Efficient parameter estimation via modified Cholesky decomposition for quantile regression with longitudinal data

Abstract

It is well known that specifying a covariance matrix is difficult in the quantile regression with longitudinal data. This paper develops a two step estimation procedure to improve estimation efficiency based on the modified Cholesky decomposition. Specifically, in the first step, we obtain the initial estimators of regression coefficients by ignoring the possible correlations between repeated measures. Then, we apply the modified Cholesky decomposition to construct the covariance models and obtain the estimator of within-subject covariance matrix. In the second step, we construct unbiased estimating functions to obtain more efficient estimators of regression coefficients. However, the proposed estimating functions are discrete and non-convex. We utilize the induced smoothing method to achieve the fast and accurate estimates of parameters and their asymptotic covariance. Under some regularity conditions, we establish the asymptotically normal distributions for the resulting estimators. Simulation studies and the longitudinal progesterone data analysis show that the proposed approach yields highly efficient estimators.

Appendix

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

(C1) The distribution function $F_{ij}(t)=p\left( {{Y_{ij}} - \varvec{X}_{ij}^T{\varvec{\beta } _\tau } \le t\left| {{\varvec{X}_{ij}}} \right. } \right)$ is absolutely continuous, with continuous densities $f_{ij}\left( \cdot \right)$ uniformly bounded, and its first derivative ${{\dot{f}}_{ij}}\left( \cdot \right)$ uniformly bounded away from 0 and $\infty$ at the points $0, i=1,..,n, j=1,\ldots ,m_i$.

(C2) The true value ${\varvec{\beta }}_\tau$ is in the interior of a bounded convex region ${\mathcal {B}}$.

(C3) When $n\rightarrow \infty$, the number of repeated measurements $m_i$ is bounded for each i.

(C4) For any positive definite matrix ${\varvec{W}}_i$, ${n^{ - 1}}\sum \nolimits _{i = 1}^n {\varvec{X}_i^T{\varvec{\Lambda } _i}{{\varvec{W}}_i}{\varvec{\Lambda } _i}{\varvec{X}_i}}$ converges to a positive definite matrix, where ${\varvec{\Lambda } _i}$ is an $m_i \times m_i$ diagonal matrix with the jth diagonal element ${f_{ij}}\left( 0 \right)$. In addition, ${\sup _i}\left\| {{\varvec{X}_i}} \right\| < \infty$, where $\left\| \cdot \right\|$ denotes the Euclidean norm.

(C5) Matrix $\varvec{\Omega }$ is positive definite and $\varvec{\Omega }=O\left( {{1 / n}} \right)$.

(C6) The differentiation of ${{\tilde{\varvec{U}}}_{w\tau } }({\varvec{\beta }}_\tau )$, $-\frac{{\partial {{\tilde{\varvec{U}}}_{w\tau } }({\varvec{\beta }}_\tau )}}{{\partial \varvec{\beta }_\tau }}$ is positive definite with probability 1.

(C7) For ${{\varvec{ \upsilon } }_{ij}} = {\left( {\sum \limits _{l = 1}^{j - 1} {{\psi _\tau }\left( {{{ \varepsilon }_{il}}} \right) W_{j,l,1}^{(i)}} ,\ldots ,\sum \limits _{l = 1}^{j - 1} {{\psi _\tau }\left( {{{ \varepsilon }_{il}}} \right) W_{j,l,s}^{(i)}} } \right) ^T}$, we have

$$\begin{aligned} {\left( {N - n} \right) ^{ - 1}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{\varvec{ \upsilon }_{ij}}\varvec{ \upsilon } _{ij}^T} } \mathop \rightarrow \limits ^p \varvec{\Lambda } > 0, \end{aligned}$$

where $\mathop \rightarrow \limits ^p$ denotes convergence in probability.

