Quantile estimations via modified Cholesky decomposition for longitudinal single-index models

Abstract

Quantile regression is a powerful complement to the usual mean regression and becomes increasingly popular due to its desirable properties. In longitudinal studies, it is necessary to consider the intra-subject correlation among repeated measures over time to improve the estimation efficiency. In this paper, we focus on longitudinal single-index models. Firstly, we apply the modified Cholesky decomposition to parameterize the intra-subject covariance matrix and develop a regression approach to estimate the parameters of the covariance matrix. Secondly, we propose efficient quantile estimating equations for the index coefficients and the link function based on the estimated covariance matrix. Since the proposed estimating equations include a discrete indicator function, we propose smoothed estimating equations for fast and accurate computation of the index coefficients, as well as their asymptotic covariances. Thirdly, we establish the asymptotic properties of the proposed estimators. Finally, simulation studies and a real data analysis have illustrated the efficiency of the proposed approach.

Appendix

In the proofs, C denotes a positive constant that might assume different values at different places. For any matrix ${\varvec{A}} = \left( {{A_{ij}}} \right) _{i = 1,j = 1}^{s,t}$, denote ${\left\| {\varvec{A}} \right\| _\infty } = {\max _{1 \le i \le s}}\sum _{j = 1}^t {\left| {{A_{ij}}} \right| }$. To establish the asymptotic properties of the proposed estimators, the following regularity conditions are needed in this paper.

(C1) Let $\mathscr {U}=\left\{ {u:u = {\varvec{X}}_{ij}^T{\beta } ,{{\varvec{X}}_{ij}} \in A,i = 1,\ldots ,n,j = 1,\ldots ,{m_i}} \right\}$ and A be the support of ${\varvec{X}}_{ij}$ which is assumed to be compact. Suppose that the density function $f_{{\varvec{X}}_{ij}^T{\beta }}(u)$ of ${\varvec{X}} _{ij}^T{\beta }$ is bounded away from zero and infinity on $\mathscr {U}$ and satisfies the Lipschitz condition of order 1 on $\mathscr {U}$ for ${\beta }$ in a neighborhood of ${\beta }_0$.

(C2) The function $g_0\left( \cdot \right)$ has the dth bounded and continuous derivatives for some $d\ge 2$ and $g_{1s}(\cdot )$ satisfies the Lipschitz condition of order 1, where $g_{1s}\left( u\right)$ is the sth component of ${\varvec{g}}_1\left( u\right) =E\left( {{{\varvec{X}}_{ij}}\left| {{\varvec{X}}_{ij}^T{{\beta } _0} = u} \right. } \right)$, $s=1,\ldots ,p$.

(C3) Let the distance between neighboring knots be ${H_{i}} = {\xi _{i}} - {\xi _{i - 1}}$ and $H = {\max _{1 \le i \le {N_n} + 1}}\left\{ {{H_{i}}} \right\}$. Then, there exists a constant $C_{0}$ such that $\frac{{{H}}}{{{{\min }_{1 \le i \le {N_n} + 1}}\left\{ {{H_{i}}} \right\} }} < {C_{0}}, {\max _{1 \le i \le {N_n}}}\left\{ {{H_{i + 1}} - {H_{i}}} \right\} = o({N_n^{ - 1}})$.

(C4) The distribution function $F_{ij}(t)=p\left( {{Y_{ij}} - g_0\left( {\varvec{X}}_{ij}^T{{\beta } _0} \right) \le t} \right)$ is absolutely continuous, with continuous densities $f_{ij}\left( \cdot \right)$ uniformly bounded, and its first derivative ${f'_{ij}}\left( \cdot \right)$ uniformly bounded away from 0 and $\infty$ at the point $0, i=1,\ldots ,n, j=1,\ldots ,m_i$.

(C5) The eigenvalues of ${\varvec{\varSigma }}_{\tau i}$ are uniformly bounded and bounded away from zero.

(C6) $K\left( \cdot \right)$ is bounded and compactly supported on $[-1,1]$. For some constant $C_K\ne 0$, $K\left( \cdot \right)$ is a $\nu$th-order kernel, i.e., $\int {{u^j}} K\left( u \right) du = 1$ if $j=0$; 0 if $1 \le j \le \nu - 1$; $C_K$ if $j=\nu$, where $\nu \ge 2$.

(C7) The positive bandwidth parameter h satisfies $n{h^{2 \nu }} \rightarrow 0$.

Lemma 1

Under conditions (C1)–(C7), and $N_n\rightarrow \infty$ and $n N_n^{-1} \rightarrow \infty$, as $n\rightarrow \infty$, we have (i) $\left| {{{\hat{g}}}({u};{{\beta } _0}) - {g_0}({u})} \right| = O_p\left( {\sqrt{{{{N_n}} \big / n}} + N_n^{ - d}} \right)$ uniformly for any $u\in [a,b]$; and (ii) under $N_n\rightarrow \infty$ and $n N_n^{-3} \rightarrow \infty$, as $n\rightarrow \infty$, $\left| {{{\hat{g}'}}({u};{{\beta } _0}) - {g'_0}({u})} \right| = O_p\left( {\sqrt{{{N_n^3} \big / n}} + N_n^{ - d + 1}} \right)$ uniformly for any $u\in [a,b]$.

Proof

Suppose $g^0(u)={\varvec{B}}_q(u)^T {\varvec{\theta }}^0$ is the best approximating spline function for ${g}_0(u)$. According to the result on page 149 of de Boor (2001) for ${g}_0(u)$ satisfying condition (C2), we have

$$\begin{aligned} {\sup _{u \in [a,b]}}\left| {g_0(u) - {g^0}(u)} \right| = {O}(N_n^{ - d}). \end{aligned}$$

(11)

Let ${\alpha _n} =N_n n^{-1/2} + N_n^{ - d+1/2}$ and set $\left\| {{{\varvec{u}}_n}} \right\| = C$, where C is a large enough constant. Our aim is to show that for any given $\delta > 0,$ there is a large constant C such that, for large n, we have

$$\begin{aligned} P\left\{ {\mathop {\inf }\limits _{\left\| {{{\varvec{u}}_n}} \right\| = C} {L_n}\left( {{\beta }_0 ;{{\varvec{\theta }} ^0} + {\alpha _n}{{\varvec{u}}_n}} \right) > {L_n}\left( {{\beta }_0 ;{{\varvec{\theta }} ^0}} \right) } \right\} \ge 1 -\delta . \end{aligned}$$

(12)

This implies that there is local minimum ${{\hat{{\varvec{\theta }}}} }$ in the ball $\left\{ {{{\varvec{\theta }} ^0} + {\alpha _n}{\varvec{u}}_n:\left\| {\varvec{u}}_n \right\| \le C} \right\}$ with probability tending to one, such that $\left\| {{\hat{{\varvec{\theta }}}} - {{\varvec{\theta }} ^0}} \right\| = {O_p}\left( {{\alpha _n}} \right)$. Define ${\varDelta _{ij}} = {{\varvec{B}}_q}{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) ^T}{{\varvec{\theta }} ^0} - {g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right)$. Applying the identity

$$\begin{aligned} {\rho _\tau }\left( {r - v} \right) - {\rho _\tau }\left( r \right) = - v\left( { \tau -I\left( {r < 0} \right) } \right) + \int _0^v {\left[ {I\left( {r \le t} \right) - I\left( {r \le 0} \right) } \right] } dt, \end{aligned}$$

we have

$$\begin{aligned} \begin{array}{l} {L_n}\left( {{{\beta } _0};{{\varvec{\theta }} ^0} + {\alpha _n}{{\varvec{u}}_n}} \right) - {L_n}\left( {{{\beta } _0};{{\varvec{\theta }} ^0}} \right) \\ = \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{\rho _\tau }\left( {{\varepsilon _{ij}} - {\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n} - {\varDelta _{ij}}} \right) } } - \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{\rho _\tau }\left( {{\varepsilon _{ij}} - {\varDelta _{ij}}} \right) } } \\ = -\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}\left( {\tau - I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} < 0} \right) } \right) } } \\ ~~~~+ \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {\int _0^{{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} {\left[ {I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le t} \right) - I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le 0} \right) } \right] } dt} } \\ \buildrel \varDelta \over = I + II. \\ \end{array} \end{aligned}$$

