Smoothed empirical likelihood inference via the modified Cholesky decomposition for quantile varying coefficient models with longitudinal data

Abstract

It is essential to deal with the within-subject correlation among repeated measures over time to improve statistical inference efficiency. However, it is a challenging task to correctly specify a working correlation in quantile regression with longitudinal data. In this paper, we first develop an adaptive approach to estimate the within-subject covariance matrix of quantile regression by applying a modified Cholesky decomposition. Then, weighted kernel GEE-type quantile estimating equations are proposed for varying coefficient functions. Note that the proposed estimating equations include a discrete indicator function, which results in some problems for computation and asymptotic analysis. Thus, we construct smoothed estimating equations by introducing a bounded kernel function. Furthermore, we develop a smoothed empirical likelihood method to improve the accuracy of interval estimation. Finally, simulation studies and a real data analysis indicate that the proposed method has superior advantages over the existing methods in terms of coverage accuracies and widths of confidence intervals.

Appendix

To establish the asymptotic properties of the proposed estimators, the following regularity conditions are needed in this paper.

(C1) The bandwidth satisfies $h_1 = {N^{{{ - 1} \big / 5}}}{h_0}$ for some constant $h_0>0$.

(C2) ${\lim _{n \rightarrow \infty }}{N^{{{ - 6} \big / 5}}}\sum \nolimits _{i = 1}^n {m_i^2} = \kappa$ for some $0\le \kappa <\infty$.

(C3) The kernel function $K\left( \cdot \right)$ has a compact support on ${\mathscr {R}}$ and satisfies $\int {K\left( u \right) \hbox {d}u = 1}$, $\int {{K^2}\left( u \right) \hbox {d}u < \infty }$, $\int {{u^2}K\left( u \right) \hbox {d}u < \infty }$, $\int {uK\left( u \right) \hbox {d}u = 0}$ and $\int {{u^4}K\left( u \right) \hbox {d}u < \infty }$.

(C4) There exists a constant $\delta \in \left( {{2 \big / {5,\left. 2 \right] }}} \right.$, and we have ${\sup _t}E\left( {{{\left| {{\psi _\tau }\left( {{\varepsilon _1}\left( t_{ij} \right) } \right) } \right| }^{2 + \delta }}} \left| t_{ij}\right. \right. \left. =t \right) < \infty$ and ${\sup _t}E\left( {{{\left| {{\psi _{\tau h} }\left( {{\varepsilon _1}\left( t_{ij} \right) } \right) } \right| }^{2 + \delta }}} \left| { t_{ij}=t} \right. \right) < \infty$ for all $i=1,\ldots ,n, j=1,\ldots ,m_i, l=1,\ldots ,p$ and $t\in S(f_T)$.

(C5) For all $l, r=1,\ldots ,p$, $\beta _r\left( {{t}} \right)$, $\eta _{lr}\left( {{t}} \right)$ and $f_T\left( {{t}} \right)$ have continuous second derivatives at $t_0$.

(C6) $\left\{ {{{\left( {{Y_{ij}},{\varvec{X}_i^\mathrm{T}}\left( {{t_{ij}}} \right) } \right) }^\mathrm{T}},j = 1,\ldots ,{m_i}} \right\}$ are independently and identically distributed for $i=1,\ldots ,n$. We assume that the dimension p of the covariates ${\varvec{X}_i}\left( {{t_{ij}}} \right)$ is fixed and there is a positive constant M such that $\left| {{X_{il}}\left( t \right) } \right| \le M$ for all t and $i=1,...,n,l=1,...,p$.

(C7) The covariance function $\rho _\varepsilon \left( {{t}} \right)$ is continuous at $t_0$, and $\varvec{\varPhi }\left( {{t_0}} \right) = \left( {{\eta _{lr}}\left( {{t_0}} \right) _{l,r = 1}^p} \right)$ is positive definite matrixes.

(C8) The distribution function $F_{ij}\left( x \right) =p\left( {Y_{ij}} - \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \varvec{\beta } \left( {{t_{ij}}} \right) <x|t_{ij}\right)$ is absolutely continuous, with continuous densities $f_{ij}\left( \cdot \right)$ uniformly bounded, and its first derivative uniformly bounded away from 0 and $\infty$ at the points $0, i=1,\ldots ,n, j=1,\ldots ,m_i$.

(C9) $K_1\left( \cdot \right)$ is a symmetric density function with a bounded support in ${\mathscr {R}}$. For some constant $C_K\ne 0$, $K_1\left( \cdot \right)$ is a $\nu$ th-order kernel, i.e., $\int {{u^j}} K_1\left( u \right) \hbox {d}u = 1$ if $j=0$; 0 if $1 \le j \le \nu - 1$; $C_K$ if $j=\nu$, where $\nu \ge 2$ is an integer.

(C10) The positive bandwidth parameter h satisfies $n{h^{2 \nu }} \rightarrow 0$.

Lemma 1

Suppose that conditions (C2)–(C10) hold and that the bandwidth satisfies $\sup {\lim _{n \rightarrow \infty }}N{h_1^5} < \infty$. If ${\varvec{\beta } \left( {{t_0}} \right) }$ is the true parameter, then ${\max _{1 \le i \le n}}\left\| {{ Z_i}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \right\| = {o_p}\left( {\sqrt{Nh_1} } \right)$ and ${\max _{1 \le i \le n}}\left\| {{ Z_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \right\| = {o_p}\left( {\sqrt{Nh_1} } \right)$, where $\left\| \cdot \right\|$ is the Euclidean norm.

Proof

The proof of Lemma 1 is omitted since it is similar to the proof of Lemma A.1 in Xue and Zhu (2007). $\square$

Lemma 2

Suppose that conditions (C1)–(C8) hold, we have

$$\begin{aligned} \sqrt{Nh_1} \left( {\hat{\varvec{ \beta }} \left( {{t_0};h_1} \right) - \varvec{\beta } \left( {{t_0}} \right) - \varvec{B}\left( {{t_0}} \right) } \right) \mathop \rightarrow \limits ^d N\left( {\varvec{0},\bar{\varvec{ D}}\left( {{t_0}} \right) } \right) , \end{aligned}$$

where $\bar{\varvec{D}}\left( {{t_0}} \right) = {\left( {{f_T}\left( {{t_0}} \right) } {{\bar{f}}}\left( 0 \right) \right) ^{ - 2}}{\upsilon ^2}\left( {{t_0}} \right) {\varvec{\varPhi }^{ - 1}}\left( {{t_0}} \right)$ and ${{\bar{f}}}\left( 0 \right) =\mathop {\lim }\nolimits _{n \rightarrow \infty } {N^{ - 1}}\sum \nolimits _{i = 1}^n {\sum \nolimits _{j = 1}^{{m_i}} {{f_{ij}}\left( 0 \right) } }$, $\varvec{B}\left( {{t_0}} \right)$ and $\varvec{\varPhi }\left( {{t_0}} \right)$ are defined in Theorem 3 and condition (C7), ${\upsilon ^2}\left( {{t_0}} \right) = \tau \left( 1-\tau \right) {f_T}\left( {{t_0}} \right) {\nu _0} + \kappa {h_0}{\rho _\varepsilon }\left( {{t_0}} \right) f_T^2\left( {{t_0}} \right) ,$${h_0}$ and $\kappa$ are defined by conditions (C1) and (C2).

