃₁(𝑡),…,𝜃_𝑞(𝑡))^𝜏 is a 𝑞-dimensional vector of smooth functions of time 𝑡, and 𝜖(𝑡) is a zero-mean stochastic process. Due to the curse of dimensionality, for simplicity, we assume that 𝑡 is univariate.

Obviously, model (1) contains many usual parametric, nonparametric, and semiparametric models. Model (1) has been studied by many authors. Zhang et al. [1] suggested a two-step method for estimating it. Li et al. [2] suggested a local least-squares procedure with a kernel weight function. Fan and Huang [3] developed a profile least-squares technique for estimating parametric model. You and Zhou [4] and Huang and Zhang [5] suggested the estimator of the parametric and nonparametric models, respectively. Fan et al. [6] proposed a semiparametric estimation of the working correlation matrix and applied a profile weighted least-squares approach.

However, in many practical situations, these variables are often measured with error. In this paper, we consider this case where the variable 𝑋(𝑡) is measured with additive error and both 𝑍(𝑡) and 𝑡 are measured exactly. That is, 𝑋(𝑡) cannot be observed, but an unbiased measure of 𝑋(𝑡), denoted by 𝑊(𝑡), can be obtained as follows:𝑊(𝑡)=𝑋(𝑡)+𝑈(𝑡), (2) where 𝑈(𝑡) is the measurement error, which is independent of (𝑋^𝜏(𝑡), 𝑍^𝜏(𝑡), 𝜖(𝑡), 𝑡), with mean zero and covariance matrix Σ_𝑢𝑢. We can assume that Σ_𝑢𝑢 is known. If Σ_𝑢𝑢 is unknown, we estimate it by repeatedly measuring 𝑊(𝑡) by Liang et al. [7]. For errors-in-variables models (1) and (2), Liang et al. [8] developed a profile least-squares procedure to estimate the parametric component and derived the asymptotic normality of the resulting estimator.

The empirical likelihood, which is a nonparametric approach for constructing confidence regions, was introduced by Owen [9] and has many nice statistical properties (see Owen [10]). Owen [11] applied empirical likelihood to linear regression models and Kolaczyk [12] made further extensions to generalized linear models. Recently, Xue and Zhu [13] considered the varying coefficient models. You and Zhou [4], Huang and Zhang [5], and Zhao and Xue [14] investigated the empirical likelihood confidence regions for varying-coefficient partially linear models. Other related papers contain Yang and Li [15], Hu et al. [16], Wang et al. [17], and Fan et al. [18, 19].

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 𝛽 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.

2. Methodology

In this section, we are to extend the result of Hu [21] to the semivarying coefficient errors-in-variables model with longitudinal data.

We apply longitudinal data (𝑌_𝑖(𝑡_𝑖𝑗), 𝑋_𝑖(𝑡_𝑖𝑗), 𝑍_𝑖(𝑡_𝑖𝑗), 𝑡_𝑖𝑗). 𝑖=1,…,𝑛, and 𝑗=1,…,𝑛_𝑖 which are generated from semivarying coefficient errors-in-variables model through the following equation: 𝑌_𝑖𝑗=𝑋^𝜏_𝑖𝑗𝛽+𝑍^𝜏_𝑖𝑗𝜃𝑡_𝑖𝑗+𝜖_𝑖𝑗,𝑊_𝑖𝑗=𝑋_𝑖𝑗+𝑈_𝑖𝑗, (3) where 𝑌_𝑖𝑗=𝑌_𝑖(𝑡_𝑖𝑗), 𝑋_𝑖𝑗=𝑋_𝑖(𝑡_𝑖𝑗), 𝑍_𝑖𝑗=𝑍_𝑖(𝑡_𝑖𝑗) and 𝜖_𝑖𝑗=𝜖_𝑖(𝑡_𝑖𝑗), and 𝑈_𝑖𝑗=𝑈_𝑖(𝑡_𝑖𝑗), 𝑖=1,…,𝑛, 𝑗=1,…,𝑛_𝑖. We use counting process 𝑁_𝑖(𝑡)≡∑^𝑛𝑖_𝑗=1𝐼(𝑡_𝑖𝑗≤𝑡) to describe the number of observations of the 𝑖th subject. We assume that 𝑛_𝑖 is bounded, but the number of subjects 𝑛 goes to infinity.