The observed covariates vector is written as $\mathscr {D}=\Big \{ {\varvec{X}}_{11}^T,\ldots ,{\varvec{X}}_{1{m_1}}^T,\ldots ,{\varvec{X}}_{n1}^T,\ldots ,{\varvec{X}}_{n{m_n}}^T \Big \}^T$. Moreover, we have

$$\begin{aligned}&E\left( {II} \mid \mathscr {D}\right) \\&\quad = E\left( {\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {\int _0^{{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} {\Big [ {I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le t} \right) - I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le 0} \right) } \Big ]} dt} } }\mid \mathscr {D} \right) \\&\quad = \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {\int _0^{{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} {\left[ {{F_{ij}}\left( {{\varDelta _{ij}} + t} \right) - {F_{ij}}\left( {{\varDelta _{ij}}} \right) } \right] } dt} }\\&\quad = \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {\int _0^{{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} {{f_{ij}}\left( {{\varDelta _{ij}}} \right) t\left( {1 + o\left( 1 \right) } \right) } dt} } \\&\quad = \frac{1}{2}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{f_{ij}}\left( {{\varDelta _{ij}}} \right) {{\left( {{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} \right) }^2}} } \left( {1 + o\left( 1 \right) } \right) \\&\quad = \frac{1}{2}\alpha _n^2{\varvec{u}}_n^T\left( {\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{f_{ij}}\left( 0 \right) {{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) {{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}} } } \right) {{\varvec{u}}_n}\\&\qquad +\, o_p\left( {{{n\alpha _n^2{{\left\| {{{\varvec{u}}_n}} \right\| }^2}} \Big / {{N_n}}}} \right) , \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} Var\left( {II}\mid \mathscr {D} \right) \\ \quad = Var\left( {\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {\int _0^{{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} {\left[ {I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le t} \right) - I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le 0} \right) } \right] } dt} } }\mid \mathscr {D} \right) \\ \quad \le \sum \limits _{i = 1}^n E\left[ {{\left( {\sum \limits _{j = 1}^{{m_i}} {\int _0^{{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} {\left[ {I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le t} \right) - I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le 0} \right) } \right] } dt} } \right) }^2} \mid \mathscr {D}\right] \\ \quad \le \sum \limits _{i = 1}^n {{m_i}\sum \limits _{j = 1}^{{m_i}} {E\left[ {{{\left( {\int _0^{{\alpha _n}{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} {\left[ {I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le t} \right) - I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le 0} \right) } \right] } dt} \right) }^2}\left| {\mathscr {D}} \right. } \right] } } \\ \quad \le \sum \limits _{i = 1}^n {m_i}\sum \limits _{j = 1}^{{m_i}} \int _0^{\left| {{\alpha _n}{{\varvec{B}}_q^T}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }}{{\varvec{u}}_n}} \right| } \int _0^{\left| {{\alpha _n}{{\varvec{B}}_q^T}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }}{{\varvec{u}}_n}} \right| }\\ \qquad \quad {\left[ {{F_{ij}}\left( {{\varDelta _{ij}} + \left| {{\alpha _n}{{\varvec{B}}_q^T}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }}{{\varvec{u}}_n}} \right| } \right) - {F_{ij}}\left( {{\varDelta _{ij}}} \right) } \right] d{t_1}d{t_2}} \\ \quad \le o_p\left( {{{n\alpha _n^2{{\left\| {{{\varvec{u}}_n}} \right\| }^2}} \Big / {{N_n}}}} \right) . \\ \end{array} \end{aligned}$$

In addition,

$$\begin{aligned} I = E\left( I \right) + {O_p}\left( {\sqrt{Var\left( I \right) } } \right) . \end{aligned}$$

Moreover, the condition that $\varepsilon _{ij}$ has the $\tau$th quantile zero implies $E\left( {{\psi _\tau }\left( {{\varepsilon _{ij}}} \right) } \right) = 0$. By (11) and condition (C4), we have $E\left( I\right) = o\left( 1\right)$ and

$$\begin{aligned} \begin{array}{l} Var\left( I\mid \mathscr {D} \right) \le \sum \limits _{i = 1}^n {E\left[ {{{\left( {{\alpha _n}\sum \limits _{j = 1}^{{m_i}} {\left( {\tau - I\left( {{\varepsilon _{ij}} - {\varDelta _{ij}} \le 0} \right) } \right) {{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} } \right) }^2}\left| {\mathscr {D}} \right. } \right] } \\ ~~~~~~~~~~~~~~~\le C{\varvec{u}}_n^T\alpha _n^2\sum \limits _{i = 1}^n m_i{\sum \limits _{j = 1}^{{m_i}} { {{{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) {{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{u}}_n}} } } \\ ~~~~~~~~~~~~~~~=O_p\left( {{{n\alpha _n^2{{\left\| {{{\varvec{u}}_n}} \right\| }^2}} \Big / {{N_n}}}} \right) \\ \end{array} \end{aligned}$$

implies that $I= {O_p}\left( {\sqrt{{{n\alpha _n^2} \big / {{N_n}}}} } \right) \left\| {{{\varvec{u}}_n}} \right\|$. Based on all the above, ${L_n}\left( {{{\beta } _0};{{\varvec{\theta }} ^0} + {\alpha _n}{{\varvec{u}}_n}} \right) - {L_n}\left( {{{\beta } _0};{{\varvec{\theta }} ^0}} \right)$ is dominated by $\frac{1}{2}\alpha _n^2{\varvec{u}}_n^T\left( {\sum _{i = 1}^n {\sum _{j = 1}^{{m_i}} {{f_{ij}}\left( 0 \right) {{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) {{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}} } } \right) {{\varvec{u}}_n}$ by choosing a sufficiently large $\left\| {{{\varvec{u}}_n}} \right\| =C$. Therefore, (12) holds and there exists a local minimizer ${\hat{{\varvec{\theta }}}}$ such that

$$\begin{aligned} \left\| {{\hat{{\varvec{\theta }}}} - {{\varvec{\theta }} ^0}} \right\| = {O_p}\left( {{\alpha _n}} \right) ={O_p}\left( {N_n^{ - d+1/2} +N_n n^{-1/2} } \right) . \end{aligned}$$

(13)

Since $\left\| {{{\varvec{B}}_q}\left( u \right) {{\varvec{B}}_q}{{\left( u \right) }^T}} \right\| = O\left( 1/N_n \right)$, together with (13), we have

$$\begin{aligned} \begin{array}{l} \left| {\hat{g}\left( {u;{{\beta } _0}} \right) - {g^0}\left( u \right) } \right| ^2 \\ = \left| {{{\varvec{B}}_q}{{\left( u \right) }^T}{\hat{{\varvec{\theta }}}} - {{\varvec{B}}_q}{{\left( u \right) }^T}{{\varvec{\theta }} ^0}} \right| ^2\\ \le \left\| {{{\varvec{B}}_q}\left( u \right) {{\varvec{B}}_q}{{\left( u \right) }^T}} \right\| \left\| {{\hat{{\varvec{\theta }}}} - {{\varvec{\theta }} ^0}} \right\| ^2 ={O_p}\left( {N_n^{ - 2d} +N_n n^{-1} } \right) .\\ \end{array} \end{aligned}$$

(14)

By the triangle inequality, $\left| {\hat{g}\left( {u;{{\beta } _0}} \right) - {g_0}\left( u \right) } \right| \le \left| {\hat{g}\left( {u;{{\beta } _0}} \right) - {g^0}\left( u \right) } \right| + \big | {g^0}\left( u \right) - {g_0}\left( u \right) \big |$. Therefore, by (11) and (14), we have $\left| {\hat{g}\left( {u;{{\beta } _0}} \right) - g_0\left( u \right) } \right| = {O_p}\left( {N_n^{ - d} + \sqrt{{{{N_n}} \big / n}} } \right)$ uniformly for every $u\in [a,b]$.