Proof

Because

$$\begin{aligned}&\sum \limits _{i = 1}^n {\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{K}}_i}\left( {{t};{h_1}} \right) {\psi _\tau }\left( {{\varvec{Y}_i-\varvec{X}_i\varvec{\beta }(t)}} \right) }\nonumber \\&\quad = \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{m_i}} {{\psi _\tau }\left( {{Y_{ij}} - \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \varvec{\beta } \left( t \right) } \right) {\varvec{X}_i}\left( {{t_{ij}}} \right) {K_{h_1}}\left( {t - {t_{ij}}} \right) } \nonumber \\&\quad = \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{m_i}} {{\psi _\tau }\left( {{\varepsilon _i}\left( {{t_{ij}}} \right) + \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \varvec{\beta } \left( {{t_{ij}}} \right) - \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \varvec{\beta } \left( t \right) } \right) {\varvec{X}_i}\left( {{t_{ij}}} \right) {K_{h_1}}\left( {t - {t_{ij}}} \right) } \nonumber \\&\quad = \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{m_i}} {{\varvec{X}_i}\left( {{t_{ij}}} \right) {K_{h_1}}\left( {t - {t_{ij}}} \right) \left[ {\tau - I\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right)< \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \left( {\varvec{\beta } \left( t \right) - \varvec{\beta } \left( {{t_{ij}}} \right) } \right) } \right\} } \right] } \nonumber \\&\quad = \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} } {\varvec{X}_i}\left( {{t_{ij}}} \right) {K_{{h_1}}}\left( {t - {t_{ij}}} \right) \left[ {{\psi _\tau }\left( {{\varepsilon _i}\left( {{t_{ij}}} \right) } \right) + I\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right)< 0} \right\} } \right. \nonumber \\&\quad \left. - I\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right) < \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \left( {\varvec{\beta } \left( t \right) - \varvec{\beta } \left( {{t_{ij}}} \right) } \right) } \right\} \right] \nonumber \\&\quad = \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{m_i}} {{\varvec{X}_i}\left( {{t_{ij}}} \right) {K_{h_1}}\left( {t - {t_{ij}}} \right) {\psi _\tau }\left( {{\varepsilon _i}\left( {{t_{ij}}} \right) } \right) } \nonumber \\&\qquad - \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{m_i}} {{\varvec{X}_i}\left( {{t_{ij}}} \right) {K_{h_1}}\left( {t - {t_{ij}}} \right) {\varUpsilon _i}\left( {{t_{ij}}} \right) } \nonumber \\&\quad \buildrel \varDelta \over = \varvec{I} - \varvec{II} , \end{aligned}$$

(A.1)

where ${\varUpsilon _i}\left( {{t_{ij}}} \right) = \left[ {I\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right)< \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \left( {\varvec{\beta } \left( t \right) - \varvec{\beta } \left( {{t_{ij}}} \right) } \right) } \right\} - I\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right) < 0} \right\} } \right]$. In addition, we define

By condition (C8), we have

$$\begin{aligned} \varvec{I}I_1= & {} \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{m_i}} {{\varvec{X}_i}\left( {{t_{ij}}} \right) {K_{h_1}}\left( {t - {t_{ij}}} \right) \left[ {{F_{ij}}\left( {\varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \left( {\varvec{\beta } \left( t \right) - \varvec{\beta } \left( {{t_{ij}}} \right) } \right) } \right) - {F_{ij}}\left( {0} \right) } \right] } \nonumber \\= & {} \sum \limits _{i = 1}^{n}\sum \limits _{j = 1}^{{m_i}}{\varvec{X}_i}\left( {{t_{ij}}} \right) {K_{h_1}}\left( {t - {t_{ij}}} \right) {f_{ij}} \left( 0 \right) \left\{ {1 + {o}\left( 1 \right) } \right\} \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \nonumber \\&\times \left( {\varvec{\beta } \left( t \right) - \varvec{\beta } \left( {{t_{ij}}} \right) } \right) . \end{aligned}$$

(A.2)

By Cauchy–Schwartz inequality and conditions (C6) and (C8), for all $\varvec{\zeta }\in {{\mathscr {R}}^p}$ with $\varvec{\zeta }^\mathrm{T}\varvec{\zeta }=1$,

$$\begin{aligned} E{\left( {{\varvec{\zeta }^\mathrm{T}}\varvec{II}_2} \right) ^2}= & {} \sum \limits _{i = 1}^n {E{{\left\| {{\varvec{\zeta }^\mathrm{T}}\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{K}}_i}\left( {t;{h_1}} \right) \left[ {{\varvec{\varUpsilon } _i} - E\left( {{\varvec{\varUpsilon } _i}} \right) } \right] } \right\| }^2}} \nonumber \\\le & {} \sum \limits _{i = 1}^n {\varvec{\zeta }^\mathrm{T}}\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{K}}_i}\left( {t;{h_1}} \right) {\varvec{\mathrm{K}}_i}\left( {t;{h_1}} \right) {\varvec{X}_i}\varvec{\zeta }\nonumber \\&\times E\left\{ {{{\left( {{\varvec{\varUpsilon } _i} - E\left( {{\varvec{\varUpsilon } _i}} \right) } \right) }^\mathrm{T}}\left( {{\varvec{\varUpsilon } _i} - E\left( {{\varvec{\varUpsilon } _i}} \right) } \right) } \right\} \nonumber \\\le & {} C\max \left\{ {\left| {\varvec{X}_{i}^\mathrm{T} \left( t_{ij} \right) \left( {\varvec{\beta } \left( t \right) - \varvec{\beta } \left( {{t_{ij}}} \right) } \right) } \right| } \right\} \nonumber \\&\times \sum \limits _{i = 1}^n {{\varvec{\zeta }^\mathrm{T}}\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{K}}_i}\left( {t;{h_1}} \right) {\varvec{\mathrm{K}}_i}\left( {t;{h_1}} \right) {\varvec{X}_i}\varvec{\zeta }} \nonumber \\= & {} o\left( Nh_1 \right) , \end{aligned}$$

(A.3)

where $\varvec{\varUpsilon } _i=(\varUpsilon _i\left( {{t_{i1}}} \right) ,\ldots ,\varUpsilon _i\left( {{t_{im_i}}} \right) )^\mathrm{T}$. According to McCullagh (1983), we have

$$\begin{aligned}&\varvec{{\hat{\beta }}} \left( {{t_0};{h_1}} \right) - \varvec{\beta } \left( {{t_0}} \right) \nonumber \\&\quad = {\left\{ {\frac{1}{Nh_1}\sum \limits _{i = 1}^n {\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{K}}_i}\left( {{t_0};{h_1}} \right) {\varvec{\varLambda }_i}{\varvec{X}_i}} } \right\} ^{ - 1}}\nonumber \\&\qquad \times \left\{ {\frac{1}{Nh_1}\sum \limits _{i = 1}^n {\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{K}}_i}\left( {{t_0};{h_1}} \right) {\psi _\tau }\left( {{\varvec{Y}_i-\varvec{X}_i \varvec{\beta }(t_0)}} \right) } } \right\} \nonumber \\&\qquad +\, {o_p}\left( (Nh_1)^{-1/2} \right) , \end{aligned}$$

(A.4)

where $\varvec{\varLambda }_i=\hbox {diag}\left( {{f_{i1}}\left( 0 \right) ,\ldots ,{f_{i{m_i}}}\left( 0 \right) } \right)$. Then, by the law of large numbers together with (A.1)–(A.4), we have

$$\begin{aligned} {\varvec{{\hat{\beta }}} \left( {{t_0};{h_1}} \right) - \varvec{\beta } \left( {{t_0}} \right) } = f_T^{ - 1}\left( {{t_0}} \right) {\bar{ f}^{-1}}\left( 0 \right) {\varvec{\varPhi }^{ - 1}} \left( {{t_0}} \right) \varvec{{\hat{R}}}\left( {{t_0};{h_1}} \right) +{o_p}\left( (Nh_1)^{-1/2} \right) . \nonumber \\ \end{aligned}$$

(A.5)

where $\varvec{{\hat{R}}}\left( {{t_0};{h_1}} \right) = \frac{1}{{Nh_1}}\sum \nolimits _{i = 1}^n \big \{{\varvec{X}_i^\mathrm{T}{\varvec{\varvec{\mathrm{K}}}_i}\left( {{t_0};{h_1}} \right) {\psi _\tau }\left( {{\varvec{\varepsilon } _i}} \right) + } \varvec{X}_i^\mathrm{T}{\varvec{\mathrm{K}}_i}\left( {{t_0};{h_1}} \right) {\varvec{\varLambda }_i}{\varvec{X}_i}\left[ \varvec{\beta } \left( {{t_{ij}}} \right) \right. \left. - \varvec{\beta } \left( {{t_0}} \right) \right] \big \}.$ It can be shown that the lth element of $\varvec{{\hat{R}}}\left( {{t_0};{h_1}} \right)$ can be written as

$$\begin{aligned} {{{\hat{R}}}_{l}}\left( {{t_0};{h_1}} \right) = \frac{1}{{Nh_1}}\sum \limits _{i = 1}^n {{\varphi _{il}} \left( {{t_0};{h_1}} \right) } , \end{aligned}$$

