Functional partially linear quantile regression model

Abstract

This paper considers estimation of a functional partially quantile regression model whose parameters include the infinite dimensional function as well as the slope parameters. We show asymptotical normality of the estimator of the finite dimensional parameter, and derive the rate of convergence of the estimator of the infinite dimensional slope function. In addition, we show the rate of the mean squared prediction error for the proposed estimator. A simulation study is provided to illustrate the numerical performance of the resulting estimators.

Appendix

Let ${{\varvec{Z}}}=({{\varvec{Z}}}_1^T,\ldots , {{\varvec{Z}}}^T_n)^T$ and ${{\varvec{U}}}=({{\varvec{U}}}_1,\ldots , {{\varvec{U}}}_n)^T$ be the $n$ by $p$ design matrix and the $n$ by $m$ design matrix for the constant slope parametric and the functional slope parametric component, respectively. Also let ${{\varvec{P}}}= {{\varvec{U}}}({{\varvec{U}}}^T{{\varvec{U}}})^{-1}{{\varvec{U}}}^T, {{\varvec{Z}}}^*=({{\varvec{I}}}-{{\varvec{P}}}){{\varvec{Z}}}$, ${{\varvec{Z}}}^* =({{\varvec{Z}}}^*_1,\ldots ,{{\varvec{Z}}}^*_n)^T$, ${{\varvec{S}}}_n={{{\varvec{Z}}}^*}^T{{\varvec{Z}}}^*$.

Lemma 1

Under conditions C1–C5, one has

$$\begin{aligned} {\frac{1}{n}{{\varvec{S}}}_{n}}=\Sigma +o_p(1). \end{aligned}$$

Proof

Let ${\varvec{\eta }}=({\varvec{\eta }}_1,\ldots ,{\varvec{\eta }}_n)^T$ and ${{\varvec{Z}}} = ({{\varvec{Z}}}- {\varvec{\Delta }}) +{\varvec{\Delta }}={\varvec{\eta }}+{\varvec{\Delta }}$, where ${\varvec{\Delta }}=(\langle {{\varvec{g}}}, X_1 \rangle ,\ldots \langle {{\varvec{g}}}, X_n \rangle )^T$ and ${{\varvec{g}}}=(g_1,\ldots ,g_p)^T$ is defined in Condition C5. We can write

$$\begin{aligned} {{\varvec{S}}}_{n}&= ( ({{\varvec{I}}}-{{\varvec{P}}}){{\varvec{Z}}})^{T}(({{\varvec{I}}}-{{\varvec{P}}}){{\varvec{Z}}}) \\&= ({\varvec{\eta }}+{\varvec{\Delta }})^{T}({{\varvec{I}}}-{{\varvec{P}}})^{T}({{\varvec{I}}}-{{\varvec{P}}})({\varvec{\eta }}+{\varvec{\Delta }})\\&= {\varvec{\eta }}^{T}{\varvec{\eta }}+{\varvec{\Delta }}^{T}({{\varvec{I}}}-{{\varvec{P}}})^{T}({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\Delta }}+{\varvec{\Delta }}^{T}({{\varvec{I}}}-{{\varvec{P}}})^{T}({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\eta }}\\&\quad +{\varvec{\eta }}^{T}({{\varvec{I}}}-{{\varvec{P}}})^{T}({{\varvec{I}}}-{{\varvec{P}}}) {\varvec{\Delta }}-{\varvec{\eta }}^{T}{{\varvec{P}}}^{T}{{\varvec{P}}}{\varvec{\eta }}\\&= {\varvec{\eta }}^{T}{\varvec{\eta }}+I_{n1}+I_{n2}+I_{n3}+I_{n4}, \end{aligned}$$

where

$$\begin{aligned}&I_{n1}={\varvec{\Delta }}^{T}({{\varvec{I}}}-{{\varvec{P}}})^{T}({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\Delta }},\\&I_{n2}={\varvec{\eta }}^{T}({{\varvec{I}}}-{{\varvec{P}}})^{T}({{\varvec{I}}}-{{\varvec{P}}})\Delta ,\\&I_{n3}={\varvec{\Delta }}^{T}({{\varvec{I}}}-{{\varvec{P}}})^{T}({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\eta }},\\&I_{n4}={\varvec{\eta }}^{T}{{\varvec{P}}}^{T}{{\varvec{P}}}{\varvec{\eta }}. \end{aligned}$$