Since ${{\hat{g}'}}({u};{\beta } _0)={\varvec{B}}_{q-1}(u)^T {\varvec{D}}_1 {\hat{{\varvec{\theta }}}} ({\beta }_0)$, where ${\varvec{B}}_{q-1}(u)=\{B_{s,q}(u): 2\le s \le J_n\}^T$ is the $(q-1)$th-order B-spline basis and ${\varvec{D}}_1$ is defined in Sect. 2.1. It is easy to prove that ${\left\| {{{\varvec{D}}_1}} \right\| _\infty } = O({N_n})$. Then, employing similar techniques to that used in the proof of ${{\hat{g}}}({u};{\beta } _0)$, we obtain that

$$\begin{aligned} \left| {{{\hat{g}'}}({u};{{\beta } _0}) - g'_0({u})} \right| = O_p\left( {\sqrt{{{N_n^3} \big / n}} + N_n^{ - d + 1}} \right) \end{aligned}$$

uniformly for any $u\in [a,b]$. This completes the proof. $\square$

Lemma 2

Under conditions (C1)–(C7), and the number of knots satisfies ${n^{{1 / {(2d + 1)}}}} \ll {N_n} \ll {n^{{1 / 4}}}$, then for any $J_n \times 1$-vector ${\varvec{c}}_n$ whose components are not all 0, we have

$$\begin{aligned} \bar{\sigma } _n^{-1}\left( u \right) {{\varvec{c}}_n^T}\left( {{\hat{{\varvec{\theta }}}} \left( {{{\beta } _0}} \right) - {{\varvec{\theta }} ^0}\left( {{{\beta } _0}} \right) } \right) \mathop \rightarrow \limits ^d N\left( {0,1} \right) , \end{aligned}$$

where $\bar{\sigma } _n^2\left( u \right) = {\varvec{c}}_n^T{{\varvec{V}}^{ - 1}}\left( {{{\beta } _0}} \right) \sum _{i = 1}^n {{\varvec{B}}_q^T\left( {{{\varvec{X}}_i}{{\beta } _0}} \right) {{\varvec{{\varLambda } }}_i}{{\varvec{\varSigma }} _{\tau i}}{{{\varvec{\varLambda }} }_i}{{\varvec{B}}_q}\left( {{{\varvec{X}}_i}{{\beta } _0}} \right) } {{\varvec{V}}^{ - 1}}\left( {{{\beta } _0}} \right) {{\varvec{c}}_n}$ and the definition of ${\varvec{V}}\left( {{{\beta } _0}} \right)$ is given in subsection 2.4.

Proof

When ${\beta }={\beta }_0$, the minimizer ${\hat{{\varvec{\theta }}} }$ of (1) satisfies the score equations

$$\begin{aligned} \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) f_{ij} \left( 0 \right) \left[ {I\left( {{Y_{ij}} - {{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{\varvec{\theta }} < 0} \right) - \tau } \right] } } = {\varvec{0}}. \end{aligned}$$

(15)

Then, the left-hand side of Eq. (15) becomes

$$\begin{aligned}&\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) f_{ij} \left( 0 \right) \left[ {I\left( {{\varepsilon _{ij}}< 0} \right) - I\left( {{\varepsilon _{ij}}< 0} \right) + I\left( {\varepsilon _{ij}}< {\zeta _{ij}} \right) - \tau } \right] } } \nonumber \\&\quad = \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) f_{ij} \left( 0 \right) \left[ {I\left( {{\varepsilon _{ij}}< 0} \right) - \tau } \right] } } \nonumber \\&\qquad + \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) f_{ij} \left( 0 \right) \left[ {{F_{ij}}\left( 0 \right) - {F_{ij}}\left( {{\zeta _{ij}}} \right) } \right] } } \nonumber \\&\qquad - \sum \limits _{i = 1}^n \sum \limits _{j = 1}^{{m_i}} {{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) f_{ij} \left( 0 \right) \Bigg [ {F_{ij}}\left( 0 \right) - {F_{ij}}\left( {{\zeta _{ij}}} \right) \nonumber \\&\quad - \left\{ {I\left( {{\varepsilon _{ij}}< {\zeta _{ij}}} \right) - I\left( {{\varepsilon _{ij}} < 0} \right) } \right\} \Bigg ] \nonumber \\&\quad \buildrel \varDelta \over = I + II + III, \end{aligned}$$

(16)

where ${\zeta _{ij}} = {{\varvec{B}}_q}{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) ^T}{\varvec{\theta }} - {g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right)$. By (11), taking Taylor’s explanation for ${{F_{ij}}\left( {{\zeta _{ij}}} \right) }$ at 0 gives

$$\begin{aligned} II= & {} - \sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) {f_{ij}^2}\left( 0 \right) {\zeta _{ij}}\left( {1 + {o}\left( 1 \right) } \right) } } \nonumber \\= & {} - \sum \limits _{i = 1}^n \sum \limits _{j = 1}^{{m_i}} {{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) {f_{ij}^2}\left( 0 \right) \nonumber \\&\times \left[ {{{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}\left( {{\varvec{\theta }} - {{\varvec{\theta }} ^0}} \right) + {{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{{\varvec{\theta }} ^0} - {g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) } \right] \left( {1 + {o}\left( 1 \right) } \right) \nonumber \\= & {} -\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{{\varvec{B}}_q}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) {{\varvec{B}}_q}{{\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) }^T}{f_{ij}^2}\left( 0 \right) \left( {{\varvec{\theta }} - {{\varvec{\theta }} ^0}} \right) \left( {1 + {o}\left( 1 \right) } \right) } }. \end{aligned}$$

(17)

By direct calculation of the mean and variance, we can show that $III = {o_p}\left( {\sqrt{n/{N_n}} } \right)$. This combined with (15)–(17) leads to

$$\begin{aligned} \left( {{\varvec{\theta }} - {{\varvec{\theta }} ^0}} \right) \left( {1 + {o_p}\left( 1 \right) } \right) = {{\varvec{V}}^{ - 1}}\left( {{{\beta } _0}} \right) \sum \limits _{i = 1}^n {{\varvec{B}}_q^T\left( {{{\varvec{X}}_i}{{\beta } _0}} \right) {\varvec{\varLambda }}_i{\psi _\tau }\left( {{{\varvec{\varepsilon }} _i}} \right) }. \end{aligned}$$

It is easy to derive that $I = \sum _{i = 1}^n {{\varvec{B}}_q^T\left( {{{\varvec{X}}_i}{{\beta } _0}} \right) {\varvec{\varLambda }}_i{\psi _\tau }\left( {{{\varvec{\varepsilon }}_i}} \right) }$ is a sum of independent vector, $E\left( I\right) =0$ and $Cov\left( I \right) = \sum _{i = 1}^n {{\varvec{B}}_q^T\left( {{{\varvec{X}}_i}{{\beta } _0}} \right) {\varvec{\varLambda }}_i {{\varvec{\varSigma }} _{\tau i}} {\varvec{\varLambda }}_i {{\varvec{B}}_q}\left( {{{\varvec{X}}_i}{{\beta } _0}} \right) } .$ By the multivariate central limit theorem and the Slutsky’s theorem, we can complete the proof. $\square$

Lemma 3

Under conditions (C1)–(C7) and the number of knots satisfies ${n^{{1 /{(2d + 2)}}}} \ll {N_n} \ll {n^{{1 / 4}}}$, we have

$$\begin{aligned} \sqrt{n} \left( {\hat{{\beta }}} - {\beta }_0\right) \mathop \rightarrow \limits ^d N\left( {{\varvec{0}},{\varvec{J}}_{{{\beta }_0 ^{(r)}}}{{\varvec{\varPhi }} ^{ -1}}{\varvec{\varPsi }} {{ {{{\varvec{\varPhi }} ^{ -1}}} }}} {\varvec{J}}_{{{\beta }_0^{(r)}}}^T\right) . \end{aligned}$$