Proof of Theorem 1

By the definition of ${{\hat{\varvec{\upsilon }} }_{ij}}$, we have

$$\begin{aligned} {{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}= & {} {\left( {\sum \limits _{l = 1}^{j - 1} {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{il}}} \right) W_{j,l,1}^{(i)}} ,\ldots ,\sum \limits _{l = 1}^{j - 1} {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{il}}} \right) W_{j,l,s}^{(i)}} } \right) ^T} \\&- {\left( {\sum \limits _{l = 1}^{j - 1} {{\psi _\tau }\left( {{\varepsilon _{il}}} \right) W_{j,l,1}^{(i)}} ,\ldots ,\sum \limits _{l = 1}^{j - 1} {{\psi _\tau }\left( {{\varepsilon _{il}}} \right) W_{j,l,s}^{(i)}} } \right) ^T} \\= & {} \left( \sum \limits _{l = 1}^{j - 1} {\left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{il}}} \right) - {\psi _\tau }\left( {{\varepsilon _{il}}} \right) } \right) W_{j,l,1}^{(i)}} ,\ldots ,\right. \\&\left. \sum \limits _{l = 1}^{j - 1} {\left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{il}}} \right) - {\psi _\tau }\left( {{\varepsilon _{il}}} \right) } \right) W_{j,l,s}^{(i)}} \right) ^T, \end{aligned}$$

where ${{\hat{\varepsilon }} _{ij}}=Y_{ij}-\varvec{X}_{ij}^T{\hat{\varvec{\beta }}}_I$.

Similar to Fu and Wang (2012), we have

$$\begin{aligned} \sqrt{n} \left\{ {{{\hat{\varvec{\beta }} }_I} - {{\varvec{\beta }} _\tau }} \right\} \mathop \rightarrow \limits ^d N\left( {\mathbf {0},{\varvec{V}_I}} \right) , \end{aligned}$$

(18)

where

$$\begin{aligned} {\varvec{V}_I}= & {} \mathop {\lim }\limits _{n \rightarrow \infty } n{\left( {\sum \limits _{i = 1}^n {\varvec{X}_i^T{\varvec{\Lambda } _i}{\varvec{\Lambda } _i}{\varvec{X}_i}} } \right) ^{ - 1}}\left( {\sum \limits _{i = 1}^n {\varvec{X}_i^T{\varvec{\Lambda } _i}Cov \left( {{\varvec{S}_i}} \right) {\varvec{\Lambda } _i}{\varvec{X}_i}} } \right) \\&\times {\left( {\sum \limits _{i = 1}^n {\varvec{X}_i^T{\varvec{\Lambda } _i}{\varvec{\Lambda } _i}{\varvec{X}_i}} } \right) ^{ - 1}}. \end{aligned}$$

Thus, by conditions (C1) and (C4), we have

$$\begin{aligned} \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) {{\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) }^T}} } = {O_p}\left( {{n^{ - 1}}} \right) . \end{aligned}$$

(19)

Similarly, we have

$$\begin{aligned} \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) {\hat{\varvec{\upsilon }}} _{ij}^T} }= & {} \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) \varvec{\upsilon } _{ij}^T} } \\&+ \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) {{\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) }^T}} } \\= & {} \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) \varvec{\upsilon } _{ij}^T} } + {O_p}\left( {{n^{ - 1}}} \right) , \end{aligned}$$

and

$$\begin{aligned}&\left| {\frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) \varvec{\upsilon } _{ij}^T} } } \right| \\&\quad \le {o_p}\left( 1 \right) \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left( {\left| {\sum \limits _{l = 1}^{j - 1} {{\psi _\tau }\left( {{\varepsilon _{il}}} \right) W_{j,l,1}^{(i)}} } \right| ,\ldots ,\left| {\sum \limits _{l = 1}^{j - 1} {{\psi _\tau }\left( {{{{{\varepsilon }} }_{il}}} \right) W_{j,l,s}^{(i)}} } \right| } \right) } } \\&\quad = {o_p}\left( 1 \right) . \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) {\hat{\varvec{\upsilon }}} _{ij}^T} } = {o_p}\left( 1 \right) . \end{aligned}$$