Invoking the central limit theorem, one has

$$\begin{aligned} \frac{1}{n}{\varvec{\eta }}^{T}{\varvec{\eta }}\rightarrow \Sigma , \quad a.s. \end{aligned}$$

(7)

Invoking (7), to prove Lemma 1, it is enough to show that

$$\begin{aligned} \Vert I_{nl}\Vert =o_{p}(n),\quad l=1,2,3,4. \end{aligned}$$

By Condition C2, the fact $\Vert \phi _{j}-\hat{\phi }_{j}\Vert ^2=O_p(n^{-1}j^2)$ and the Karhunen-Loève representation, one has

$$\begin{aligned} \Vert {\varvec{\Delta }}-{{\varvec{U}}}{{\varvec{M}}}\Vert ^{2}&= \sum _{i=1}^{n}\sum _{l=1}^{p}\left\| \langle x_{i},g_{l}\rangle -\sum _{j=1}^{m}\langle x_{i},\hat{\phi }_{j}\rangle \lambda _{lj}^{^{\prime }}\right\| ^{2} \\&= \sum _{i=1}^{n}\sum _{l=1}^{p}\left\| \sum _{j=1}^{\infty }\phi _{j}\lambda _{lj}-\sum _{j=1}^{m}\hat{\phi }_{j}\lambda _{lj}^{^{\prime }}\right\| ^{2}\\&= \sum _{i=1}^{n}\sum _{l=1}^{p}\left\| \sum _{j=m+1}^{\infty }\phi _{j}\lambda _{lj}+\sum _{j=1}^{m}\left( \phi _{j}\lambda _{lj}-\hat{\phi }_{j}\lambda _{lj}^{^{\prime }}\right) \right\| ^{2}\\&\le 2\sum _{i=1}^{n}\sum _{l=1}^{p}\sum _{j=m+1}^{\infty }\left\| \langle x_{i},\phi _{j}\rangle \lambda _{lj}\Vert ^{2}\!+\!2\sum _{i=1}^{n}\sum _{l=1}^{p}\Vert \sum _{j=1}^{m}\left( \phi _{j}\lambda _{lj}\!-\! \hat{\phi }_{j}\lambda _{lj}^{^{\prime }}\right) \right\| ^{2}\\&\le 2\sum _{i=1}^{n}\sum _{l=1}^{p}\sum _{j=m+1}^{\infty }j^{-2b}\!+\!2\sum _{i=1}^{n}\sum _{l=1}^{p}\sum _{j=1}^{m}\Vert \phi _{j}(\lambda _{lj}-\lambda _{lj}^{^{\prime }})\!+\!(\phi _{j}\!-\!\hat{\phi }_{j})\lambda _{lj}^{^{\prime }}\Vert ^2\\&= O_{p}\left( n^{-\frac{2b-1}{a+2b}}\right) n+O_{p}\left( n\sum _{j=1}^{m} \Vert \phi _{j}-\hat{\phi }_{j}\Vert ^{2}\right) \\&= O_{p}\left( n^{-\frac{2b-1}{a+2b}}\right) n+O_{p}\left( n\sum _{j=1}^{m}n^{-1}j^2\right) \\&= O_{p}\left( nn^{-\frac{2b-1}{a+2b}}\right) +O_{p}\left( \frac{m(2m+1)(m+1)}{6}\right) \\&= O_{p}\left( n^{\frac{a+1}{a+2b}}\right) +O_{p}(m^{3}), \end{aligned}$$

where $\lambda _{lj}^{^{\prime }}=\langle \hat{\phi }_{j},g_{l}\rangle$ and $\lambda _{lj}=\langle \phi _{j},g_{l}\rangle , l=1,\ldots ,p.$