Proof

Define ${\varvec{H}}_i^T = {{\varvec{J}}_{{{\beta } ^{(r)}}}^T{\hat{{\varvec{X}}}}_i^T {\hat{{\varvec{G}}}'} \left( {{{\varvec{X}}_i}{\beta } ;{\beta } } \right) }$, ${\varvec{S_{i}}}=(S_{i1},\ldots ,S_{im_i})^T$ with ${S_{ij}} ={S_{ij}}\left( {\beta } \right) ={\psi _\tau }\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{\beta } ;{\beta } } \right) } \right) =I\left( {Y_{ij}}-{\hat{g}\left( {{\varvec{X}}_{ij}^T{\beta } ;{\beta } } \right) }\le 0 \right) - \tau$ being a discontinuous function, then $R\left( {\beta } ^{(r)} \right) = \sum _{i = 1}^n { {{\varvec{H}}_i^T {\varvec{\varLambda }} _i{{\varvec{S}}_{i}}\left( {\beta } \right) } }$. Let $\bar{R}\left( {\beta } ^{(r)} \right) = \sum _{i = 1}^n { {{\varvec{H}}_i^T{\varvec{\varLambda }} _i{{\varvec{P}}_{i}}\left( {\beta } \right) } }$, where ${\varvec{P_{i}}}=(P_{i1},\ldots ,P_{im_i})^T$ with $P_{ij}={P_{ij}}\left( {\beta } \right) =p\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{\beta } ;{\beta } } \right) \le 0} \right) - \tau .$ For any ${\beta }^{(r)}$ satisfying $\left\| {\beta }^{(r)}-{\beta }_0^{(r)}\right\| \le C{n^{{{ - 1} / 2}}}$, we have

$$\begin{aligned} \begin{array}{l} { R\left( {\beta }^{(r)} \right) - R\left( {\beta }_0^{(r)} \right) } \\ \quad = \sum \limits _{i = 1}^n { {{\varvec{H}}_i^T\left( {\beta } ^{(r)} \right) {\varvec{\varLambda }} _i{{\varvec{S}}_{i}}\left( {\beta } \right) } } - \sum \limits _{i = 1}^n { {{\varvec{H}}_i^T\left( {{{\beta } _0^{(r)}}} \right) {\varvec{\varLambda }} _i{{\varvec{S}}_{i}}\left( {{{\beta } _0}} \right) } } \\ \quad = \sum \limits _{i = 1}^n { {{\varvec{H}}_i^T\left( {\beta }^{(r)} \right) {\varvec{\varLambda }} _i\left\{ {{{\varvec{S}}_{i}}\left( {\beta } \right) - {{\varvec{S}}_{i}}\left( {{{\beta } _0}} \right) } \right\} } } \\ \qquad + \sum \limits _{i = 1}^n { {{{\left\{ {{{\varvec{H}}_i}\left( {\beta }^{(r)} \right) - {{\varvec{H}}_i}\left( {{{\beta } _0^{(r)}}} \right) } \right\} }^T}{\varvec{\varLambda }} _i{{\varvec{S}}_{i}}\left( {{{\beta } _0}} \right) } }. \\ \end{array} \end{aligned}$$

At first, the first term can be written as

$$\begin{aligned} \begin{array}{l} \sum \limits _{i = 1}^n { {{\varvec{H}}_i^T\left( {\beta }^{(r)} \right) {\varvec{\varLambda }} _i\left\{ {{{\varvec{S}}_{i}}\left( {\beta } \right) - {{\varvec{S}}_{i}}\left( {{{\beta } _0}} \right) } \right\} } } \\ \quad = \sum \limits _{i = 1}^n { {{\varvec{H}}_i^T\left( {\beta } ^{(r)} \right) {\varvec{\varLambda }} _i{{\varvec{P}}_{i}}\left( {\beta }^{(r)} \right) } } + \sum \limits _{i = 1}^n { {{\varvec{H}}_i^T\left( {\beta }^{(r)} \right) {\varvec{\varLambda }} _i\left( {{{\varvec{S}}_{i}}\left( {\beta } \right) - {{\varvec{S}}_{i}}\left( {{{\beta } _0}} \right) - {{\varvec{P}}_{i}}\left( {\beta } \right) } \right) } } \\ \quad = \bar{R}\left( {\beta } ^{(r)} \right) +\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^{{m_i}} {{{\varvec{h}}_{ij}}{f_{ij}}(0)\left[ {I\left( {{Y_{ij}} - \hat{g}\left( {X_{ij}^T\beta ;\beta } \right) \le 0} \right) } \right. } } \\ \qquad - I\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) \le 0} \right) \left. { - p\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{\beta } ;{\beta } } \right) \le 0} \right) + \tau } \right] \\ \quad \triangleq \bar{R}\left( {\beta } ^{(r)} \right) +\varUpsilon , \end{array} \end{aligned}$$

where ${\varvec{H}}_i^T = \left( {{{\varvec{h}}_{i1}},\ldots ,{{\varvec{h}}_{i{m_i}}}} \right)$ and ${\varvec{h}}_{ij}$ is a $(p-1)\times 1$ vector. According to Lemma 3 in Jung (1996) and Lemma 1, we have $\sup \left| \varUpsilon \right| = {o_p}\left( {\sqrt{n} } \right)$. Then, the first term becomes

$$\begin{aligned} \sum \limits _{i = 1}^n { {{\varvec{H}}_i^T\left( {\beta }^{(r)} \right) {\varvec{\varLambda }} _i\left\{ {{{\varvec{S}}_{i}}\left( {\beta } \right) - {{\varvec{S}}_{i}}\left( {{{\beta } _0}} \right) } \right\} } } = \bar{R}\left( {\beta } ^{(r)} \right) + {o_p}\left( {\sqrt{n} } \right) . \end{aligned}$$

By the law of large numbers (Pollard 1990), together with Lemma 1, the second term becomes

$$\begin{aligned} \begin{array}{l} \sum \limits _{i = 1}^n { {{{\left\{ {{{\varvec{H}}_i}\left( {\beta }^{(r)} \right) - {{\varvec{H}}_i}\left( {{{\beta } _0^{(r)}}} \right) } \right\} }^T}{\varvec{\varLambda }} _i{{\varvec{S}}_{i}}\left( {{{\beta } _0}} \right) } } \\ = \sum \limits _{i = 1}^n \sum \limits _{j = 1}^{{m_i}} \left( {{{\varvec{h}}_{ij}}\left( {{{\beta } ^{(r)}}} \right) - {{\varvec{h}}_{ij}}\left( {{\beta } _0^{(r)}} \right) } \right) f_{ij}(0)\\ ~~~~\times \left[ {I\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) \le 0} \right) - p\left( {{Y_{ij}} -g_0\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) \le 0} \right) } \right] \\ = {o_p}\left( {\sqrt{n} } \right) .\\ \end{array} \end{aligned}$$

Therefore, ${ R\left( {\beta }^{(r)} \right) - R\left( {\beta }_0^{(r)} \right) } =\bar{R}\left( {\beta } ^{(r)} \right) + {o_p}\left( {\sqrt{n} } \right)$. By Taylor’s expansion of $\bar{R}\left( {\beta } ^{(r)} \right)$, we can obtain

$$\begin{aligned} R\left( {{{\beta } ^{(r)}}} \right) - R\left( {{\beta } _0^{(r)}} \right) = \left[ {{{\partial \bar{R}\left( {{{\beta } ^{(r)}}} \right) } \Big / {\partial {{\beta } ^{(r)}}}}} \right] \left| {_{{{\beta } ^{(r)}} = {\beta } _0^{(r)}}} \right. \left( {{{{\beta } }^{(r)}} - {\beta } _0^{(r)}} \right) + {o_p}\left( {\sqrt{n} } \right) . \end{aligned}$$