Suppose that 𝛽 is known; then, model (3) can be reduced to a varying-coefficient regression model: 𝑌_𝑖𝑗−𝑋^𝜏_𝑖𝑗𝛽=^𝑞_𝑘=1𝑍_𝑖𝑗𝑘𝜃_𝑘𝑡_𝑖𝑗+𝜖_𝑖𝑗,𝑖=1,…,𝑛,𝑗=1,…,𝑛_𝑖. (4) Here, the local linear regression method is applied to estimate the coefficient function {𝜃_𝑗(⋅),𝑗=1,…,𝑞} in model (4). That is, for 𝑡 in a small neighborhood of 𝑡₀, one can approximate 𝜃_𝑗(⋅) locally by a linear function 𝜃_𝑗(𝑡)≈𝜃_𝑗𝑡₀+𝜃^′_𝑗𝑡₀𝑡−𝑡₀≡𝑎_𝑗+𝑏_𝑗𝑡−𝑡₀,𝑗=1,…,𝑞, (5) where 𝜃^′_𝑗(𝑡)=𝜕𝜃_𝑗(𝑡)/𝜕𝑡. This leads to the following weighted least-squares problem: find {(𝑎_𝑗,𝑏_𝑗),𝑗=1,…,𝑞} to minimize ^𝑛_𝑖=1¹₀𝑌_𝑖(𝑠)−𝑋^𝜏_𝑖(𝑠)𝛽−^𝑞_𝑘=1{𝑎_𝑘+𝑏_𝑘(𝑠−𝑡)}𝑍_𝑖𝑘(𝑠)²×𝐾_ℎ(𝑠−𝑡)𝑑𝑁_𝑖(𝑠), (6) where 𝐾 is a kernel function, 𝐾_ℎ(⋅)=𝐾(⋅/ℎ)/ℎ, and ℎ is a bandwidth. Let Ω_𝑡=diag𝐾_ℎ𝑡₁₁−𝑡,…,𝐾_ℎ𝑡_1𝑛1−𝑡,…,𝐾_ℎ𝑡_𝑛𝑛𝑛𝑌−𝑡𝑌=₁𝑡₁₁,…,𝑌₁𝑡_𝑛1,…,𝑌_𝑛𝑡_𝑛𝑛^𝜏,𝑋=𝑋^𝜏₁,…,𝑋^𝜏_𝑛^𝜏,𝑍𝑍=^𝜏11⋮𝑍^𝜏_1𝑛1⋮𝑍^𝜏_𝑛𝑛𝑛=𝑍₁𝑡₁₁⋯𝑍₁𝑡_1𝑞𝑍⋮⋱⋮₁𝑡_𝑛11⋯𝑍₁𝑡_𝑛1𝑞𝑍⋮⋱⋮_𝑛𝑡_𝑛𝑛1⋯𝑍_𝑛𝑡_𝑛𝑛𝑞,𝜖=𝜖₁₁,…,𝜖_1𝑛1,…,𝜖_𝑛𝑛𝑛^𝜏, (7)𝐷_𝑡=𝑍₁₁⋯𝑍_1𝑛1⋯𝑍_𝑛𝑛𝑛𝑡₁₁−𝑡ℎ𝑍₁₁⋯𝑡_1𝑛1−𝑡ℎ𝑍_1𝑛1⋯𝑡_𝑛𝑛𝑛−𝑡ℎ𝑍_𝑛𝑛𝑛^𝜏. (8)