(A.6)

where ${\varphi _{il}}\left( {{t_0};{h_1}} \right) = \sum \nolimits _{j = 1}^{{m_i}} {{\xi _{il}}\left( {{t_0},{t_{ij}}} \right) {K_{{h_1}}}\left( {{t_0} - {t_{ij}}} \right) }$ and

$$\begin{aligned} {\xi _{il}}\left( {{t_0},{t_{ij}}} \right) = {X_{il}}\left( {{t_{ij}}} \right) {\psi _\tau }\left( {{\varepsilon _i}\left( {{t_{ij}}} \right) } \right) + {X_{il}}\left( {{t_{ij}}} \right) {f_{ij}}\left( 0 \right) \sum \limits _{r = 1}^p {{X_{ir}}\left( {{t_{ij}}} \right) \left[ {{\beta _r}\left( {{t_{ij}}} \right) - {\beta _r}\left( {{t_0}} \right) } \right] }. \end{aligned}$$

Then (A.6) implies that $\varvec{{\hat{R}}}\left( {{t_0};{h_1}} \right)$ is a sum of independent vectors

$$\begin{aligned} \varvec{{\hat{R}}}\left( {{t_0};{h_1}} \right) = \frac{1}{{Nh_1}}\sum \limits _{i = 1}^n \varvec{\varPsi }_i\left( {{t_0};{h_1}} \right) , \end{aligned}$$

where $\varvec{\varPsi }_i\left( {{t_0};{h_1}} \right) ={\left( {{\varphi _{i1}}\left( {{t_0};{h_1}} \right) ,\ldots ,{\varphi _{ip}}\left( {{t_0};{h_1}} \right) } \right) ^\mathrm{T}}$. Because $E\left( {{\psi _\tau }\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right) } \right\} } \right) = 0$ and the design points $t_{ij}, i=1,\ldots ,n, j=1,\ldots ,m_i$ are independent, direct calculation and the change of variables show that

$$\begin{aligned} E\left( {{\varphi _{il}}\left( {{t_0};{h_1}} \right) } \right)= & {} \sum \limits _{j = 1}^{{m_i}} {\int {E\left( {{\xi _{il}}\left( {{t_0},{t_{ij}}} \right) \left| {{t_{ij}} = v} \right. } \right) K\left( {\frac{{{t_0} - v}}{{{h_1}}}} \right) {f_T}\left( v \right) \hbox {d}v} } \\= & {} \sum \limits _{j = 1}^{{m_i}} f_{ij}\left( 0\right) {h_1}\sum \limits _{r = 1}^p \int \left[ {{\beta _r}\left( {{t_0} - {h_1}u} \right) - {\beta _r}\left( {{t_0}} \right) } \right] \\&\times \eta _{lr}\left( {{t_0} - {h_1}u} \right) {f_T}\left( {{t_0} - {h_1}u} \right) K\left( u \right) \hbox {d}u \\= & {} \sum \limits _{j = 1}^{{m_i}} {f_{ij}}\left( 0 \right) {\mu _2}h_1^3\sum \limits _{r = 1}^p \left[ {{{{\dot{\beta }} }_r}\left( {{t_0}} \right) {{{\dot{\eta }} }_{lr}}\left( {{t_0}} \right) {f_T}\left( {{t_0}} \right) } \right. \\&\left. + \frac{1}{2}{{\ddot{\beta }}_r}\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) {f_T}\left( {{t_0}} \right) + {{{\dot{\beta }}}_r}\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) {{\dot{f}}_T}\left( {{t_0}} \right) \right] \left\{ {1 + o\left( 1 \right) }\right\} . \end{aligned}$$

Then, by (A.6) and assumptions (C1), (C3) and (C6), and taking the Taylor expansions on the right side of the foregoing equation, we have

$$\begin{aligned} E\left( {{{{\hat{R}}}_l}\left( {{t_0};{h_1}} \right) } \right)= & {} \frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n {E\left( {{\varphi _{il}}\left( {{t_0};{h_1}} \right) } \right) } \rightarrow {\mu _2}h_1^2{\bar{ f}}\left( 0 \right) \sum \limits _{r = 1}^p \left[ {{\dot{\beta }} _r}\left( {{t_0}} \right) {{\dot{\eta }} _{lr}}\left( {{t_0}} \right) {f_T}\left( {{t_0}} \right) \right. \\&\left. + \frac{1}{2}{\ddot{\beta }_r}\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) {f_T}\left( {{t_0}} \right) + {{\dot{\beta }} _r}\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) {\dot{f}_T}\left( {{t_0}} \right) \right] . \end{aligned}$$

Therefore,

$$\begin{aligned} E\left( {\varvec{{\hat{\beta }}} \left( {{t_0};{h_1}} \right) - \varvec{\beta } \left( {{t_0}} \right) } \right) = f_T^{ - 1}\left( {{t_0}} \right) {\varvec{\varPhi }^{ - 1}} \left( {{t_0}} \right) \varvec{b}\left( {{t_0}} \right) =\varvec{B}\left( {{t_0}} \right) . \end{aligned}$$

For the covariance of $\sqrt{Nh_1} \varvec{{\hat{R}}}\left( {{t_0};{h_1}} \right)$, because

$$\begin{aligned}&\hbox {Cov}\left[ \sqrt{Nh_1} {{{{\hat{R}}}_l}\left( {{t_0};{h_1}} \right) ,\sqrt{Nh_1} {{{\hat{R}}}_r}\left( {{t_0};{h_1}} \right) } \right] \nonumber \\&\qquad = E\left[ \sqrt{Nh_1} {{{{\hat{R}}}_l}\left( {{t_0};{h_1}} \right) \sqrt{Nh_1} {{{\hat{R}}}_r}\left( {{t_0};{h_1}} \right) } \right] \nonumber \\&\qquad \quad - E\left[ \sqrt{Nh_1} {{{{\hat{R}}}_l}\left( {{t_0};{h_1}} \right) } \right] E\left[ \sqrt{Nh_1} {{{{\hat{R}}}_r}\left( {{t_0};{h_1}} \right) } \right] , \end{aligned}$$

(A.7)

and

$$\begin{aligned}&E\left[ {\sqrt{Nh_1} {{{\hat{R}}}_l}\left( {{t_0};{h_1}} \right) \sqrt{Nh_1} {{{\hat{R}}}_r}\left( {{t_0};{h_1}} \right) } \right] \nonumber \\&\quad = E\left\{ {\left[ {\frac{1}{{\sqrt{Nh_1} }}\sum \limits _{i = 1}^n {{\varphi _{il}}\left( {{t_0};{h_1}} \right) } } \right] \left[ {\frac{1}{{\sqrt{Nh_1} }}\sum \limits _{i = 1}^n {{\varphi _{ir}}\left( {{t_0};{h_1}} \right) } } \right] } \right\} \nonumber \\&\quad = \frac{1}{{Nh_1}} \sum \limits _{i = 1}^n E\left[ {{\varphi _{il}}\left( {{t_0};{h_1}} \right) {\varphi _{ir}} \left( {{t_0};{h_1}} \right) } \right] \nonumber \\&\qquad + \frac{1}{{Nh_1}}\sum \limits _{{i_1} \ne {i_2}} {E\left[ {{\varphi _{{i_1}l}} \left( {{t_0};{h_1}} \right) {\varphi _{{i_2}r}}\left( {{t_0};{h_1}} \right) } \right] } . \end{aligned}$$

(A.8)