By the fact that

$$\begin{aligned} \sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}}{\hat{\varvec{\upsilon }} }_{ij}}{\hat{\varvec{\upsilon }}} _{ij}^T= & {} \sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left\{ {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) + {\varvec{\upsilon } _{ij}}} \right\} \left\{ {\left( {{\hat{\varvec{\upsilon }}} _{ij}^T - \varvec{\upsilon } _{ij}^T} \right) + \varvec{\upsilon } _{ij}^T} \right\} } } \\= & {} \sum \limits _{i = 1}^n \sum \limits _{j = 2}^{{m_i}} \left\{ \left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) {{\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) }^T} + \left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) {\hat{\varvec{\upsilon }}} _{ij}^T \right. \\&\left. + {\varvec{\upsilon } _{ij}}{{\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) }^T} + {\varvec{\upsilon } _{ij}}\varvec{\upsilon } _{ij}^T \right\} . \end{aligned}$$

Together with the condition (C7), we obtain

$$\begin{aligned} \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{{\hat{\varvec{\upsilon }} }_{ij}}{\hat{\varvec{\upsilon }}} _{ij}^T = } } \frac{1}{{N - n}}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{\varvec{\upsilon } _{ij}}\varvec{\upsilon } _{ij}^T + {o_p}} } \left( 1 \right) \mathop \rightarrow \limits ^p \varvec{\Lambda } \end{aligned}$$

(20)

as $n\rightarrow \infty$. In addition

$$\begin{aligned}&\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{{\hat{\varvec{\upsilon }} }_{ij}}{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) } } \\&\quad =\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {\left\{ {\left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) \left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) } \right) } \right. } } \\&\qquad \left. { + \left( {{{\hat{\varvec{\upsilon }} }_{ij}} - {\varvec{\upsilon } _{ij}}} \right) {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) + {\varvec{\upsilon } _{ij}}\left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) } \right) + {\varvec{\upsilon } _{ij}}{\psi _\tau }\left( {{\varepsilon _{ij}}} \right) } \right\} \\&\quad \mathop {=}\limits ^{\Delta } \sqrt{N - n} \left( {{I_1} + {I_2} + {I_3} + {I_4}} \right) . \end{aligned}$$

Similar to the proof of (19), we have $I_1=o_p\left( 1\right)$, $I_2=o_p\left( 1\right)$ and $I_3=o_p\left( 1\right)$. Furthermore,

$$\begin{aligned} {I_4}= & {} \frac{1}{{\sqrt{N - n} }}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{\varvec{\upsilon } _{ij}}{\psi _\tau }\left( {{\varepsilon _{ij}}} \right) } } \\= & {} \frac{1}{{\sqrt{N - n} }}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{\varvec{\upsilon } _{ij}}\varvec{\upsilon } _{ij}^T\varvec{\theta }_\tau } } + \frac{1}{{\sqrt{N - n} }}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{\varvec{\upsilon } _{ij}}{e_{ij,\tau }}} }. \end{aligned}$$

It remains to show that

$$\begin{aligned} \frac{1}{{\sqrt{N - n} }}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{\varvec{\upsilon } _{ij}}{e_{ij,\tau }}} } \mathop \rightarrow \limits ^d N\left( {\mathbf {0},\varvec{\Delta } } \right) . \end{aligned}$$

(21)

Then, combine (20) and (21) and use the Slutsky’s theorem, it follows that

$$\begin{aligned} \sqrt{N - n} \left( {{\hat{\varvec{\theta }}}_\tau - {\varvec{\theta }_\tau }} \right) \mathop \rightarrow \limits ^d N\left( {\mathbf {0},{\varvec{\Lambda } ^{ - 1}}\varvec{\Delta } {\varvec{\Lambda } ^{ - 1}}} \right) . \end{aligned}$$