Then, the solution to problem (6) is given by 𝑎₁(𝑡),…,𝑎_𝑞(𝑡),ℎ𝑏̂₁(𝑡),…,ℎ𝑏̂_𝑞(𝑡)^𝜏=𝐷^𝜏_𝑡Ω_𝑡𝐷^𝜏_𝑇⁻¹𝐷^𝜏_𝑡Ω_𝑡𝑌−𝑋^𝜏𝛽. (9) Then, 𝜃(𝑡) can be given by 𝜃(𝑡)=𝐼_𝑞,0_𝑞𝐷^𝜏_𝑡Ω_𝑡𝐷^𝜏_𝑇⁻¹𝐷^𝜏_𝑡Ω_𝑡𝑌−𝑋^𝜏𝛽, (10) where 𝐼_𝑞 is 𝑞×𝑞 identity matrix and 0_𝑞 is 𝑞×𝑞 zero matrix. Denote 𝐼_𝑞,0_𝑞𝐷^𝜏_𝑡Ω_𝑡𝐷^𝜏_𝑇⁻¹𝐷^𝜏_𝑡Ω_𝑡≡𝑆₁₁(𝑡),…,𝑆_𝑛𝑛𝑛(𝑡); (11) then, 𝜃𝑡_𝑖𝑗=^𝑛𝑛_𝑙=1^𝑙_𝑚=1𝑆_𝑙𝑚𝑡_𝑖𝑗𝑌_𝑙𝑚−𝑋^𝜏_𝑙𝑚𝛽. (12) Substituting (12) into (4), we can obtain the approximate residuals as the following: 𝑟̂_𝑖𝑗𝛽=𝑌_𝑖𝑗−𝑋^𝜏_𝑖𝑗𝛽−𝑍^𝜏𝑛_𝑖𝑗^𝑛_𝑙=1^𝑙_𝑚=1𝑆_𝑙𝑚𝑡_𝑖𝑗𝑌_𝑙𝑚−𝑋^𝜏_𝑙𝑚𝛽=𝑌_𝑖𝑗−𝑋^𝜏_𝑖𝑗𝛽, (13) where 𝑌_𝑖𝑗=𝑌_𝑖𝑗−𝑍^𝜏𝑛_𝑖𝑗^𝑛_𝑙=1^𝑙_𝑚=1𝑆_𝑙𝑚𝑡_𝑖𝑗𝑌_𝑙𝑚,𝑋_𝑖𝑗=𝑋_𝑖𝑗−^𝑛𝑛_𝑙=1^𝑙_𝑚=1𝑆_𝑙𝑚𝑡_𝑖𝑗𝑋_𝑙𝑚^𝜏𝑍_𝑖𝑗. (14)

Similar to Owen [10], {𝑟̂_𝑖𝑗(𝛽),𝑖=1,…,𝑛;𝑗=1,…,𝑛_𝑖} can be treated as a random sieve approximation of the random error sequence {𝜖_𝑖𝑗,𝑖=1,…,𝑛;𝑗=1,…,𝑛_𝑖}. In order to deal with the correlation within group, we use the block empirical likelihood method. The block empirical likelihood procedure takes the ‘‘data” 𝑟̂_𝑖𝑗(𝛽),𝑗=1,…,𝑛_𝑖 into account as a whole. Hence, similar to Xue and Zhu [13], we introduce the auxiliary random vector 𝜂_𝑖𝛽=¹₀𝑋_𝑖(𝑡)𝑌_𝑖(𝑡)−𝑋^𝜏_𝑖(𝑡)𝛽𝑑𝑁_𝑖(𝑡). (15)