For the first term on the right side of (A.8), we consider the further decomposition

$$\begin{aligned} {\varphi _{il}}\left( {{t_0};{h_1}} \right) {\varphi _{ir}}\left( {{t_0};{h_1}} \right)= & {} \sum \limits _{j = 1}^{{m_i}} {{\xi _{il}}\left( {{t_0},{t_{ij}}} \right) {\xi _{ir}}\left( {{t_0},{t_{ij}}} \right) {{\left[ {K\left( {\frac{{{t_0} - {t_{ij}}}}{{{h_1}}}} \right) } \right] }^2}} \nonumber \\&\!+\! \sum \limits _{{{j}_1} \ne {j_2}} {{\xi _{il}}\left( {{t_0},{t_{i{j_1}}}} \right) {\xi _{ir}} \left( {{t_0},{t_{i{j_2}}}} \right) K\left( {\frac{{{t_0} \!-\! {t_{i{j_1}}}}}{{{h_1}}}} \right) K\left( {\frac{{{t_0} \!-\! {t_{i{j_2}}}}}{{{h_1}}}} \right) }\nonumber \\ \end{aligned}$$

(A.9)

The change of variables, and the fact that ${{\psi _\tau }\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right) } \right\} }$ is a mean 0 and independent of $\varvec{X}_i\left( t_{ij} \right)$ , it can be shown by direct calculation that, as $n\rightarrow \infty$ and $v \rightarrow {t_0}$,

$$\begin{aligned}&E\left( {{\xi _{il}}\left( {{t_0},{t_{ij}}} \right) {\xi _{ir}}\left( {{t_0},{t_{ij}}} \right) \left| {{t_{ij}} = v} \right. } \right) \\&\quad = \tau \left( 1-\tau \right) E\left( {{X_{il}}\left( {{t_{ij}}} \right) {X_{ir}}\left( {{t_{ij}}} \right) \left| {{t_{ij}} = v} \right. } \right) \\&\qquad + \sum \limits _{c = 1}^p {{{\left[ {{\beta _c}\left( v \right) - {\beta _c}\left( {{t_0}} \right) } \right] }^2}f_{ij}^2\left( 0 \right) E\left( {{X_{il}}\left( {{t_{ij}}} \right) {X_{ir}}\left( {{t_{ij}}} \right) X_{ic}^2\left( {{t_{ij}}} \right) \left| {{t_{ij}} = v} \right. } \right) } \\&\qquad + \sum \limits _{{c_1} \ne {c_2}} \left[ {{\beta _{{c_1}}}\left( v \right) - {\beta _{{c_1}}}\left( {{t_0}} \right) } \right] \left[ {{\beta _{{c_2}}}\left( v \right) - {\beta _{{c_2}}}\left( {{t_0}} \right) } \right] f_{ij}^2\left( 0 \right) \\&\qquad \times E\left( {{X_{il}}\left( {{t_{ij}}} \right) {X_{ir}}\left( {{t_{ij}}} \right) {X_{i{c_1}}}\left( {{t_{ij}}} \right) {X_{i{c_2}}}\left( {{t_{ij}}} \right) \left| {{t_{ij}} = v} \right. } \right) \\&\qquad \rightarrow \tau \left( 1-\tau \right) {\eta _{lr}}\left( {{t_0}} \right) . \\ \end{aligned}$$

Then, we have

$$\begin{aligned}&E\left[ {\sum \limits _{j = 1}^{{m_i}} {{\xi _{il}}\left( {{t_0},{t_{ij}}} \right) {\xi _{ir}}\left( {{t_0},{t_{ij}}} \right) {K^2}\left( {\frac{{{t_0} - {t_{ij}}}}{{{h_1}}}} \right) } } \right] \nonumber \\&\quad = \sum \limits _{j = 1}^{{m_i}} \int {\left[ {E\left( {{\xi _{il}}\left( {{t_0},{t_{ij}}} \right) {\xi _{ir}}\left( {{t_0},{t_{ij}}} \right) \left| {{t_{ij}} = v} \right. } \right) {K^2}\left( {\frac{{{t_0} - v}}{{{h_1}}}} \right) } \right] {f_T}\left( v \right) \hbox {d}v} \nonumber \\&\quad = m_i {{h_1}\tau \left( 1-\tau \right) {\eta _{lr}}\left( {{t_0}} \right) {\nu _0}{f_T}\left( {{t_0}} \right) }+o(m_i {h_1}). \end{aligned}$$

(A.10)

Similarly, it can be shown by direct calculation that as $n \rightarrow \infty ,{v_1} \rightarrow {t_0},{v_2} \rightarrow {t_0}$

$$\begin{aligned}&E\left( {{\xi _{il}}\left( {{t_0},{t_{i{j_1}}}} \right) {\xi _{ir}}\left( {{t_0},{t_{i{j_2}}}} \right) \left| {{t_{i{j_1}}} = {v_1},{t_{i{j_2}}} = {v_2}} \right. } \right) \\&\quad = {\rho _\varepsilon }\left( {{v_1},{v_2}} \right) E\left( {{X_{il}}\left( {{t_{i{j_1}}}} \right) {X_{ir}}\left( {{t_{i{j_2}}}} \right) \left| {{t_{i{j_1}}} = {v_1},{t_{i{j_2}}} = {v_2}} \right. } \right) \\&\qquad + \sum \limits _{c = 1}^p {\left[ {{\beta _c}\left( {{v_1}} \right) - {\beta _c}\left( {{t_0}} \right) } \right] \left[ {{\beta _c}\left( {{v_2}} \right) - {\beta _c}\left( {{t_0}} \right) } \right] f_{ij}^2\left( 0 \right) } \\&\qquad \times E\left( {{X_{il}}\left( {{t_{i{j_1}}}} \right) {X_{ir}}\left( {{t_{i{j_2}}}} \right) {X_{ic}}\left( {{t_{i{j_1}}}} \right) {X_{ic}}\left( {{t_{i{j_2}}}} \right) \left| {{t_{i{j_1}}} = {v_1},{t_{i{j_2}}} = {v_2}} \right. } \right) \\&\qquad + \sum \limits _{{c_1} \ne {c_2}} {\left[ {{\beta _{{c_1}}}\left( {{v_1}} \right) - {\beta _{{c_1}}}\left( {{t_0}} \right) } \right] \left[ {{\beta _{{c_2}}}\left( {{v_2}} \right) - {\beta _{{c_2}}}\left( {{t_0}} \right) } \right] f_{ij}^2\left( 0 \right) } \\&\qquad \times E\left( {{X_{il}}\left( {{t_{i{j_1}}}} \right) {X_{ir}}\left( {{t_{i{j_2}}}} \right) {X_{i{c_1}}}\left( {{t_{i{j_1}}}} \right) {X_{i{c_2}}}\left( {{t_{i{j_2}}}} \right) \left| {{t_{i{j_1}}} = {v_1},{t_{i{j_2}}} = {v_2}} \right. } \right) \\&\qquad \rightarrow {\rho _\varepsilon }\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) . \end{aligned}$$

Therefore, the expectation of the second term on the right side of (A.9) is

$$\begin{aligned}&E\left[ {\sum \limits _{{j_1} \ne {j_2}} {{\xi _{il}}\left( {{t_0},{t_{i{j_1}}}} \right) {\xi _{ir}}\left( {{t_0},{t_{i{j_2}}}} \right) K\left( {\frac{{{t_0} - {t_{i{j_1}}}}}{{{h_1}}}} \right) K\left( {\frac{{{t_0} - {t_{i{j_2}}}}}{{{h_1}}}} \right) } } \right] \nonumber \\&\quad = \sum \limits _{{j_1} \ne {j_2}} {\left\{ {\int \int {E\left( {{\xi _{il}}\left( {{t_0},{t_{i{j_1}}}} \right) {\xi _{ir}}\left( {{t_0},{t_{i{j_2}}}} \right) \left| {{t_{i{j_1}}} = {v_1},{t_{i{j_2}}} = {v_2}} \right. } \right) } } \right. } \nonumber \\&\qquad \times \left. {K\left( {\frac{{{t_0} - {v_1}}}{{{h_1}}}} \right) K\left( {\frac{{{t_0} - {v_2}}}{{{h_1}}}} \right) {f_T}\left( {{v_1}} \right) {f_T}\left( {{v_2}} \right) \hbox {d}{v_1}\hbox {d}{v_2}} \right\} \nonumber \\&\quad = {m_i}\left( {{m_i} - 1} \right) h_1^2{\rho _\varepsilon }\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) f_T^2\left( {{t_0}} \right) + o\left( {{m_i}\left( {{m_i} - 1} \right) h_1^2} \right) . \end{aligned}$$