By Condition C3, there exists a matrix ${{\varvec{M}}}$ such that $\Vert {\varvec{\Delta }}-{{\varvec{U}}}{{\varvec{M}}}\Vert ^{2}=O_p\left( n^{a_1/(a+2b)}\right)$, where $a_1=\max (3, a+1)$. In addition, as ${{\varvec{P}}}$ is a projection matrix, we have

$$\begin{aligned} \Vert ({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\Delta }}\Vert ^{2}&= \Vert {\varvec{\Delta }} -{{\varvec{U}}}{{\varvec{M}}} \Vert ^{2}+\Vert {{\varvec{U}}}{{\varvec{M}}} - {{\varvec{P}}} {\varvec{\Delta }}\Vert ^{2}\nonumber \\&\le 2\Vert {\varvec{\Delta }} -{{\varvec{U}}}{{\varvec{M}}} \Vert ^{2}=O_p\left( n^{a_1/(a+2b)}\right) . \end{aligned}$$

(8)

By Condition C4 and the strong law of large numbers, one has $\frac{1}{n}{\varvec{\eta }}^T{\varvec{\eta }}$ converges almost surely to $\Sigma$. For $k\ne l$, one has

$$\begin{aligned} E\left\{ ({\varvec{\eta }}^T{{\varvec{P}}}{\varvec{\eta }})_{kl}\right\} ^2&= (\Sigma _{kl})^2(\text{ trace }({{\varvec{P}}}))^2+\left\{ \Sigma _{kk}\Sigma _{ll}+(\Sigma _{kl})^2\right\} \text{ trace }({{\varvec{P}}}{{\varvec{P}}}^{T})\nonumber \\&\quad +\left\{ E[{\varvec{\eta }}_{1k}^2{\varvec{\eta }}_{1l}^2]-2(\Sigma _{kl})^2-\Sigma _{kk} \Sigma _{ll}\right\} \sum _{s}{{\varvec{P}}}_{ss}. \end{aligned}$$

(9)

In addition, as ${{\varvec{P}}}$ is a projection matrix, this expression is $O(m)$. Since ${{\varvec{P}}}$ is a positive semidefine matrix, when $k=l$,

$$\begin{aligned} E\left\{ ({\varvec{\eta }}^T{{\varvec{P}}}{\varvec{\eta }})_{kk}\right\} =\Sigma _{kk}\text{ trace }({{\varvec{P}}})=O(m). \end{aligned}$$

(10)

Invoking (9) and (10), we have

$$\begin{aligned} \Vert {{\varvec{P}}}{\varvec{\eta }}\Vert ^{2}=O_{p}(m). \end{aligned}$$

(11)

Similarly, we have

$$\begin{aligned} \Vert ({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\eta }}\Vert ^{2}=O_{p}(m). \end{aligned}$$

(12)

Invoking (11) and (12) and Condition C2, we have

$$\begin{aligned} \Vert I_{n1}\Vert = \Vert ({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\Delta }}\Vert ^{2}&= O_p\left( n^{a_1/(a+2b)}\right) =o_{p}(n),\\ \Vert I_{n2}\Vert = \Vert I_{n3}\Vert&= \Vert {\varvec{\eta }}^{T}\left( {{\varvec{I}}}-{{\varvec{P}}}\right) ^{T}\left( {{\varvec{I}}}-{{\varvec{P}}}\right) {\varvec{\Delta }}\Vert ^{2}\\&\le \Vert ({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\eta }}\Vert ^{2}\Vert ({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\Delta }}\Vert ^{2}\\&= O_{p}(m)O_{p}\left( n^{\frac{a_1}{a+2b}}\right) \\&= o_{p}(n), \end{aligned}$$

and

$$\begin{aligned} \Vert I_{n4}\Vert = \Vert {\varvec{\eta }}^{T}{{\varvec{P}}}^{T}{{\varvec{P}}}{\varvec{\eta }}\Vert ^{2} =O_{p}(m)=o_{p}(n). \end{aligned}$$

The proof is hence complete. $\square$

Lemma 2

Under conditions C1–C5, we have

$$\begin{aligned} {{\varvec{S}}}_{n}^{-\frac{1}{2}}{{\varvec{Z}}}^{*T}{ {\varvec{\psi }}({\varvec{\varepsilon }})}\rightarrow N(0,\tau (1-\tau )I_{p}), \end{aligned}$$

where ${ {\varvec{\psi }}({\varvec{\varepsilon }})}=(\psi (\varepsilon _{1}),...,\psi (\varepsilon _{n}))^{T}.$