Following (13), if 𝛽 is true, then 𝐸{𝜂_𝑖(𝛽)}=𝑜(1). If one ignores the measurement error and replaces 𝑋_𝑖𝑗 by 𝑊_𝑖𝑗 in 𝜂_𝑖(𝛽), 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 so-called 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 𝜂̌_𝑖𝛽=¹₀𝑊_𝑖(𝑡)𝑌_𝑖(𝑡)−𝑊^𝜏_𝑖(𝑡)𝛽+Σ_𝑢𝑢𝛽𝑑𝑁_𝑖(𝑡), (16) where 𝑊_𝑖𝑗=𝑊_𝑖𝑗−(∑^𝑛_𝑙=1∑^𝑛𝑙_𝑚=1𝑆_𝑙𝑚(𝑡_𝑖𝑗)𝑊_𝑙𝑚)^𝜏𝑍_𝑖𝑗. The term Σ_𝑢𝑢𝛽 aims to avoid the underestimating for the parameter caused by the measurement error. Therefore, the empirical likelihood ratio function for 𝛽 is defined as ℛ𝛽=max^𝑛_𝑖=1𝑛𝑝_𝑖∣𝑝_𝑖≥0,^𝑛_𝑖=1𝑝_𝑖=1,^𝑛_𝑖=1𝑝_𝑖𝜂̌_𝑖𝛽=0. (17) A unique value for ℛ(𝛽) exists, provided that 0 is inside the convex hull of the point (𝜂̌₁(𝛽),…,𝜂̌_𝑛(𝛽)). Using the Lagrange multiplier technique, the optimal value for 𝑝_𝑖 is 𝑝_𝑖=1𝑛1+𝜆^𝜏𝜂̌_𝑖(𝛽)⁻¹,𝑖=1,…,𝑛, (18) where 𝜆=(𝜆₁,…,𝜆_𝑛)^𝜏 is the solution of the equation 1𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽1+𝜆^𝜏𝜂̌_𝑖𝛽=0. (19) Then, the block empirical log-likelihood ratio function is ℒℛ𝛽=−2logℛ𝛽=2^𝑛_𝑖=1log1+𝜆^𝜏𝜂̌_𝑖𝛽. (20) In addition, by maximizing ℒℛ(𝛽), we can obtain the maximum empirical likelihood estimator (MELE) 𝛽̌. Let Γ̌=1𝑛^𝑛_𝑖=1¹₀𝑊_𝑖(𝑡)𝑊^𝜏_𝑖(𝑡)−Σ_𝑢𝑢𝑑𝑁_𝑖(𝑡). (21) If the matrix Γ̌ is invertible, then the MELE of 𝛽 can be given by 𝛽̌=Γ̌⁻¹1𝑛^𝑛_𝑖=1¹₀𝑊_𝑖(𝑡)𝑌_𝑖(𝑡)+𝑜_𝑝𝑛^−1/2. (22) According to 𝛽̌, we can define the estimator {𝜃_𝑗(⋅),𝑗=1,…,𝑞} as 𝜃̌(𝑡)=𝜃̌₁(𝑡),…,𝜃̌_𝑞(𝑡)^𝜏=𝐼_𝑞,0_𝑞𝐷^𝜏_𝑡Ω_𝑡𝐷^𝜏_𝑇⁻¹𝐷^𝜏_𝑡Ω_𝑡𝑌−𝑋^𝜏𝛽̌. (23)

3. Main Results

To establish asymptotic properties of the block empirical log-likelihood ratio, we make the following assumptions. These assumptions are made by You and Zhou [4]. We use ‖⋅‖ to denote the Euclidean norm with ‖𝑎‖=(𝑎²₁+⋯+𝑎²_𝑛)^1/2 and 𝑎=(𝑎₁,…,𝑎_𝑛)^𝜏.

Assumption 1. The random variable 𝑡 has a compact support Ξ. The density function 𝑓(⋅) of 𝑡 has a continuous second derivative and is uniformly bounded away from zero.

Assumption 2. The 𝑝×𝑝 matrix 𝐸(𝑋𝑋^𝜏∣𝑡) is nonsingular for each 𝑡∈Ξ. 𝐸(𝑋𝑋^𝜏∣𝑡), 𝐸(𝑋𝑋^𝜏∣𝑡)⁻¹, and 𝐸(𝑋𝑍^𝜏∣𝑡) are all Lipschitz continuous.

Assumption 3. There is a 𝑠>2 such that 𝐸‖𝑋‖^2𝑠<∞, 𝐸‖𝑍‖^2𝑠<∞, 𝐸‖𝜖‖^2𝑠<∞, and 𝐸‖𝑡‖^2𝑠<∞ and for some 𝜖<2−𝑠⁻¹ such that 𝑛^2𝜖−1ℎ→∞ as 𝑛→∞.

Assumption 4. 𝛼_𝑗(⋅),𝑗=1,…,𝑞 have the continuous second derivative in 𝑡∈Ξ.