Because $R\left( {\hat{{\beta }}}^{(r)} \right) = 0$ and ${\hat{{\beta }}}^{(r)}$ is in the $n^{-1/2}$ neighborhood of ${\beta }_0^{(r)}$, we have

$$\begin{aligned} \sqrt{n} \left( {{{\hat{{\beta }} }^{(r)}} - {\beta } _0^{(r)}} \right) = - \left( \frac{1}{{ n }}D\left( {{{\beta } _0^{(r)}}} \right) \right) ^{-1}\frac{1}{{\sqrt{n} }}R\left( {{\beta } _0^{(r)}} \right) + {o_p}\left( 1 \right) , \end{aligned}$$

(18)

where $R\left( {{\beta }_0 ^{(r)}} \right) =\sum _{i = 1}^n {{\varvec{J}}_{{{\beta }^{(r)}}}^T{\hat{{\varvec{X}}}}_i^T{\hat{{\varvec{G'}}}}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {\varvec{\varLambda }} _i{{\varvec{S}}_{i}}\left( {\beta } \right) } |_{{\beta } ^{(r)}={\beta }_0^{(r)}}$, ${{\varvec{S}}_{i}}\left( {\beta }_0 \right) = {\left( {{S_{i1}}\left( {\beta }_0 \right) ,\ldots ,{S_{i{m_i}}}\left( {\beta }_0 \right) } \right) ^T}$ with ${S_{ij}}\left( {\beta } _0 \right) = {I\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) \le 0} \right) }-\tau$ and

$$\begin{aligned} {D}\left( {{{\beta }_0 ^{(r)}}} \right) =\sum \limits _{i = 1}^n {{\varvec{J}}_{{{\beta } ^{(r)}}}^T {\hat{{\varvec{X}}}}_i^T{\hat{{\varvec{G}}}'}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {\varvec{\varLambda }} _i{{\varvec{\varLambda }} _i}{\hat{{\varvec{G}}}'}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {{\hat{{\varvec{X}}}}_i}{{\varvec{J}}_{{{\beta } ^{(r)}}}}} |_{{\beta } ^{(r)}={\beta }_0 ^{(r)}}. \end{aligned}$$

Thus, we have

$$\begin{aligned} \begin{array}{l} {S_{ij}}\left( {{{\beta } _0}} \right) = {S_{ij}}\left( {{{\beta } _0}} \right) - S_{ij}^*\left( {{{\beta } _0}} \right) + S_{ij}^*\left( {{{\beta } _0}} \right) \\ ~~~~~~~~~~= I\left( {{Y_{ij}} -\hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) \le 0} \right) - I\left( {{Y_{ij}} - g_0\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) \le 0} \right) \\ ~~~~~~~~~~~~~~ + I\left( {{Y_{ij}} - g_0\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) \le 0} \right) - \tau , \\ \end{array} \end{aligned}$$

where $S_{ij}^*\left( {{{\beta } _0}} \right) =I\left( {{Y_{ij}} - g_0\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) \le 0} \right) - \tau$. Moreover, $\Big | \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) - g_0\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) \Big | = {O_p}\left( {\sqrt{{{{N_n}} \big / n}} + N_n^{ - d}} \right)$ and $E\left\{ {I\left( {{Y_{ij}} - {g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) \le 0} \right) } \right\} = p\left( {{Y_{ij}} - {g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) \le 0} \right) = \tau$, we have $E\left( {S_{ij}}\left( {{{\beta } _0}} \right) \right) =o\left( 1\right)$. Therefore, we have $E (\varvec{S}_i (\beta _0)) = o (1)$ and

$$\begin{aligned} \begin{array}{l} Var\left( {\frac{1}{{\sqrt{n} }}R\left( {{\beta } _0^{(r)}} \right) } \right) \\ \quad = \frac{1}{n}\sum \limits _{i = 1}^n {{\varvec{J}}_{{{\beta }_0 ^{(r)}}}^T{\hat{{\varvec{X}}}}_i^T{\hat{{\varvec{G}}}'}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {\varvec{\varLambda }}_i{{\varvec{\varSigma }} _{\tau i}}} {\varvec{\varLambda }}_i{\hat{{\varvec{G}}}'}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {{\hat{{\varvec{X}}}}_i}{{\varvec{J}}_{{{\beta } ^{(r)}}}} |_{{\beta } ^{(r)}={\beta }_0 ^{(r)}} \left( 1+o(1)\right) . \end{array} \end{aligned}$$

Based on Lemma 1, together with ${\varvec{S}}_{i}\left( {{{\varvec{\beta }}_0}} \right)$ are the independent random variables, the multivariate central limit theorem implies that

$$\begin{aligned} {\frac{1}{{\sqrt{n} }}R\left( {{\beta } _0^{(r)}} \right) }\mathop \rightarrow \limits ^d N\left( {{\varvec{0}},{\varvec{\varPsi }} } \right) . \end{aligned}$$

(19)

By the law of large numbers and Lemma 1, we have

$$\begin{aligned} \frac{1}{n}D\left( {{\beta } _0^{(r)}} \right) \mathop \rightarrow \limits ^p {\varvec{\varPhi }}. \end{aligned}$$

(20)

Then, combine (18)–(20) and use the Slutsky’s theorem; it follows that

$$\begin{aligned} \sqrt{n} \left( {{{\hat{{\beta }} }^{(r)}} - {\beta } _0^{(r)}} \right) \mathop \rightarrow \limits ^d N\left( {{\varvec{0}},{\varvec{\varPhi }}^{ - 1}{{\varvec{\varPsi }}}{{ {\varvec{\varPhi }}^{ - 1}} }} \right) . \end{aligned}$$

(21)

According to the result of (21) and the multivariate delta method, we have

$$\begin{aligned} \sqrt{n} \left( {\hat{{\beta }}} - {\beta }_0\right) \mathop \rightarrow \limits ^d N\left( {{\varvec{0}},{\varvec{J}}_{{{\beta }_0 ^{(r)}}}{{\varvec{\varPhi }} ^{ -1}}{\varvec{\varPsi }} {{ {{\varvec{\varPhi }} ^{ -1}}} }} {\varvec{J}}_{{{\beta }_0^{(r)}}}^T\right) . \end{aligned}$$

$\square$

Proof of Theorem 1

Using conditions (C4), (C6), and (C7), similar to Lemma 3 (k) of Horowitz (1998), we obtain ${n^{{{ - 1} / 2}}}\tilde{R}\left( {{\beta }_0 ^{(r)}} \right) = {n^{{{ - 1} / 2}}}R\left( {{\beta }_0 ^{(r)}} \right) + {o_p}\left( 1 \right)$. In order to prove the asymptotic normality of $\tilde{{\beta }}^{(r)}$, we need to prove ${n^{ - 1}}\left\{ {\tilde{D}\left( {{\beta } _0^{(r)}} \right) - D\left( {{\beta } _0^{(r)}} \right) } \right\} \mathop \rightarrow \limits ^p 0$, where

$$\begin{aligned} \begin{array}{l} \frac{1}{n}\tilde{D}\left( {{\beta } _0^{(r)}} \right) \buildrel \varDelta \over =\frac{1}{n}\frac{{\partial \tilde{R}\left( {{{\beta } ^{(r)}}} \right) }}{{\partial {{\beta } ^{(r)}}}}\left| {_{{{\beta } ^{(r)}} = {\beta } _0^{(r)}}} \right. \\ ~~~~~~~~~~~~~~~~~= \frac{1}{n}\sum \limits _{i = 1}^n {{\varvec{J}}_{{{\beta }_0 ^{(r)}}}^T{\hat{{\varvec{X}}}}_i^T{\hat{{\varvec{G}}}'}\left( {{{\varvec{X}}_i}{\beta }_0 ;{\beta }_0 } \right) {\varvec{\varLambda }} _i{{\tilde{{\varvec{\varLambda }}} }_i}\left( {\beta }_0\right) {\hat{{\varvec{G}}}'}\left( {{{\varvec{X}}_i}{\beta }_0 ;{\beta }_0 } \right) {{\hat{{\varvec{X}}}}_i}{{\varvec{J}}_{{{\beta }_0 ^{(r)}}}}}. \end{array} \end{aligned}$$