Proof

Invoking ${{\varvec{Z}}}={\varvec{\eta }}+{\varvec{\Delta }},$ we have

$$\begin{aligned} {{\varvec{S}}}_{n}^{-\frac{1}{2}}{{\varvec{Z}}}^{*T}{\varvec{\psi }}({\varvec{\varepsilon }})&= {{\varvec{S}}}_{n}^{-\frac{1}{2}}(({{\varvec{I}}}-{{\varvec{P}}}){{\varvec{Z}}})^{T}{\varvec{\psi }}({\varvec{\varepsilon }})\\&= {{\varvec{S}}}_{n}^{-\frac{1}{2}}(({{\varvec{I}}}-{{\varvec{P}}})({\varvec{\eta }}+{\varvec{\Delta }}))^{T}{\varvec{\psi }}({\varvec{\varepsilon }})\\&= {{\varvec{S}}}_{n}^{-\frac{1}{2}}{\varvec{\eta }}^{T}{\varvec{\psi }}({\varvec{\varepsilon }})-{{\varvec{S}}}_{n}^{-\frac{1}{2}}({{\varvec{P}}}{\varvec{\eta }})^{T}{\varvec{\psi }}({\varvec{\varepsilon }})+{{\varvec{S}}}_{n}^{-\frac{1}{2}}(({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\Delta }})^{T}{\varvec{\psi }}({\varvec{\varepsilon }}). \end{aligned}$$

By Lemma 1, (8) and (11), we have

$$\begin{aligned} \Vert {{\varvec{S}}}_{n}^{-\frac{1}{2}}({{\varvec{P}}}{\varvec{\eta }})^{T}\Vert ^2=o_{p}(1)\quad \text{ and } \quad \Vert {{\varvec{S}}}_{n}^{-\frac{1}{2}}(({{\varvec{I}}}-{{\varvec{P}}}){\varvec{\Delta }})^{T}{\varvec{\psi }}({\varvec{\varepsilon }})\Vert ^2=o_{p}(1). \end{aligned}$$

Thus,

$$\begin{aligned} {{\varvec{S}}}_{n}^{-\frac{1}{2}}{{\varvec{Z}}}^{*T}{\varvec{\psi }}({\varvec{\varepsilon }})={{\varvec{S}}}_{n}^{-\frac{1}{2}}{\varvec{\eta }}^{T}{\varvec{\psi }}({\varvec{\varepsilon }})+o_{p}(1). \end{aligned}$$

By condition C5 and the central limit theorem, one has

$$\begin{aligned} {{\varvec{S}}}_{n}^{-\frac{1}{2}}{{\varvec{Z}}}^{*T}{\varvec{\psi }}({\varvec{\varepsilon }})\rightarrow N\left( 0,\tau (1-\tau )I_{p}\right) \!. \end{aligned}$$

$\square$

Proof of Theorem 1

Let

$$\begin{aligned} {\varvec{\xi }}\left( \begin{array}{c} {\varvec{\theta }} \\ {\varvec{\gamma }} \end{array} \right) =\left( \begin{array}{c} {\varvec{\xi }}_{1}\\ {\varvec{\xi }}_{2} \end{array} \right) =\left( \begin{array}{c} f(0)S_{n}^{\frac{1}{2}}({\varvec{\theta }}-{\varvec{\theta }}_{0}) \\ H_{m}^{\frac{1}{2}}({\varvec{\gamma }}-{\varvec{\gamma }}_{m})+{{\varvec{H}}}_{m}^{-\frac{1}{2}}{{\varvec{U}}}{{\varvec{Z}}}(\theta -\theta _{0}) \end{array} \right) , \end{aligned}$$

where ${{\varvec{H}}}_{m}=m{{\varvec{U}}}^{T}{{\varvec{U}}}$. Let $\hat{{\varvec{\xi }}}={\varvec{\xi }}(\hat{{\varvec{\theta }}},\hat{{\varvec{\gamma }}})=(\hat{{\varvec{\xi }}}_{1}^{T},\hat{{\varvec{\xi }}}_{2}^{T})^{T}.$ Now, we show that $\Vert \hat{{\varvec{\xi }}}\Vert =O_{p}({\varvec{\delta }}_{n}).$ To do so, let $\tilde{{{\varvec{Z}}}}_{i}=\frac{1}{f(0)}{{\varvec{S}}}_{n}^{-\frac{1}{2}}{{\varvec{Z}}}_{i},$ $\tilde{{{\varvec{U}}}}_{i}={{\varvec{H}}}_{m}^{-1}{{\varvec{U}}}_{i}, R_{i}=\sum _{j=1}^{m}\langle x_{i},\hat{\phi }_{j}\rangle \gamma _{j0}-\int \nolimits _{0}^{1}\beta (t)x(t)dt$.