Assumption 5. The kernel 𝐾(⋅) is a symmetric probability density function and is a bounded variation function on its support.

Assumption 6. The bandwidth ℎ satisfies 𝑛ℎ⁸→0 and 𝑛ℎ²/(log𝑛)²→∞ as 𝑛→∞.

The following theorem gives the asymptotic distribution of ℒℛ(𝛽).

Theorem 1. Assume that the Assumptions 1–6 hold; if 𝛽 is the true value of the parameter, then ℒℛ𝛽⟶^𝒟𝜒²_𝑝𝑎𝑠𝑛⟶∞, (24) where −−→^𝒟 denotes the convergence in distribution and 𝜒²_𝑝 is a chi-square distribution with 𝑝 degrees of freedom.

Then, we can construct the confidence regions for the parameter 𝛽. More precisely, for any 0<𝛼<1, let 𝐶_𝛼 be such that 𝑝(𝜒²_𝑝>𝐶_𝛼)≤1−𝛼. Then, ℋ(𝛼)=𝛽∈𝑅^𝑝∶ℒℛ𝛽≤𝐶_𝛼 (25) constitute a confidence region for 𝛽 with asymptotic coverage 1−𝛼.

Theorem 2. Assume that the Assumptions 1–6 hold. Then, one has √𝑛𝛽̌−𝛽⟶^𝒟𝑁0,Γ⁻¹ΣΓ⁻¹, (26) where Γ=𝐸¹₀𝑊(𝑡)−𝜇^𝜏_𝑡𝑍(𝑡)^⊗2𝑑𝑁(𝑡),𝐵=𝐸¹₀[𝑊(𝑡)−𝜇^𝜏_𝑡𝑍(𝑡)]𝜀(𝑡)^⊗2,𝐴^⊗2=𝐴𝐴^𝜏,𝜇(𝑡)=𝐸𝑍^𝜏(𝑡)𝑊(𝑡)∣𝑡⁻¹𝐸𝑍(𝑡)𝑍^𝜏(𝑡)∣𝑡. (27)

4. Simulation Results

In this section, we will conduct some simulations to the empirical likelihood (EL) method. The data are generated from 𝑦_𝑖𝑡_𝑖𝑗=𝑥_𝑖𝑡_𝑖𝑗𝛽+𝑧_1𝑖𝑡_𝑖𝑗𝜃₁𝑡_𝑖𝑗+𝑧_1𝑖𝑡_𝑖𝑗𝜃₂𝑡_𝑖𝑗+𝜖_𝑖𝑡_𝑖𝑗,𝑤_𝑖𝑡_𝑖𝑗=𝑥_𝑖𝑡_𝑖𝑗+𝑢_𝑖𝑡_𝑖𝑗, (28) where 𝑤_𝑖(𝑡)∼𝑁(0,1), 𝛽=1.5, 𝑧_1𝑖(𝑡)∼𝑁(0,1), 𝑧_2𝑖(𝑡)∼𝑁(0,1), 𝑡∼𝑈(0,1), 𝜃₁(𝑡)=sin(2𝜋𝑡), 𝜃₂(𝑡)=cos(2𝜋𝑡), 𝜖_𝑖(𝑡)∼𝑁(0,1), 𝑢_𝑖(𝑡)=𝑏𝑒_𝑖(𝑡_𝑗)+𝑒_𝑖(𝑡_𝑗−1), and 𝑒_𝑖(𝑡)∼𝑁(0,1).

In the simulation studies, for each combination of 𝑛_𝑖, and 𝑏, we draw 1,000 random samples of sizes 100 or 200 from the above model, respectively. For each sample, a 95% confidence interval for 𝛽=1.5 is computed using our block empirical likelihood method. The kernel function is taken as the Gauss kernel 𝐾_ℎ(𝑡)=(1/√2𝜋ℎ)exp(−(𝑡)²/2ℎ²). The “leave-one-sample-out” method is used to select the bandwidth ℎ. We define the score of ℎ as follows:(ℎ)=1CV𝑛^𝑛_𝑖=1¹₀𝑌_𝑖𝑗−𝑊^𝜏_𝑖𝑗𝛽̌_−𝑖−𝑍^𝜏_𝑖𝑗𝜃̌_−𝑖𝑡_𝑖𝑗²−𝛽̌^𝜏_−𝑖Σ_𝑣𝑣𝛽̌_−𝑖𝑑𝑁_𝑖(𝑡). (29) Then cross-validation smoothing parameter ℎ_CV is the minimizer of CV(ℎ). Some representative coverage probabilities are reported in Table 1.