It is easy to get that

$$\begin{aligned}&E\left\{ {\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) } \right\} - D\left( {{{\beta } _0^{(r)}}} \right) = \sum \limits _{i = 1}^n \sum \limits _{j = 1}^{{m_i}} {{\varvec{h}}_{ij}}\left( {{\beta } _0^{(r)}} \right) \\&\quad \times f_{ij}(0)\left\{ {{h^{ - 1}}E\left[ {K\left( {{{\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{\beta _0}} \right) } \right) } \Big / h}} \right) } \right] - {f_{ij}}\left( 0 \right) } \right\} {\varvec{h}}_{ij}^T\left( {{\beta } _0^{(r)}} \right) , \end{aligned}$$

where ${\varvec{h}}_{ij}$ is given in the proof of Lemma 3. Because

$$\begin{aligned} \begin{array}{l} \left| {{h^{ - 1}}E\left[ {K\left( {{{\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) }\right) } \big / h}} \right) } \right] - {f_{ij}}\left( 0 \right) } \right| \\ \quad = \left| {{h^{ - 1}}\int _{ - \infty }^{ + \infty } {K\left( {{{\left( {\epsilon + {g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) - \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) } \right) } \Big / h}} \right) } {f_{ij}}\left( \epsilon \right) d\epsilon - {f_{ij}}\left( 0 \right) } \right| \\ \quad = \left| {\int _{ - \infty }^{ + \infty } {K\left( t \right) } {f_{ij}}\left( { \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) -{g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) + ht} \right) dt - {f_{ij}}\left( 0 \right) } \right| \\ \quad = \left| {\int _{ - \infty }^{ + \infty } {K\left( t \right) } \left\{ {{f_{ij}}\left( { \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) -{g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) } \right) + ht{f'_{ij}}\left( {{\varsigma _t}} \right) } \right\} dt - {f_{ij}}\left( 0 \right) } \right| \\ \quad \le \left| {{f_{ij}}\left( { \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) -{g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) } \right) - {f_{ij}}\left( 0 \right) } \right| + h\int _{ - \infty }^{ + \infty } {\left| {K\left( t \right) t{f'_{ij}}\left( {{\varsigma _t}} \right) } \right| } dt, \\ \end{array} \end{aligned}$$

where $\varsigma _t$ is between $\hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) -{g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right)$ and $ht+ \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) -{g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right)$. By condition (C4), ${f'_{ij}}\left( \cdot \right)$ is uniformly bounded; hence, there exists a constant M satisfying $\left| {{f'_{ij}}\left( {{\varsigma _t}} \right) } \right| \le M$. Combining $\left| {\hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) - g_0\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) } \right| = {O_p}\left( {\sqrt{{{{N_n}} \big / n}} + N_n^{ - r}} \right)$ with conditions (C4), (C6), and (C7), we have

$$\begin{aligned} \begin{array}{l} \left| {{h^{ - 1}}E\left[ {K\left( {{{\left( {{Y_{ij}} - \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) } \right) } \Big / h}} \right) } \right] - {f_{ij}}\left( 0 \right) } \right| \\ \quad \le \left| {{f_{ij}}\left( \hat{g}\left( {{\varvec{X}}_{ij}^T{{\beta } _0};{{\beta } _0}} \right) -{g_0}\left( {{\varvec{X}}_{ij}^T{{\beta } _0}} \right) \right) - {f_{ij}}\left( 0 \right) } \right| + hC\rightarrow 0. \end{array} \end{aligned}$$

So we can obtain $\left| {{n^{ - 1}}\left\{ E\left\{ {\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) } \right\} - D\left( {{{\beta } _0^{(r)}}} \right) \right\} } \right| \rightarrow 0$. By the strong law of large number, we have ${{n^{ - 1}}\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) \rightarrow E\left( {{n^{ - 1}}\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) } \right) }$. Using the triangle inequality, we have

$$\begin{aligned} \begin{array}{l} \left| {{n^{ - 1}}\left\{ {\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) - D\left( {{{\beta } _0^{(r)}}} \right) } \right\} } \right| \\ \quad \le \left| {{n^{ - 1}}\left\{ {\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) - E\left( {\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) } \right) } \right\} } \right| + \left| {{n^{ - 1}}\left\{ {E\left( {\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) } \right) - D\left( {{{\beta } _0^{(r)}}} \right) } \right\} } \right| \rightarrow 0. \end{array} \end{aligned}$$

Furthermore, by the Taylor series expansion of $\tilde{R}\left( {{{\beta } ^{(r)}}} \right)$ around ${\beta }_0^{(r)}$, we have

$$\begin{aligned} \tilde{R}\left( {{{\beta } ^{(r)}}} \right) - \tilde{R}\left( {{\beta } _0^{(r)}} \right) = \tilde{D}\left( {{{\beta } ^*}} \right) \left( {{{\beta } ^{(r)}} -{\beta } _0^{(r)}} \right) , \end{aligned}$$

where ${\beta } ^*$ lies between ${\beta }^{(r)}$ and ${\beta }_0^{(r)}$. Let ${\beta }^{(r)}={\tilde{{\beta }}}^{(r)}$, we have

$$\begin{aligned} \sqrt{n} \left( {{\tilde{{\beta }}} ^{(r)} - {{\beta } _0^{(r)}}} \right) = - \left( \frac{1}{n}\tilde{D}\left( {{{\beta } ^*}} \right) \right) ^{-1}{n^{{{ - 1} / 2}}}\tilde{R}\left( {{{\beta } _0^{(r)}}} \right) \end{aligned}$$

because of $\tilde{R}\left( {\tilde{{\beta }} }^{(r)} \right) =0$. Since ${\tilde{{\beta }}}^{(r)}\rightarrow {\beta }_0^{(r)}$, we can obtain ${\beta }^*\rightarrow {\beta }_0^{(r)}$ and ${{\tilde{D}}^{ - 1}}\left( {{{\beta } ^*}} \right) \rightarrow {{\tilde{D}}^{ - 1}}\left( {{{\beta } _0^{(r)}}} \right)$. Since ${n^{ - 1}}\left\{ {\tilde{D}\left( {{{\beta } _0^{(r)}}} \right) - D\left( {{{\beta } _0^{(r)}}} \right) } \right\} \mathop \rightarrow \limits ^p 0$ and ${n^{{{ - 1} / 2}}}\tilde{R}\left( {{\beta } _0 ^{(r)}} \right) = {n^{{{ - 1} / 2}}}R\left( {{\beta } _0 ^{(r)}} \right) + {o_p}\left( 1 \right)$, we have

$$\begin{aligned} \sqrt{n} \left( {{\tilde{{\beta }}}^{(r)} - {{\beta } _0^{(r)}}} \right) = -\left( \frac{1}{n} D\left( {{{\beta } _0^{(r)}}} \right) \right) ^{-1}{n^{{{ - 1} / 2}}}R\left( {{{\beta } _0^{(r)}}} \right) + {o_p}\left( 1 \right) . \end{aligned}$$

Next, similar to the proof of Lemma 3, we can complete the proof of Theorem 1. $\square$

Proof of Theorem 2

Since $\left\| {{\tilde{{\beta }}}- {{\beta } _0}} \right\| ={O_p}\left( {{n^{{{ - 1} / 2}}}} \right)$, Theorem 2 (i) follows from this result and Lemma 1. Based on Lemma 2, we have

$$\begin{aligned} \bar{\sigma } _n^{-1}\left( u \right) {{\varvec{c}}_n^T}\left( {{\hat{{\varvec{\theta }}}}\left( {{{\beta } _0}} \right) - {{\varvec{\theta }} ^0}\left( {{{\beta } _0}} \right) } \right) \mathop \rightarrow \limits ^d N\left( {0,1} \right) . \end{aligned}$$