Note that $\Vert \phi _{j}-\hat{\phi }_{j}\Vert ^2=O_p(n^{-1}j^2)$, one has

$$\begin{aligned} \Vert R_{i}\Vert ^{2}&= \left\| \sum _{j=1}^{m}\langle x_{i},\hat{\phi }_{j}\rangle \gamma _{j0}-\sum _{j=1}^{\infty }\langle x_{i},\phi _{j}\rangle \gamma _{j0}\right\| ^{2}\\&\le \left\| \sum _{j=1}^{m}\langle x_{i},\hat{\phi }_{j}\rangle \gamma _{j0}-\sum _{j=1}^{m}\langle x_{i},\phi _{j}\rangle \gamma _{j0}\right\| ^{2}+\left\| \sum _{j=m+1}^{\infty }\langle x_{i},\phi _{j}\rangle \gamma _{j0}\right\| ^{2}\\&\le \sum _{j=1}^{m}\left\| \hat{\phi }_{j}-\phi _{j}\right\| ^{2}|\gamma _{j0}|^{2}+\sum _{j=m+1}^{\infty }|\gamma _{j0}|^{2}\\&= \sum _{j=1}^{m}O_{p}\left( n^{-1} j^{2-2b}\right) +\sum _{j=m+1}^{\infty }j^{-2b}\\&= O_{p_{}}\left( n^{-\frac{2b-1}{a+2b}}\right) +O\left( n^{-(2b-1)/(a+2b)}\right) \\&= O_{p}\left( n^{-\frac{2b-1}{a+2b}}\right) \!. \end{aligned}$$

Thus, one has

$$\begin{aligned} \sum _{i=1}^{n}\rho _{\tau }(Y_{i}-{{\varvec{Z}}}_{i}^{T}{\varvec{\theta }}-{{\varvec{U}}}_{i}^{T}\gamma )=\sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i} -\tilde{{{\varvec{Z}}}}_{i}^{T}{\varvec{\xi }}_{1}-\tilde{{{\varvec{U}}}}_{i}^{T}{\varvec{\xi }}_{2}-R_{i}\right) \!, \end{aligned}$$

which is minimized at $\hat{{\varvec{\xi }}}$.

By similar arguments to these of Lemma 1 of Cardot et al. (2005) for any $\kappa >0$, there exists $L_{\kappa }$ such that

$$\begin{aligned} P\left\{ \inf \limits _{\Vert {\varvec{\xi }}\Vert >L_{\kappa }\delta _{n}}\sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i} -\tilde{{{\varvec{Z}}}}_{i}^{T}{\varvec{\xi }}_{1}-\tilde{{{\varvec{U}}}}_{i}^{T}{\varvec{\xi }}_{2}-R_{i}\right) >\sum _{i=1}^{n}\rho _{\tau }(\varepsilon _{i}-R_{i})\right\} >1-\kappa . \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i} -\tilde{{{\varvec{Z}}}}_{i}^{T}\hat{{\varvec{\xi }}}_{1}-\tilde{{{\varvec{U}}}}_{i}^{T}\hat{{\varvec{\xi }}}_{2}-R_{i}\right) =\inf \limits _{{\varvec{\xi }}\in R^{m+p}} \rho _{\tau }\left( \varepsilon _{i} -\tilde{{{\varvec{Z}}}}_{i}^{T}{\varvec{\xi }}_{1}-\tilde{{{\varvec{U}}}}_{i}^{T}{\varvec{\xi }}_{2}-R_{i}\right) .\qquad \end{aligned}$$

(13)

Thus, we have

$$\begin{aligned} \sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i} -\tilde{{{\varvec{Z}}}}_{i}^{T}\hat{{\varvec{\xi }}}_{1}-\tilde{{{\varvec{U}}}}_{i}^{T}\hat{{\varvec{\xi }}}_{2}-R_{i}\right) <\sum _{i=1}^{n}\rho _{\tau }(\varepsilon _{i}-R_{i}). \end{aligned}$$