Table 1 

Coverage probabilities (CP) and average lengths (AL) of the confidence intervals for 𝛽=1.5 and 𝜎²=0.2.

5. Proof of the Main Results

In order to prove the main results, we first introduce several lemmas. Let 𝑢_𝑘=∫𝑡^𝑘𝐾(𝑡)𝑑𝑡, 𝑣_𝑘=∫𝑡^𝑘𝐾²(𝑡)𝑑𝑡, 𝑘=0,1,2,4, 𝑐_𝑛=ℎ²+((log𝑛/𝑛)ℎ)^1/2, 𝐺(𝑇)=𝐸(𝑍𝑍^𝜏∣𝑇), Ψ(𝑇)=𝐸(𝑋𝑍^𝜏∣𝑇), and 𝑀=(𝑍^𝜏₁𝜃(𝑡₁),…,𝑍^𝜏_𝑛𝜃(𝑡_𝑛))^𝜏.

Lemma 3. Let (𝑋₁,𝑌₁),…,(𝑋_𝑛,𝑌_𝑛) be i.i.d random vector, where 𝑌_𝑖 is scalar random variable. Further, assume that 𝐸|𝑌₁|^𝑠<∞, sup_𝑥∫|𝑦|^𝑠𝑓(𝑥,𝑦)𝑑𝑦<∞, where 𝑓(⋅,⋅) denotes the joint density of (𝑋,𝑌). Let 𝐾(⋅) be a bounded positive function with a bounded support, satisfying a Lipschitz condition. Given that 𝑛^2𝜖−1ℎ→∞ for some 𝜖<1−𝑠⁻¹, then size sup_𝑥1𝑛^𝑛_𝑖=1𝐾_ℎ𝑋_𝑖−𝑥𝑌_𝑖−𝐸𝐾_ℎ𝑋_𝑖−𝑥Y_𝑖=𝑂_𝑝log(1/ℎ)𝑛ℎ^1/2. (30)

Proof. This lemma can be found in Mack and Silverman [23].

Lemma 4. Let 𝜖_𝑖,𝑖=1,…,𝑛, be a sequence of multi-independent random variate with 𝐸(𝜖_𝑖)=0 and 𝐸(𝜖²_𝑖)<𝑐<∞. Then, max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝜖_𝑖=𝑂_𝑝√𝑛log𝑛. (31) Further, let (𝑗₁,…,𝑗_𝑛) be a permutation of (1,…,𝑛). Then, one has max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝜖_𝑗𝑖=𝑂_𝑝√𝑛log𝑛. (32)

Proof. We can prove this lemma immediately by Kolmogorov inequality.

Lemma 5. Let 𝐷₁,…,𝐷_𝑛 be i.i.d random variables. If 𝐸|𝐷_𝑖|^𝑠 are uniformly bounded for 𝑠>1, then one has max_{1≤𝑖≤𝑛}𝐷_𝑖=𝑜𝑛^1/𝑠. (33)

Proof. This lemma can be found in Shi and Lau [24].

Lemma 6. Suppose that Assumptions 1–6 hold; one has 𝐷^𝜏_𝑡Ω_𝑡𝐷_𝑡=𝑁𝑓(𝑡)Γ⊗𝐺(𝑡)1+𝑂_𝑝𝑐_𝑛,𝐷^𝜏_𝑡Ω_𝑡𝑋_t=𝑁𝑓(𝑡)(1,0)^𝜏⊗Ψ(𝑡)1+𝑂_𝑝𝑐_𝑛,𝐷^𝜏_𝑡Ω_𝑡𝑍_𝑡=𝑁𝑓(𝑡)(1,0)^𝜏⊗𝐺(𝑡)1+𝑂_𝑝𝑐_𝑛, (34) which hold for all 𝑡∈[𝑎,𝑏]⊂[0,1], where Γ=diag(1,𝑢₂).