(A.11)

Combining (A.9)–(A.11), it follows immediately that when n is sufficiently large

$$\begin{aligned}&\frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n {E\left[ {{\varphi _{il}}\left( {{t_0};{h_1}} \right) {\varphi _{ir}}\left( {{t_0};{h_1}} \right) } \right] } \nonumber \\&\quad = \tau \left( 1-\tau \right) {\eta _{lr}}\left( {{t_0}} \right) {\nu _0}{f_T}\left( {{t_0}} \right) + {N^{ - 1}}{h_1}\left( {\sum \limits _{i = 1}^n {m_i^2 - N} } \right) {\rho _\varepsilon }\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) f_T^2\left( {{t_0}} \right) \nonumber \\&\qquad + o\left( {{N^{ - 1}}{h_1}\left( {\sum \limits _{i = 1}^n {m_i^2 - N} } \right) } \right) +o\left( 1\right) \nonumber \\&\qquad \rightarrow \tau \left( 1-\tau \right) {\eta _{lr}}\left( {{t_0}} \right) {\nu _0}{f_T}\left( {{t_0}} \right) + \kappa {h_0} {\rho _\varepsilon }\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) f_T^2\left( {{t_0}} \right) \end{aligned}$$

(A.12)

because $h_1 = {N^{{{ - 1} \big / 5}}}{h_0}$ and ${\lim _{n \rightarrow \infty }}{N^{{{ - 6} \big / 5}}}\sum \nolimits _{i = 1}^n {m_i^2} = \kappa$, and it is easy to see that as $n\rightarrow \infty$

$$\begin{aligned} {N^{ - 1}}{h_1}\left( {\sum \limits _{i = 1}^n {m_i^2 - N} } \right) = {N^{{{ - 6} \big / 5}}}\left( {\sum \limits _{i = 1}^n {m_i^2 - N} } \right) {h_0} \rightarrow \kappa {h_0}. \end{aligned}$$

Similar to the proof of (A.13) in Wu et al. (1998), as $n \rightarrow \infty$, we have

$$\begin{aligned}&\left| \frac{1}{{N{h_1}}}\sum \limits _{{i_1} \ne {i_2}} {E\left[ {{\varphi _{{i_1}l}}\left( {{t_0};{h_1}} \right) {\varphi _{{i_2}r}}\left( {{t_0};{h_1}} \right) } \right] }\right. \nonumber \\&\qquad \left. - E\left[ {\frac{1}{{\sqrt{N{h_1}} }}\sum \limits _{i = 1}^n {{\varphi _{il}}\left( {{t_0};{h_1}} \right) } } \right] E\left[ {\frac{1}{{\sqrt{N{h_1}} }}\sum \limits _{i = 1}^n {{\varphi _{ir}}\left( {{t_0};{h_1}} \right) } } \right] \right| \nonumber \\&\qquad \rightarrow 0. \end{aligned}$$

(A.13)

Based on (A.7), (A.8), (A.12) and (A.13), we have

$$\begin{aligned}&\hbox {Cov}\left[ {\sqrt{Nh_1} {{{\hat{R}}}_l}\left( {{t_0};{h_1}} \right) ,\sqrt{Nh_1} {{{\hat{R}}}_r}\left( {{t_0};{h_1}} \right) } \right] \\&\quad = \tau \left( 1-\tau \right) {\eta _{lr}}\left( {{t_0}} \right) {\nu _0}{f_T}\left( {{t_0}} \right) + \kappa {h_0}{\rho _\varepsilon }\left( {{t_0}} \right) {\eta _{lr}}\left( {{t_0}} \right) f_T^2\left( {{t_0}} \right) + o\left( 1 \right) .\\ \end{aligned}$$

The multivariate central limit theorem and the Slutsky’s theorem imply that

$$\begin{aligned} \sqrt{N{h_1}} \left( {\varvec{{\hat{\beta }}} \left( {{t_0};{h_1}} \right) -\varvec{\beta } \left( {{t_0}} \right) - \varvec{B}\left( {{t_0}} \right) } \right) \mathop \rightarrow \limits ^d N\left( {\varvec{0},{\upsilon ^2}\left( {{t_0}} \right) f_T^{ - 2}\left( {{t_0}} \right) {\bar{f}}^{ - 2}\left( 0 \right) \varvec{\varPhi }{{\left( {{t_0}} \right) }^{ - 1}}} \right) . \end{aligned}$$

Proof of Theorem 1

According to McCullagh (1983), we have

$$\begin{aligned}&\varvec{{\hat{\gamma }}} - {\varvec{\gamma } _0} = - {\left( {\frac{1}{n}\sum \limits _{i = 1}^n {\varvec{V}_i^\mathrm{T}} {\varvec{V}_i}} \right) ^{ - 1}}\frac{1}{n}\sum \limits _{i = 1}^n {\varvec{V}_i^\mathrm{T}} {\varvec{\tilde{e}}_i} + {o_p}\left( {{n^{{{ - 1} \big / 2}}}} \right) ,\\&\quad {\varvec{{\tilde{e}}}_i} =\varvec{L}_i^{ - 1}{\psi _\tau }\left( {{\varvec{{\hat{\varepsilon }} }_i}} \right) = \varvec{L}_i^{ - 1}\left[ {\tau \varvec{1}_{m_i} - I\left( {{\varvec{\varepsilon } _i}< 0} \right) + I\left( {{\varvec{\varepsilon } _i}< 0} \right) - I\left( {{\varvec{{\hat{\varepsilon }} }_i} < 0} \right) } \right] \\&\qquad \,\,= \varvec{e}_i - \varvec{L}_i^{ - 1}{\varvec{\varDelta }_i}, \\ \end{aligned}$$

where $\varvec{1}_{m_i}$ is an $m_i \times 1$ vector with all elements being 1 and ${\varvec{\varDelta }_i} = {\left( {{\varDelta _{i1}},\ldots ,{\varDelta _{i{m_i}}}} \right) ^\mathrm{T}}$ with ${\varDelta _{ij}} = \left[ {I\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right)< \varvec{X}_i^\mathrm{T}\left( {{t_{ij}}} \right) \left( {\varvec{{\hat{\beta }}} \left( {{t_{ij}};{h_1}} \right) - \varvec{\beta } \left( {{t_{ij}}} \right) } \right) } \right\} - I\left\{ {{\varepsilon _i}\left( {{t_{ij}}} \right) < 0} \right\} } \right]$. Because $\varvec{e}_i$ are independent random variables with $E\left( {{\varvec{e}_{i}}} \right) = \varvec{0}$ and $\hbox {Cov}\left( {{\varvec{e}_{i}}} \right) = {\varvec{D}_{i}}$. In addition, ${\varDelta _{ij}}=O_p\left( {{1 \big / {\sqrt{N{h_1}} }}} +h_1^2\right)$ by Lemma 2. The multivariate central limit theorem and the Slutsky’s theorem imply that

$$\begin{aligned} \sqrt{n} \left( {\varvec{{\hat{\gamma }}} - {\varvec{\gamma }_0}} \right) \mathop \rightarrow \limits ^d N\left( {\varvec{0},{\varvec{\varXi }^{ - 1}}\varvec{\varGamma } {\varvec{\varXi }^{ - 1}}} \right) . \end{aligned}$$

$\square$

Proof of Theorem 2

Following the same line of argument of Theorem 1 of Fan and Yao (1998), we have

$$\begin{aligned} {{{\hat{d}}}^2}\left( t \right) - {d^2}\left( t \right) = \frac{1}{{Nh_2{f_T}\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{K_{h_2}}\left( t- {{t_{ij}}} \right) \left\{ {{\hat{e}}_{ij}^2 - {d^2}\left( t \right) - {{\dot{d}}^2}\left( t \right) \left( {{t_{ij}} - t} \right) } \right\} } }. \end{aligned}$$