Next, we prove (21). Note that for any vector $s\times 1$ constant vector $\varvec{\kappa }= {\left( {{\kappa _1},\ldots ,{\kappa _s}} \right) ^T}$ whose components are not all zero. Let $\varvec{\Psi } = {\left( {N - n} \right) ^{ - 1}}\sum \nolimits _{i = 1}^n {\sum \nolimits _{j = 2}^{{m_i}} {\left( {{\varvec{\kappa }^T}{\varvec{\upsilon } _{ij}}} \right) {e_{ij,\tau }}} }$. It is easy to show that $E\left( \varvec{\Psi }\right) = 0$ and

$$\begin{aligned} Var\left( {\sqrt{N - n} \varvec{\Psi } } \right) = \frac{1}{{N - n}}\sum \limits _{i = 1}^n {E{{\left\{ {\sum \limits _{j = 2}^{{m_i}} {\left( {{\varvec{\kappa }^T}{\varvec{\upsilon } _{ij}}} \right) {e_{ij,\tau }}} } \right\} }^2}} . \end{aligned}$$

Denote ${\xi _i} = \sum \nolimits _{j = 2}^{{m_i}} {\left( {{\varvec{\kappa }^T}{\varvec{\upsilon } _{ij}}} \right) {e_{ij,\tau }}}$. Then

$$\begin{aligned} {S^2} =Var\left( \sum \limits _{i = 1}^n { {{\xi _i}} }\right) = \sum \limits _{i = 1}^n {Var\left( {{\xi _i}} \right) } = \sum \limits _{i = 1}^n E{\left( \left( \sum \limits _{j = 2}^{{m_i}} {\left( {{\varvec{\kappa }^T}{\varvec{\upsilon } _{ij}}} \right) {e_{ij,\tau }}} \right) ^2\right) } . \end{aligned}$$

It follows easily by checking Lyapunov condition that if

$$\begin{aligned} \frac{{\sum \nolimits _{i = 1}^n {E{{\left| {{\xi _i}} \right| }^3}} }}{{{S^3}}} \rightarrow 0. \end{aligned}$$

(22)

Thus, we can conclude that (21) holds. Now, we only need to show (22) holds. By the condition (C3), we have

$$\begin{aligned} \frac{{\sum \limits _{i = 1}^n {E{{\left| {{\xi _i}} \right| }^3}} }}{{{S^3}}} \le \frac{{CnE{{\left( {\sum \limits _{k = 1}^s {\left| {{\kappa _k}} \right| \left| {{\upsilon _{ijk}}} \right| } } \right) }^3}}}{{{n^{{3 /2}}}{{\left[ {E{{\left\{ {\sum \limits _{k = 1}^s {{\kappa _k}{\upsilon _{ijk}}} } \right\} }^2}} \right] }^{{3 /2}}}}} = {O_p}\left( {{n^{{{ - 1} /2}}}} \right) = {o_p}\left( 1 \right) . \end{aligned}$$

Thus we complete the proof of Theorem 1. $\square$

Proof of Theorem 2

Note that

$$\begin{aligned} {\hat{e}}_{ij}^2= & {} {\left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\hat{\varvec{\upsilon }}} _{ij}^T{\hat{\varvec{\theta }}}_\tau } \right) ^2} \\= & {} {\left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\hat{\varvec{\upsilon }}} _{ij}^T{\hat{\varvec{\theta }}}_\tau + {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) } \right) ^2} \\= & {} {\left[ {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) - \left( {{\hat{\varvec{\upsilon }}} _{ij}^T{\hat{\varvec{\theta }}}_\tau - \varvec{\upsilon } _{ij}^T\varvec{\theta } _\tau } \right) + {d_{ij,\tau }}{\varsigma _{ij}}} \right] ^2} \\= & {} d_{ij,\tau }^2\varsigma _{ij}^2 + 2{d_{ij,\tau }}{\varsigma _{ij}}\left[ {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) - \left( {{\hat{\varvec{\upsilon }}} _{ij}^T{\hat{\varvec{\theta }}} _\tau - \varvec{\upsilon } _{ij}^T\varvec{\theta }_\tau } \right) } \right] \\&+ {\left[ {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) - \left( {{\hat{\varvec{\upsilon }}} _{ij}^T{\hat{\varvec{\theta }}}_\tau - \varvec{\upsilon } _{ij}^T\varvec{\theta }_\tau } \right) } \right] ^2} \end{aligned}$$