Proof. This follows immediately from the result that was obtained by Yang and Li [15].

Lemma 7. Suppose that Assumptions 1–6 hold; one has, when 𝑡_𝑖𝑗∈[𝑎,𝑏], 𝑆^𝜏_𝑖𝑡_𝑖𝑗𝑋=𝑍^𝜏_𝑖𝑡_𝑖𝑗𝐺⁻¹𝑡_𝑖𝑗Ψ𝑡_𝑖𝑗1+𝑂_𝑝𝑐_𝑛,𝑆^𝜏_𝑖𝑡_𝑖𝑗𝑀=𝑍^𝜏_𝑖𝑡_𝑖𝑗𝜃𝑡_𝑖𝑗1+𝑂_𝑝𝑐_𝑛. (35)

Proof. Let 𝑆^𝜏_𝑖(𝑡_𝑖𝑗)=[𝑍^𝜏_𝑖(𝑡_𝑖𝑗),0](𝐷^𝜏_𝑡𝑖𝑗Ω_𝑡𝑖𝑗𝐷_𝑡𝑖𝑗)⁻¹𝐷^𝜏_𝑡𝑖𝑗Ω_𝑡𝑖𝑗; then, Lemma 7 can be directly attained by Lemma 6.

Lemma 8. Suppose that the Assumptions 1–6 hold, one has 1√𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽^𝒟(),1⟶𝑁0,Σ𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽𝜂̌^𝜏_𝑖𝛽⟶^𝒫Σ,max_{1≤𝑥≤𝑛}𝜂̌_𝑖𝛽=𝑂_𝑝𝑛^1/2, (36) where Σ is defined by (26).

Proof of Theorem 1. From (36), using the same arguments as were used in the proof of Owen [10], we have ‖𝜆‖=𝑂_𝑝𝑛^−1/2, (37) where 𝜆 is defined in (19). Then, we have size 10=𝑛^𝑛_𝑖=1𝜂̌𝛽1+𝜆^𝜏𝜂̌_𝑖𝛽=1𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽−1𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽𝜂̌^𝜏_𝑖𝛽1𝜆+𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽𝜆^𝜏𝜂̌_𝑖𝛽²1+𝜆^𝜏𝜂̌_𝑖𝛽. (38) By using Lemma 8, we obtain ^𝑛_𝑖=1𝜆^𝜏𝜂̌_𝑖𝛽²=^𝑛_𝑖=1𝜆^𝜏𝜂̌_𝑖𝛽+𝑂_𝑝(1),𝜆=^𝑛_𝑖=1𝜂̌_𝑖(𝛽)𝜂̌^𝜏_𝑖(𝛽)^𝑛−1_𝑖=1𝜂̌_𝑖𝛽+𝑜_𝑝𝑛^−1/2. (39) Applying the Taylor expansion to (20), we get that ℒℛ𝛽=2^𝑛_𝑖=1𝜆^𝜏𝜂̌_𝑖𝛽−12𝜆^𝜏𝜂̌_𝑖𝛽²+𝑜_𝑝(1). (40) Hence, together with (39), we have size 𝛽=1ℒℛ√𝑛^𝑛_𝑖=1𝜂̌_𝑖(𝛽)^𝜏×1√𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽𝜂̌^𝜏_𝑖𝛽⁻¹1√𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽+𝑜_𝑝(1). (41)
Together with Lemma 8, this proves Theorem 1.

Proof of Theorem 2. Following the similar arguments as were used in the proof of Theorem 2 in Yang and Li [15], we have 𝛽̌−𝛽=Γ̌⁻¹1𝑛^𝑛_𝑖=1𝜂̌_𝑖𝛽+𝑜_𝑝𝑛^−1/2. (42) By (35), we can prove Γ̌−−^𝒫→Γ by the law of large numbers. Together with Lemma 8 and Slutsky’s theorem, this proves Theorem 2.