Note that

$$\begin{aligned} {\hat{e}}_{ij}^2= & {} {\left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - \sum \limits _{k = 1}^{j - 1} {{{{\hat{l}}}_{ijk}}{{{\hat{e}}}_{ik}}} } \right) ^2} \\= & {} {\left[ {{\psi _\tau }\left( {{\varepsilon _{ij}}} \right) - \left( {{\varDelta _{ij}} + \sum \limits _{k = 1}^{j - 1} {{{{\hat{l}}}_{ijk}}{{{\hat{e}}}_{ik}}} } \right) } \right] ^2} \\= & {} d_{ij}^2\varsigma _{ij}^2 + 2{d_{ij}}{\varsigma _{ij}}\left[ {\sum \limits _{k = 1}^{j - 1} {\left\{ {{l_{ijk}}\left( {{e_{ik}} - {{{\hat{e}}}_{ik}}} \right) + \left( {{l_{ijk}} - {{{\hat{l}}}_{ijk}}} \right) {{{\hat{e}}}_{ik}}} \right\} - {\varDelta _{ij}}} } \right] \\&+ {\left[ {\sum \limits _{k = 1}^{j - 1} {\left\{ {{l_{ijk}}\left( {{e_{ik}} - {{{\hat{e}}}_{ik}}} \right) + \left( {{l_{ijk}} - {{{\hat{l}}}_{ijk}}} \right) {{{\hat{e}}}_{ik}}} \right\} - {\varDelta _{ij}}} } \right] ^2}, \end{aligned}$$

where ${\varDelta _{ij}}$ is given in the proof of Theorem 1. It follows that

$$\begin{aligned} {{{\hat{d}}}^2}\left( t \right) - {d^2}\left( t \right) = {I_1} + {I_2} + {I_3} + {I_4}\left\{ 1+{o_p}\left( {{1}} \right) \right\} , \end{aligned}$$

where

$$\begin{aligned} {I_1}= & {} \frac{1}{{N{h_2}f_T\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{K_{h_2}}\left( t- {{t_{ij}}}\right) \left\{ {{d^2}\left( {{t_{ij}}} \right) - {d^2}\left( t \right) - {{\dot{d}}^2}\left( t \right) \left( {{t_{ij}} - t} \right) } \right\} } } , \\ {I_2}= & {} \frac{1}{{N{h_2}f_T\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{K_{h_2}}\left( t- {{t_{ij}}}\right) \left\{ {{d^2}\left( {{t_{ij}}} \right) \left( {\varsigma _{ij}^2 - 1} \right) } \right\} } } , \\ {I_3}= & {} 2\frac{1}{{N{h_2}{f_T}\left( t \right) }}\sum \limits _{i = 1}^n \sum \limits _{j = 1}^{{m_i}} {K_{{h_2}}}\left( t- {{t_{ij}}} \right) {d_{ij}}{\varsigma _{ij}}\\&\times \left[ {\sum \limits _{k = 1}^{j - 1} {\left\{ {{l_{ijk}}\left( {{e_{ik}} - {{{\hat{e}}}_{ik}}} \right) + \left( {{l_{ijk}} - {{{\hat{l}}}_{ijk}}} \right) {{{\hat{e}}}_{ik}}} \right\} - {\varDelta _{ij}}} }\right] , \\ {I_4}= & {} \frac{1}{{N{h_2}{f_T}\left( t \right) }}\sum \limits _{i = 1}^n {{{\sum \limits _{j = 1}^{{m_i}} {{K_{{h_2}}}\left( t- {{t_{ij}}} \right) \left[ {\sum \limits _{k = 1}^{j - 1} {\left\{ {{l_{ijk}}\left( {{e_{ik}} - {{{\hat{e}}}_{ik}}} \right) + \left( {{l_{ijk}}\! -\! {{{\hat{l}}}_{ijk}}} \right) {{{\hat{e}}}_{ik}}} \right\} \!-\! {\varDelta _{ij}}} } \right] } }^2}}. \\ \end{aligned}$$

It is easy to see that Theorem 2 follows directly from statements (a)–(d) below

• (a) ${I_1} = \frac{1}{2}{\mu _2}{h_2^2}{{\ddot{d}}^2}\left( t \right) + {o_p}\left( {{h_2^2}} \right)$,

• (b) $\sqrt{Nh_2} {I_2}\mathop \rightarrow \nolimits ^d N\left( {0,\frac{{{\nu _0}}}{{f_T\left( t \right) }}\mathop {\lim }\nolimits _{n \rightarrow \infty } \frac{1}{N}\sum \nolimits _{i = 1}^n {\sum \nolimits _{j = 1}^{{m_i}} {E\left[ {{{\left( {\varsigma _{ij}^2 \!-\! 1} \right) }^2}\left| {{t_{ij}} \!=\! t} \right. } \right] {d^4}\left( t \right) } } } \right)$,

• (c) ${I_3} = {o_p}\left( { \frac{1}{{\sqrt{Nh_2} }}} \right)$,

• (d) ${I_4} = {o_p}\left( { \frac{1}{{\sqrt{Nh_2} }}} \right)$.

It is easy to see that (a) follows from a Taylor expansion. $I_2$ is asymptotically normal with mean 0 and variance

$$\begin{aligned} \hbox {Var}\left( {{I_2}} \right) = \frac{{{\nu _0}}}{{{N^2}h_2f_T\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {E\left[ {{{\left( {\varsigma _{ij}^2 - 1} \right) }^2}\left| {{t_{ij}} = t} \right. } \right] {d^4}\left( t \right) } }. \end{aligned}$$

It follows from the definition of $I_3$ that

$$\begin{aligned} {I_3}= & {} 2\frac{1}{{N{h_2}{f_T}\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{K_{{h_2}}}\left( t- {{t_{ij}}} \right) {d_{ij}}{\varsigma _{ij}}\sum \limits _{k = 1}^{j - 1} {\left\{ {{l_{ijk}}\left( {{e_{ik}} - {{{\hat{e}}}_{ik}}} \right) + \left( {{l_{ijk}} - {{{\hat{l}}}_{ijk}}} \right) {{{\hat{e}}}_{ik}}} \right\} } } } \\&- 2\frac{1}{{N{h_2}{f_T}\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{K_{{h_2}}}\left( t- {{t_{ij}}} \right) {d_{ij}}{\varsigma _{ij}}{\varDelta _{ij}}} } \\= & {} {I_{31}} + {I_{32}}. \\ \end{aligned}$$

By Lemma 2 and condition (C1), together with $E\left( {{\varsigma _{ij}}|{t_{ij}}} \right) = 0$, ${{\hbox {Var}}}\left( {{\varsigma _{ij}}|{t_{ij}}} \right) = 1$, we have

$$\begin{aligned} {I_{31}} = {O_p}\left( {\frac{1}{{\sqrt{N{h_1}} }} + h_1^2} \right) {O_p}\left( {\frac{1}{{\sqrt{Nh_2} }}} \right) ={o_p}\left( {\frac{1}{{\sqrt{N{h_2}} }}} \right) , \end{aligned}$$

and

$$\begin{aligned} {I_{32}} = {O_p}\left( {\frac{1}{{\sqrt{N{h_1}} }} + h_1^2} \right) {O_p}\left( {\frac{1}{{\sqrt{Nh_2} }}} \right) ={o_p}\left( {\frac{1}{{\sqrt{N{h_2}} }}} \right) . \end{aligned}$$

Then ${I_{3}} = {o_p}\left( {{1 \big / {\sqrt{Nh_2} }}} \right)$. By the same arguments as proving $I_3$, we have ${I_{4}} = {o_p}\left( {{1 \big / {\sqrt{Nh_2} }}} \right)$. Under the conditions $Nh_2 \rightarrow \infty$ as $n\rightarrow \infty$ and $\lim {\sup _{n \rightarrow \infty }}N{h_2^5} < \infty$, then the proof of Theorem 2 is completed. $\square$