It follows from (A.3) of Fan and Yao (1998) that

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

(23)

where

$$\begin{aligned} {I_1}= & {} \frac{1}{{Nf_T\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{K_h}\left( {{t_{ij}} - t} \right) \left\{ {{d_\tau ^2}\left( {{t_{ij}}} \right) - {d_\tau ^2}\left( t \right) - {{\dot{d}}_\tau ^2}\left( t \right) \left( {{t_{ij}} - t} \right) } \right\} } } , \\ {I_2}= & {} \frac{1}{{Nf_T\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{K_h}\left( {{t_{ij}} - t} \right) \left\{ {{d_\tau ^2}\left( {{t_{ij}}} \right) \left( {\varsigma _{ij}^2 - 1} \right) } \right\} } } , \\ {I_3}= & {} 2\frac{1}{{Nf_T\left( t \right) }}\sum \limits _{i = 1}^n \sum \limits _{j = 2}^{{m_i}} {K_h}\left( {{t_{ij}} - t} \right) {d_{ij,\tau }}{\varsigma _{ij}}\\&\left[ {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) - \left( {{\hat{\varvec{\upsilon }} }_{ij}^T{\hat{\varvec{\theta }}}_\tau - \varvec{\upsilon } _{ij}^T\varvec{\theta }_\tau } \right) } \right] , \\ {I_4}= & {} \frac{1}{{Nf_T\left( t \right) }}{\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{K_h}\left( {{t_{ij}} - t} \right) \left[ {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) - \left( {{\hat{\varvec{\upsilon }} }_{ij}^T{\hat{\varvec{\theta }} }_\tau - \varvec{\upsilon } _{ij}^T\varvec{\theta }_\tau } \right) } \right] } } ^2}. \end{aligned}$$

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

$$\begin{aligned}\begin{array}{l} (a)\,\,{I_1} = \frac{1}{2}{\mu _2}{h^2}{{{\ddot{d}}}_\tau ^2}\left( t \right) + {o_p}\left( {{h^2}} \right) , \\ (b)\,\,\sqrt{Nh} {I_2}\mathop \rightarrow \limits ^d N\left( {0,\Xi } \right) , \\ (c)\,\,{I_3} = {o_p}\left( { \frac{1}{{\sqrt{Nh} }}} \right) , \\ (d)\,\,{I_4} = {o_p}\left( { \frac{1}{{\sqrt{Nh} }}} \right) . \\ \end{array} \end{aligned}$$

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

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

It follows from the definition of $I_3$ that

$$\begin{aligned} {I_3}&= 2\frac{1}{{Nf_T\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{K_h}\left( {{t_{ij}} - t} \right) {d_{ij,\tau }}{\varsigma _{ij}}\left[ {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) } \right] } } \\&\quad - 2\frac{1}{{Nf_T\left( t \right) }}\sum \limits _{i = 1}^n {\sum \limits _{j = 2}^{{m_i}} {{K_h}\left( {{t_{ij}} - t} \right) {d_{ij,\tau }}{\varsigma _{ij}}\left( {{\hat{\varvec{\upsilon }}} _{ij}^T{\hat{\varvec{\theta }}}_\tau -\varvec{\upsilon } _{ij}^T\varvec{\theta }_\tau } \right) } } \\&\mathop {=}\limits ^{\Delta } {I_{31}} + {I_{32}}, \end{aligned}$$

By (18), we know that ${\hat{\varvec{\beta }}_{\varvec{I}} }$ is a root-n consistent estimator of $\varvec{\beta }_\tau$, together with conditions (C1) and (C4), we have

$$\begin{aligned} {\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{ij}}} \right) - {\psi _\tau }\left( {{\varepsilon _{ij}}} \right) = {\psi _\tau }\left( {{Y_{ij}} -\varvec{X}_{ij}^T{{\hat{\varvec{\beta }} }_I}} \right) - {\psi _\tau }\left( {{Y_{ij}} - \varvec{X}_{ij}^T{\varvec{\beta } _\tau }} \right) = {O_p}\left( {{1 /{\sqrt{n} }}} \right) ,\nonumber \\ \end{aligned}$$