By the definition of ${\hat{g}\left( {u;{\beta } } \right) }$ and $\check{g} \left( u;{\beta } \right)$, choosing ${\varvec{c}}_n={\varvec{B}}_q\left( {u } \right)$ yields

$$\begin{aligned} \sigma _n^{-1}\left( u \right) \left( {\hat{g}\left( {u;{\beta }_0 } \right) - \check{g}\left( {u;{\beta }_0 } \right) } \right) \mathop \rightarrow \limits ^d N\left( {0,1} \right) . \end{aligned}$$

Thus, when ${\beta }$ is a known constant ${\beta }_0$ or estimated to the order ${{O_p}\left( {{n^{{{ - 1} / 2}}}} \right) }$, we can complete the proof of Theorem 2 (ii). $\square$

Proof of Theorem 3

Let ${\varvec{\varSigma }}_{\tau i}=\left( {{\sigma _{ijj'}}} \right) _{j,j' = 1}^{{m_i}}$ and ${\hat{{\varvec{\varSigma }}}}_{\tau i}=\left( {{\hat{\sigma } _{ijj'}}} \right) _{j,j' = 1}^{{m_i}}$ for $i=1\ldots ,n$. Based on the modified Cholesky decomposition, the diagonal elements of ${\hat{{\varvec{\varSigma }}}}_{\tau i}$ are ${\hat{\sigma } _{ijj}} = \hat{d}_{_{\tau ,ij}}^2 + \sum _{k = 1}^{j - 1} {\hat{l}_{\tau ,ijk}^2\hat{d}_{{\tau ,ik}}^2}$ for $j=1,\ldots ,m_i$, and the elements under the diagonal are ${\hat{\sigma } _{ijk}} = {{\hat{l}}_{\tau ,ijk}}\hat{d}_{_{\tau ,ik}}^2 + \sum _{k' = 1}^{k - 1} {{{\hat{l}}_{\tau ,ijk'}}{{\hat{l}}_{\tau ,ikk'}}\hat{d}_{{\tau ,ik'}}^2}$ for $j=2,\ldots ,m_i, k=1,\ldots ,j-1$. Similarly, the diagonal elements of ${\varvec{\varSigma }}_{\tau i}$ are ${\sigma _{ijj}} = d_{{\tau ,ij}}^2 + \sum _{k = 1}^{j - 1} { l_{\tau ,ijk}^2 d_{{\tau ,ik}}^2}$ for $j=1,\ldots ,m_i$, and the elements under the diagonal are ${\sigma _{ijk}} = {{ l}_{\tau ,ijk}} d_{_{\tau ,ik}}^2 + \sum _{k' = 1}^{k - 1} {{{ l}_{\tau ,ijk'}}{{l}_{\tau ,ikk'}} d_{{\tau ,ik'}}^2}$ for $j=2,\ldots ,m_i, k=1,\ldots ,j-1$. Since $\left( {{\hat{{\gamma }}}_\tau ^T,{\hat{{\varvec{\lambda }}}}_\tau ^T} \right) ^T$ are $\sqrt{n}$-consistent estimators, together with $\hat{d} _{\tau ,ij}^2= \exp \left( {\varvec{z}}_{ij}^T{\hat{{\varvec{\lambda }}}}_\tau \right)$ and $\hat{l}_{\tau ,ijk}= {\varvec{w}}_{ijk}^T{{\hat{{\gamma }}}_\tau }$ for $k<j=2,\ldots ,m_i$, we have

$$\begin{aligned} \hat{d}_{_{\tau ,ij}}^2 - d_{_{\tau ,ij}}^2 = {O_p}\left( {{n^{{{ - 1} / 2}}}} \right) ,{{\hat{l}}_{\tau ,ijk}} - {l_{\tau ,ijk}} = {O_p}\left( {{n^{{{ - 1} / 2}}}} \right) . \end{aligned}$$

Therefore, for $j=1,\ldots ,m_i$, we have

$$\begin{aligned} \begin{array}{l} {{\hat{\sigma } }_{ijj}} - {\sigma _{ijj}} = \left( {\hat{d}_{_{\tau ,ij}}^2 - d_{_{\tau ,ij}}^2} \right) + \sum \limits _{k = 1}^{j - 1} {\left( {\hat{l}_{\tau ,ijk}^2\hat{d}_{_{\tau ,ik}}^2 - l_{\tau ,ijk}^2d_{_{\tau ,ik}}^2} \right) } \\ ~~~~~~~~~~~~~~~~= \left( {\hat{d}_{_{\tau ,ij}}^2 - d_{_{\tau ,ij}}^2} \right) + \sum \limits _{k = 1}^{j - 1} {\left[ {\hat{l}_{\tau ,ijk}^2\left( {\hat{d}_{_{\tau ,ik}}^2 - d_{_{\tau ,ik}}^2} \right) + \left( {\hat{l}_{\tau ,ijk}^2 - l_{\tau ,ijk}^2} \right) d_{_{\tau ,ik}}^2} \right] } \\ ~~~~~~~~~~~~~~~~ = {O_p}\left( {{n^{{{ - 1} / 2}}}} \right) , \\ \end{array} \end{aligned}$$

and

$$\begin{aligned} {{\hat{\sigma } }_{ijk}} - {\sigma _{ijk}}= & {} \left( {{{\hat{l}}_{\tau ,ijk}} - {l_{\tau ,ijk}}} \right) \hat{d}_{_{\tau ,ik}}^2 + {l_{\tau ,ijk}}\left( {\hat{d}_{_{\tau ,ik}}^2 - d_{_{\tau ,ik}}^2} \right) \\&+\sum \limits _{k' = 1}^{k - 1} \left\{ {{\hat{l}}_{\tau ,ijk'}}{{\hat{l}}_{\tau ,ikk'}}\left( {\hat{d}_{_{\tau ,ik'}}^2 - d_{_{\tau ,ik'}}^2} \right) \right. \\&\left. + \left[ {\left( {{{\hat{l}}_{\tau ,ijk'}} - {l_{\tau ,ijk'}}} \right) {{\hat{l}}_{\tau ,ikk'}} + {{ l}_{\tau ,ijk'}}\left( {{{\hat{l}}_{\tau ,ikk'}} - {l_{\tau ,ikk'}}} \right) } \right] d_{_{\tau ,ik'}}^2 \right\} \\= & {} {O_p}\left( {{n^{{{ - 1} / 2}}}} \right) \\ \end{aligned}$$

for $j=2,\ldots ,m_i, k=1,\ldots ,j-1$. This completes the proof. $\square$

Proof of Theorem 4

Similar to the proof of Lemma 3, we have

$$\begin{aligned} \sqrt{n} \left( {{\bar{{\beta }}}^{(r)} - {{\beta } _0^{(r)}}} \right) = - {\left( {{n^{ - 1}}\bar{D}\left( {{{\beta } _0^{(r)}}} \right) } \right) ^{ - 1}}{n^{{{ - 1} / 2}}}\bar{R}\left( {{{\beta } _0^{(r)}}} \right) + {o_p}\left( 1 \right) , \end{aligned}$$