Proof of Theorem 3

Similar to the proof of Lemma 2, we have

$$\begin{aligned} \varvec{{\bar{\beta }}} \left( {{t_0};{h_1}} \right) - \varvec{\beta } \left( {{t_0}} \right)= & {} {\left\{ {\frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n { \varvec{X}_i^\mathrm{T}{ \varvec{\mathrm{{K}}}_i}\left( {{t_0};{h_1}} \right) \varvec{{\hat{\varSigma }}} _i^{ - 1}{ \varvec{{\tilde{\varLambda }} }_i}{ \varvec{X}_i}} } \right\} ^{ - 1}} \\&\times \left\{ {\frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n { \varvec{X}_i^\mathrm{T}{ \varvec{\mathrm{{K}}}_i}\left( {{t_0};{h_1}} \right) \varvec{{\hat{\varSigma }}} _i^{ - 1}{\psi _{\tau h} }\left( {{ \varvec{Y}_i} - { \varvec{X}_i} \varvec{\beta } \left( t_0 \right) } \right) } } \right\} \\&+\, {o_p}\left( {{{\left( {N{h_1}} \right) }^{{{ - 1} \big / 2}}}} \right) , \\ \end{aligned}$$

By the law of large numbers, we have

$$\begin{aligned} \frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n { \varvec{X}_i^\mathrm{T}{ \varvec{\mathrm{{K}}}_i}\left( {{t_0};{h_1}} \right) \varvec{{\hat{\varSigma }}} _i^{ - 1}{ \varvec{{\tilde{\varLambda }} }_i}{ \varvec{X}_i}} \mathop \rightarrow \limits ^p { \varvec{\varOmega }_1}, \end{aligned}$$

where $\varvec{{\tilde{\varLambda }} }_i=\hbox {diag}\left\{ {\frac{1}{h}K_1\left( {\frac{{{Y_{i1}} - \varvec{X}_i^\mathrm{T}\left( {{t_{i1}}} \right) \varvec{\beta }\, \left( t_0 \right) }}{h}} \right) ,\ldots ,\frac{1}{h}K_1\left( {\frac{{{Y_{i{m_i}}} - \varvec{X}_i^\mathrm{T}\left( {{t_{i{m_i}}}} \right) \varvec{\beta }\,\left( t_0 \right) }}{h}} \right) } \right\} .$ Using the conditions (C6), (C9) and (C10), similar to Lemma 3 (k) of Horowitz (1998), we obtain

$$\begin{aligned}&\frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n {\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{{K}}}_i}\left( {{t_0};{h_1}} \right) \varvec{{\hat{\varSigma }}} _i^{ - 1}{\psi _{\tau h}}\left( {{\varvec{Y}_i} - {\varvec{X}_i}\varvec{\beta } \left( {{t_0}} \right) } \right) } \\&\quad = \frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n {\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{{K}}}_i}\left( {{t_0};{h_1}} \right) \varvec{{\hat{\varSigma }}} _i^{ - 1}{\psi _\tau }\left( {{\varvec{Y}_i} - {\varvec{X}_i}\varvec{\beta } \left( {{t_0}} \right) } \right) } + {o_p}\left( {{{\left( {N{h_1}} \right) }^{{{ - 1} \big / 2}}}} \right) \\&\quad = \frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {\sum \limits _{j' = 1}^{{m_i}} {{\varvec{X}_i}\left( {{t_{ij}}} \right) {f_{ij'}}\left( 0 \right) K\left( {\frac{{t_0 - {t_{ij}}}}{{{h_1}}}} \right) {{{\hat{\sigma }} }^{jj'}}{\varvec{X}_i^\mathrm{T}}\left( {{t_{ij'}}} \right) \left[ {\varvec{\beta } \left( {{t_{ij'}}} \right) - \varvec{\beta } \left( {{t_0}} \right) } \right] } }} \\&\qquad + \frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {\sum \limits _{j' = 1}^{{m_i}} {{\varvec{X}_i}\left( {{t_{ij}}} \right) K\left( {\frac{{t_0 - {t_{ij}}}}{{{h_1}}}} \right) {{{\hat{\sigma }} }^{jj'}}{\psi _\tau }\left( {{\varepsilon _i}\left( {{t_{ij'}}} \right) } \right) } } } + {o_p}\left( {{{\left( {N{h_1}} \right) }^{{{ - 1} \big / 2}}}} \right) . \end{aligned}$$

Similar to the proof of Lemma 2, we have

$$\begin{aligned} E\left( {\varvec{{\bar{\beta }}} \left( {{t_0};{h_1}} \right) - \varvec{\beta } \left( {{t_0}} \right) } \right) = f_T^{ - 1}\left( {{t_0}} \right) {\varvec{\varPhi }^{ - 1}} \left( {{t_0}} \right) \varvec{b}\left( {{t_0}} \right) =\varvec{B}\left( {{t_0}} \right) , \end{aligned}$$

and

$$\begin{aligned} \hbox {Var}\left[ \frac{1}{\sqrt{N{h_1}}}\sum \limits _{i = 1}^n {\varvec{X}_i^\mathrm{T}{\varvec{\mathrm{{K}}}_i}\left( {{t_0};{h_1}} \right) \varvec{{\hat{\varSigma }}} _i^{ - 1}{\psi _{\tau h}}\left( {{\varvec{Y}_i} - {\varvec{X}_i}\varvec{\beta } \left( {{t_0}} \right) } \right) } \right] = {{\varvec{\varOmega }_2} + {\varvec{\varOmega }_3}} +o(1). \end{aligned}$$

The multivariate central limit theorem and the Slutsky’s theorem imply that

$$\begin{aligned} \sqrt{N{h_1}} \left( {\varvec{{\bar{\beta }}} \left( {{t_0};{h_1}} \right) -\varvec{\beta }\left( {{t_0}} \right) } -\varvec{B}\left( {{t_0}} \right) \right) \mathop \rightarrow \limits ^d N\left( {\varvec{0},\varvec{\varOmega }_1^{ - 1}\left( {{\varvec{\varOmega }_2} + {\varvec{\varOmega }_3}} \right) \varvec{\varOmega }_1^{ - 1}} \right) \end{aligned}$$

Therefore, we complete the proof of Theorem 3. $\square$

Proof of Theorem 4

Let $\zeta \left( {\varvec{\beta } \left( {{t_0}} \right) } \right) = \frac{1}{{\sqrt{N{h_1}} }}\sum \nolimits _{i = 1}^n {{ \varvec{Z}_{ih}}} \left( {\varvec{\beta } \left( {{t_0}} \right) } \right)$. By Theorem 3, we have $E\left[ {\zeta \left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \right] = o(1)$ and $\hbox {Cov}\left[ {\zeta \left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \right] = {{\varvec{\varOmega }_2} + {\varvec{\varOmega }_3}}+o(1)$. By Lemma 1, we know that $\zeta \left( {\varvec{\beta } \left( {{t_0}} \right) } \right)$ satisfies the conditions of the Cramer–Wold theorem (cf. Serfling 1980, theorem in sec. 1.5.2) and the Lindeberg condition (cf. Serfling 1980, theorem in sec. 1.9.2). Hence,

$$\begin{aligned} \frac{1}{{\sqrt{N{h_1}} }}\sum \limits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \mathop \rightarrow \limits ^d N\left( {\varvec{0}, {{\varvec{\varOmega }_2} + {\varvec{\varOmega }_3}}} \right) , \end{aligned}$$

(A.14)

and

$$\begin{aligned} \frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) \varvec{Z}_{ih}^\mathrm{T} \left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \mathop \rightarrow \limits ^p {{\varvec{\varOmega }_2} + {\varvec{\varOmega }_3}}. \end{aligned}$$

(A.15)

From (A.14), (A.15) and Lemma 1, and using the same arguments that are used in the proof of (2.14) in Owen (1990), we can prove that

$$\begin{aligned} \varvec{\lambda }= {O_p}\left( {{{\left( {N{h_1}} \right) }^{{{ - 1} \big / 2}}}} \right) , \end{aligned}$$

(A.16)

where $\varvec{\lambda }$ is defined in (13). Applying the Taylor expansion to (14) and invoking (A.14)–(A.16) and Lemma 1, we obtain

$$\begin{aligned} l\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) = 2\sum \limits _{i = 1}^n {\left\{ {{\varvec{\lambda }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) - {{{{\left[ {{\varvec{\lambda }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \right] }^2}} \big / 2}} \right\} } + {o_p}\left( 1 \right) . \end{aligned}$$

(A.17)