(24)

and

$$\begin{aligned} \left( {{\hat{\varvec{\upsilon }}} _{ij}^T - \varvec{\upsilon } _{ij}^T} \right) \varvec{\theta }_\tau= & {} \left( \sum \limits _{l = 1}^{j - 1} \left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{il}}} \right) - {\psi _\tau }\left( {{\varepsilon _{il}}} \right) } \right) W_{j,l,1}^{(i)},\ldots ,\right. \nonumber \\&\left. \sum \limits _{l = 1}^{j - 1} {\left( {{\psi _\tau }\left( {{{{\hat{\varepsilon }} }_{il}}} \right) - {\psi _\tau }\left( {{\varepsilon _{il}}} \right) } \right) W_{j,l,s}^{(i)}} \right) \varvec{\theta } _\tau \nonumber \\= & {} {O_p}\left( {{1 / {\sqrt{n} }}} \right) . \end{aligned}$$

(25)

Furthermore, by Theorem 1 and (24), (25), we have

$$\begin{aligned}&{\hat{\varvec{\upsilon }}} _{ij}^T{\hat{\varvec{\theta }}}_\tau - \varvec{\upsilon } _{ij}^T\varvec{\theta }_\tau \nonumber \\&\quad = \left( {\hat{\varvec{\upsilon }} _{{\varvec{ij}}}^{\varvec{T}} - \upsilon _{{\varvec{ij}}}^{\varvec{T}}} \right) \varvec{\theta } + \left( {\hat{\varvec{\upsilon }} _{{\varvec{ij}}}^{\varvec{T}} - {\varvec{\upsilon }} _{{\varvec{ij}}}^{\varvec{T}}} \right) \left( {\hat{\varvec{\theta }}}_\tau - \varvec{\theta } _\tau \right) + \varvec{\upsilon } _{ij}^T\left( {\hat{\varvec{\theta }}}_\tau - \varvec{\theta }_\tau \right) \nonumber \\&\quad = {O_p}\left( {{n^{{{ - 1} / 2}}}} \right) + {O_p}\left( {{n^{{{ - 1} /2}}}} \right) {O_p}\left( {{n^{{{ - 1} /2}}}} \right) + {O_p}\left( {{n^{{{ - 1} /2}}}} \right) \nonumber \\&\quad ={O_p}\left( {{n^{{{ - 1} / 2}}}} \right) . \end{aligned}$$

(26)

On the other hand, according to $E\left( {{\varsigma _{ij}}|{t_{ij}}} \right) = 0$, ${{Var}}\left( {{\varsigma _{ij}}|{t_{ij}}} \right) = 1$, together with (24) and (26), we have

$$\begin{aligned} {I_{31}} = {o_p}\left( {{1 /{\sqrt{Nh} }}} \right) ,{I_{32}} ={o_p}\left( {{1 /{\sqrt{Nh} }}} \right) . \end{aligned}$$

Then ${I_{3}} = {o_p}\left( {{1 / {\sqrt{Nh} }}} \right)$. By the same arguments as proving $\varvec{I}_3$, we have ${I_{4}} = {o_p}\left( {{1 / {\sqrt{Nh} }}} \right)$. Under the conditions $h\rightarrow 0$, $Nh \rightarrow \infty$ as $n\rightarrow \infty$ and $\lim {\sup _{n \rightarrow \infty }}N{h^5} < \infty$, then the proof of Theorem 2 is completed. $\square$

Proof of Theorem 3

By the similar arguments of Theorem 3.1 in Lu and Fan (2015), we can show that ${{\hat{\varvec{\beta }}_{{\varvec{w}}{\varvec{\tau }}} }}$ is a root-n consistent estimator of $\varvec{\beta }_\tau$ and has the asymptotically normal distribution, and thus omitted. $\square$

Proof of Theorem 4

By the similar arguments of Lemma 3.1 in Lu and Fan (2015), we can complete the proof of Theorem 4, and thus omitted. $\square$

Proof of Corollary 1

By the similar arguments of Theorem 3.2 in Lu and Fan (2015), we can complete the proof of Corollary 1, and thus omitted. $\square$