(22)

where ${{\bar{{\varvec{S}}}}_{i}}\left( {\beta } \right) = {\left( {{\bar{S}_{i1}}\left( {\beta } \right) ,\ldots ,{\bar{S}_{i{m_i}}}\left( {\beta } \right) } \right) ^T}$ with ${\bar{S}_{ij}}\left( {\beta } \right) =\psi _{h\tau }\left( Y_{ij} - \bar{g}\left( {{\varvec{X}}_{ij}^T{{\beta }};{{\beta } }} \right) \right)$, $\bar{R}\left( {{\beta } _0^{(r)}} \right) =\sum _{i = 1}^n {{\varvec{J}}_{{{\beta } ^{(r)}}}^T{\hat{{\varvec{X}}}}_i^T{\bar{{\varvec{G}}'}}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {\varvec{\varLambda }} _i{\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1}{{\bar{{\varvec{S}}}}_{i}}\left( {\beta } \right) } \left| {_{{{\beta } ^{(r)}} = {\beta } _0^{(r)}}} \right.$ and

$$\begin{aligned} \bar{D}\left( {{{\beta } _0^{(r)}}} \right) =\sum \limits _{i = 1}^n {{\varvec{J}}_{{{\beta } ^{(r)}}}^T {\hat{{\varvec{X}}}}_i^T{\bar{{\varvec{G}}'}}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {\varvec{\varLambda }} _i{\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1}{{\bar{{\varvec{\varLambda }}}} _i} \left( {{\beta } } \right) {\bar{{\varvec{G}}'}}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {{\hat{{\varvec{X}}}}_i}{{\varvec{J}}_{{{\beta } ^{(r)}}}}}\left| {_{{{\beta } ^{(r)}} = {\beta } _0^{(r)}}} \right. . \end{aligned}$$

Using conditions (C4), (C6), and (C7), similar to Lemma 3 (k) of Horowitz (1998), we obtain

$$\begin{aligned} {n^{{{ - 1} / 2}}} \bar{R}\left( {{\beta }_0 ^{(r)}} \right) = {n^{{{ - 1} / 2}}}R^*\left( {{\beta }_0 ^{(r)}} \right) + {o_p}\left( 1 \right) , \end{aligned}$$

(23)

where ${{\varvec{S}}_{i}^*}\left( {\beta } \right) = {\left( {{S_{i1}^*}\left( {\beta } \right) ,\ldots ,{S_{i{m_i}}^*}\left( {\beta } \right) } \right) ^T}$ with ${S_{ij}^*}\left( {\beta } \right) =\psi _{\tau }\left( Y_{ij} - \bar{g}\left( {{\varvec{X}}_{ij}^T{{\beta } };{{\beta } }} \right) \right)$ and $R^*\left( {{\beta } _0^{(r)}} \right) =\sum _{i = 1}^n {{\varvec{J}}_{{{\beta } ^{(r)}}}^T{\hat{{\varvec{X}}}}_i^T{\bar{{\varvec{G}}'}}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {\varvec{\varLambda }} _i{\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1}{{\varvec{S}}_{i}^*}\left( {\beta } \right) } \left| {_{{{\beta } ^{(r)}} = {\beta } _0^{(r)}}} \right.$. Similar to the proof of Theorem 1, we have

$$\begin{aligned} {n^{ - 1}}\left\{ {\bar{D}\left( {{{\beta } _0^{(r)}}} \right) - D^*\left( {{{\beta } _0^{(r)}}} \right) } \right\} \mathop \rightarrow \limits ^p 0, \end{aligned}$$

(24)

where $D^*\left( {{{\beta } _0^{(r)}}} \right) =\sum _{i = 1}^n {\varvec{J}}_{{{\beta } ^{(r)}}}^T {\hat{{\varvec{X}}}}_i^T{\bar{{\varvec{G}}'}}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {\varvec{\varLambda }} _i{\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1}{{\varvec{\varLambda }} _i} {\bar{{\varvec{G}}'}}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {{\hat{{\varvec{X}}}}_i}{{\varvec{J}}_{{{\beta } ^{(r)}}}}\left| {_{{{\beta } ^{(r)}} = {\beta } _0^{(r)}}} \right. .$ By (22)–(24), we have

$$\begin{aligned} \sqrt{n} \left( {{\bar{{\beta }}}^{(r)} - {{\beta } _0^{(r)}}} \right) = - {\left( {{n^{ - 1}}D^*\left( {{{\beta } _0^{(r)}}} \right) } \right) ^{ - 1}}{n^{{{ - 1} / 2}}}R^*\left( {{{\beta } _0^{(r)}}} \right) + {o_p}\left( 1 \right) . \end{aligned}$$

(25)

Similar to the proof of Lemma 1, we have

$$\begin{aligned}&\left| {{{\bar{g}}}({u};{{\beta } _0}) - {g_0}({u})} \right| \nonumber \\&\quad = O_p\left( {\sqrt{{{{N_n}} / n}} + N_n^{ - d}} \right) ,\left| {{{\bar{ g}'}}({u};{{\beta } _0}) - {g'_0}({u})} \right| = O_p\left( {\sqrt{{{N_n^3} / n}} + N_n^{ - d + 1}} \right) \nonumber \\ \end{aligned}$$

(26)

uniformly for any $u\in [a,b]$. Because ${\varvec{S}}_{i}^*\left( {{{\beta } _0}} \right)$ are the independent random variables, together with (26), we have $E\left( {{\varvec{S}}_{i}^*}\left( {{{\beta } _0}} \right) \right) =o\left( 1\right)$ and

$$\begin{aligned} \begin{array}{l} Var\left( {\frac{1}{{\sqrt{n} }} R^*\left( {{\beta } _0^{(r)}} \right) } \right) = \frac{1}{n}\sum \limits _{i = 1}^n {\varvec{J}}_{{{\beta } ^{(r)}}}^T{\hat{{\varvec{X}}}}_i^T{\bar{{\varvec{G}}'}}\\ \quad \left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {\varvec{\varLambda }}_i{\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1}{{\varvec{\varSigma }} _{\tau i}} {\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1}{\varvec{\varLambda }}_i{\bar{{\varvec{G}}'}}\left( {{{\varvec{X}}_i}{{\beta } };{{\beta } }} \right) {{\hat{{\varvec{X}}}}_i}{{\varvec{J}}_{{{\beta } ^{(r)}}}} \left| {_{{{\beta } ^{(r)}} = {\beta } _0^{(r)}}} \right. \left( 1+o(1)\right) .\\ \end{array} \end{aligned}$$

By the use of the following property (see Lemma 2 in Li 2011), let ${\varvec{A}}_n$ be a sequence of random matrices converging to an invertible matrix ${\varvec{A}}$, and then ${\varvec{A}}_n^{ - 1} = {{\varvec{A}}^{ - 1}} - {{\varvec{A}}^{ - 1}}\left( {{{\varvec{A}}_n} - {\varvec{A}}} \right) {{\varvec{A}}^{ - 1}} + {O_p}\left( {{{\left\| {{{\varvec{A}}_n} -{\varvec{A}}} \right\| }^2}} \right)$. This together with Theorem 3, we have ${\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1} - {\varvec{\varSigma }} _{\tau i}^{ - 1} = {O_p}\left( {{n^{{{ - 1} / 2}}}} \right)$ uniformly for all i. By the law of large numbers, (26), and the consistency of ${\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1}$, we have

$$\begin{aligned} n^{-1}{D^*}\left( {{{\beta } _0^{(r)}}} \right) \mathop \rightarrow \limits ^p {\varvec{\varGamma }}, ~~Var\left( {n^{-1/2}R^*\left( {{\beta } _0^{(r)}} \right) } \right) \mathop \rightarrow \limits ^p {\varvec{\varGamma }}. \end{aligned}$$

By the multivariate central limit theorem and the Slutsky’s theorem, together with (25), we can complete the proof. $\square$

Proof of Theorem 5

Similar to the proof of Theorem 2, together with the consistency ${\hat{{\varvec{\varSigma }}}} _{\tau i}^{ - 1} - {\varvec{\varSigma }} _{\tau i}^{ - 1} = {O_p}\left( {{n^{{{ - 1} / 2}}}} \right)$ of ${\varvec{ \varSigma }} _{\tau i}^{ - 1}$, when ${\beta }$ is a known constant ${\beta }_0$ or estimated to the order ${{O_p}\left( {{n^{{{ - 1} / 2}}}} \right) }$, we can complete the proof of Theorem 5. $\square$