By (13), it follows that

$$\begin{aligned} 0= & {} \sum \limits _{i = 1}^n {\frac{{{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) }}{{1 + {\varvec{\lambda }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) }}} \\= & {} \sum \limits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } - \sum \limits _{i = 1}^n {{\varvec{\lambda }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) {\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \\&+ \sum \limits _{i = 1}^n {\frac{{{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) {{\left[ {{\varvec{\lambda }^\mathrm{T}}{\varvec{Z}_{ih}} \left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \right] }^2}}}{{\left[ 1 + {\varvec{ \lambda }^\mathrm{T}}{\varvec{Z}_{ih}} \left( {\varvec{\beta } \left( {{t_0}} \right) } \right) \right] }}}. \end{aligned}$$

The application of Lemma 1 and (A.14)–(A.16) again yields

$$\begin{aligned} \sum \limits _{i = 1}^n {{{\left[ {{\varvec{\lambda }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } \right] }^2}} = \sum \limits _{i = 1}^n {{\varvec{\lambda }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } + {o_p}\left( 1 \right) , \end{aligned}$$

(A.18)

and

$$\begin{aligned} \varvec{\lambda }= {\left[ {\sum \limits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) \varvec{Z}_{ih}^\mathrm{T}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } } \right] ^{ - 1}}\sum \limits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } + {o_p}\left( {{{\left( {N{h_1}} \right) }^{{{ - 1} \big / 2}}}} \right) . \nonumber \\ \end{aligned}$$

(A.19)

Substituting (A.18) and (A.19) into (A.17), we obtain

$$\begin{aligned} l\left( {\varvec{\beta } \left( {{t_0}} \right) } \right)= & {} {\left[ {\frac{1}{{\sqrt{N{h_1}} }}\sum \limits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } } \right] ^\mathrm{T}}{\left[ {\frac{1}{{N{h_1}}}\sum \limits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) \varvec{Z}_{ih}^\mathrm{T}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } } \right] ^{ - 1}} \nonumber \\&\times \left[ {\frac{1}{{\sqrt{N{h_1}} }}\sum \limits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{\beta } \left( {{t_0}} \right) } \right) } } \right] + {o_p}\left( 1 \right) . \end{aligned}$$

(A.20)

Based on (A.14), (A.15) and (A.20), we can prove Theorem 4. $\square$

Proof of Corollary 1

Let $\varvec{Z}_{ih}^{(2)}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) = {{\partial {\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) } \big / {\partial {\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) }}$ and $\varvec{{\tilde{\lambda }}} = \varvec{\lambda }\left( {\varvec{b}\left( {{t_0}} \right) , {\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right)$, then $\varvec{{\tilde{\lambda }}}$ and ${{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) }$ satisfy

$$\begin{aligned} {Q_1}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) ,\varvec{{\tilde{\lambda }}} } \right) =\sum \limits _{i = 1}^n {\frac{{{\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) }}{{1 + {\varvec{{\tilde{\lambda }} }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) }}}=0, \end{aligned}$$

and

$$\begin{aligned} {Q_2}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) ,\varvec{{\tilde{\lambda }} }} \right) = \sum \limits _{i = 1}^n {\frac{{{\varvec{{\tilde{\lambda }} }^\mathrm{T}}\varvec{Z}_{ih}^{(2)}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) }}{{1 + {\varvec{{\tilde{\lambda }} }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) }}} =0. \end{aligned}$$

Expanding ${Q_1}\left( {\varvec{b}\left( {{t_0}} \right) \!,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) \!,\varvec{{\tilde{\lambda }}} } \right)$ and ${Q_2}\left( {\varvec{b}\left( {{t_0}} \right) \!,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) \!,\varvec{{\tilde{\lambda }}} } \right)$ at $\left( {\varvec{b}\left( {{t_0}} \right) \!,{{ \varvec{\beta } }^{(2)}}\left( t_0 \right) ,0 } \right)$, we have

$$\begin{aligned} \varvec{{\tilde{\lambda }}} = \left( {\varvec{I} -\varvec{P}} \right) \varvec{\varSigma } _n^{ - 1}\varvec{{\tilde{Z}} }+ {o_p}\left( {{{\left( {N{h_1}} \right) }^{{{ - 1} \big / 2}}}} \right) , \end{aligned}$$

and

$$\begin{aligned} {\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) - {\varvec{\beta } ^{(2)}}\left( {{t_0}} \right) = - {\left( {{\varvec{{\tilde{Z}}}^{(2)T}}\varvec{\varSigma } _n^{ - 1}{\varvec{{\tilde{Z}}}^{(2)}}} \right) ^{ - 1}}{\varvec{{\tilde{Z}}}^{(2)T}}\varvec{\varSigma } _n^{ - 1}\varvec{{\tilde{Z}}} + {o_p}\left( {{{\left( {N{h_1}} \right) }^{{{ - 1} \big / 2}}}} \right) , \end{aligned}$$

where ${\varvec{{\tilde{Z}}}^{(2)}} = \sum \nolimits _{i = 1}^n {\varvec{Z}_{ih}^{(2)}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{\beta } ^{(2)}}\left( {{t_0}} \right) } \right) }$, $\varvec{P} = \varvec{\varSigma } _n^{ - 1}{\varvec{{\tilde{Z}}}^{(2)}}{\left( {{\varvec{{\tilde{Z}}}^{(2)T}}\varvec{\varSigma } _n^{ - 1}{\varvec{{\tilde{Z}}}^{(2)}}} \right) ^{ - 1}}{\varvec{{\tilde{Z}}}^{(2)T}}$, $\varvec{{\tilde{Z}}} = \sum \nolimits _{i = 1}^n {{\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{\beta } ^{(2)}}\left( {{t_0}} \right) } \right) }$ and ${\varvec{\varSigma } _n} = \sum \nolimits _{i = 1}^n {\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{\beta } ^{(2)}}\left( {{t_0}} \right) } \right) \varvec{Z}_{ih}^\mathrm{T} \left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{\beta } ^{(2)}}\left( {{t_0}} \right) } \right)$. Because

$$\begin{aligned} l\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right)= & {} 2\sum \limits _{i = 1}^n {\log \left\{ {1 + {\varvec{{\tilde{\lambda }} }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) } \right\} } \\= & {} 2\sum \limits _{i = 1}^n {{\varvec{{\tilde{\lambda }} }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) } \\&- \sum \limits _{i = 1}^n {{{\left\{ {{\varvec{{\tilde{\lambda }} }^\mathrm{T}}{\varvec{Z}_{ih}}\left( {\varvec{b}\left( {{t_0}} \right) ,{\varvec{{\tilde{\beta }} }^{(2)}}\left( t \right) } \right) } \right\} }^2} + {o_p}\left( 1 \right) } \\= & {} {\varvec{{\tilde{Z}}}^\mathrm{T}}\varvec{\varSigma } _n^{{{ - 1} \big / 2}}\left( {\varvec{I} - \varvec{\varSigma } _n^{{1 \big / 2}}\varvec{P}\varvec{\varSigma } _n^{{{ - 1} \big / 2}}} \right) \varvec{\varSigma } _n^{{{ - 1} \big / 2}}\varvec{{\tilde{Z}}} + {o_p}\left( 1 \right) . \end{aligned}$$

Similar to the proof of Theorem 4, we have $\varvec{\varSigma } _n^{{{ - 1} \big / 2}}\varvec{{\tilde{Z}}}\mathop \rightarrow \limits ^d N\left( {\varvec{0},\varvec{I}} \right)$ and ${\varvec{\varSigma } _n^{{1 \big / 2}}\varvec{P}\varvec{\varSigma } _n^{{{ - 1} \big / 2}}}$ is symmetric and idempotent, with trace equal to $p-p_1$. Because $\varvec{Z}_{ih}\,l\left( {{\varvec{{\hat{\beta }} }^{(1)}}\left( t \right) ,{\varvec{{\hat{\beta }} }^{(2)}}\left( t \right) } \right) = 0$. Hence the empirical likelihood ratio statistic ${\bar{l}}\left( {\varvec{b}\left( {{t_0}} \right) } \right)$ converges to $\chi _{p_1}^2$. $\square$