Then connecting this with Eq. (13), we obtain

$$\begin{aligned}&P\left\{ \inf \limits _{\Vert {\varvec{\xi }}\Vert >L_{\kappa }\delta _{n}}\sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i} -\tilde{{{\varvec{Z}}}}_{i}^{T}{\varvec{\xi }}_{1}-\tilde{{{\varvec{U}}}}_{i}^{T}{\varvec{\xi }}_{2}-R_{i}\right) \right. \\&\quad \left. >\sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i} -\tilde{{{\varvec{Z}}}}_{i}^{T}\hat{{\varvec{\xi }}}_{1}-\tilde{{{\varvec{U}}}}_{i}^{T}\hat{{\varvec{\xi }}}_{2}-R_{i}\right) \right\} >1-\kappa . \end{aligned}$$

Thus, $\Vert \hat{{\varvec{\xi }}}\Vert =O_{p}(\delta _{n}).$ This together with Lemma 1, and the definition of $\hat{{\varvec{\xi }}}$, one has

$$\begin{aligned} \Vert \hat{{\varvec{\theta }}}-{\varvec{\theta }}_{0}\Vert =\left\| \frac{1}{f(0)}{{\varvec{S}}}_{n}^{-\frac{1}{2}}\hat{{\varvec{\xi }}}_{1}\right\| =O_{p}\left( n^{-\frac{1}{2}}\Vert \hat{{\varvec{\xi }}}_{1}\Vert \right) =O_{p}(n^{-\frac{1}{2}}\delta _{n}). \end{aligned}$$

Note that

$$\begin{aligned} \Vert \hat{\beta }(t)-\beta _{0}(t)\Vert ^{2}&= \left\| \sum _{j=1}^{m}\hat{\gamma }_{j}\hat{\phi }_{j}-\sum _{j=1}^{\infty }\gamma _{j}\phi _{j}\right\| ^{2}\\&\le 2\left\| \sum _{j=1}^{m}\hat{\gamma }_{j}\hat{\phi }_{j}-\sum _{j=1}^{m}\gamma _{j}\phi _{j}\right\| ^{2} +2\left\| \sum _{j=m+1}^{\infty }\gamma _{j}\phi _{j}\right\| ^{2}\\&\le 4\left\| \sum _{j=1}^{m}(\hat{\gamma }_{j}-\gamma _{j})\hat{\phi }_{j}\right\| ^{2} +4\left\| \sum _{j=1}^{m}\gamma _{j}(\hat{\phi }_{j}-\phi _{j})\right\| ^{2} +2\sum _{j=m+1}^{\infty }\gamma _{j}^{2}\\&= K_{n1}+K_{n2}+K_{n3}. \end{aligned}$$

Now we consider $K_{n1}$, by the fact that the sequences $\{\hat{\phi }_j\}$ forms an orthonormal basis in $L^2([0, 1])$, one has

$$\begin{aligned} K_{n1}&= \left\| \sum _{j=1}^{m}(\hat{\gamma }_{j}-\gamma _{j})\hat{\phi }_{j}\right\| ^{2}\\&\le \sum _{j=1}^{m}(\hat{\gamma }_{j}-\gamma _{j})^{2}\\&\le \Vert \hat{{\varvec{\gamma }}} -{{\varvec{\gamma }}_m}\Vert ^{2}. \end{aligned}$$

By Lemma 1 of Stone (1985), it is easy to show that ${{\varvec{H}}}_{m}$ is positive definite for sufficiently large $n$. Therefore, one has

$$\begin{aligned} \Vert \hat{{\varvec{\gamma }}} -{{\varvec{\gamma }}_m}\Vert ^{2}&\le C(\hat{{\varvec{\gamma }}} -{{\varvec{\gamma }}_m})^TH_{m}(\hat{{\varvec{\gamma }}} -{{\varvec{\gamma }}_m})\\&\le O_{p}(n^{-1}\Vert {\hat{\varvec{\xi }}}_2\Vert ^2)+O_p(\Vert \hat{{\varvec{\theta }}}-{\varvec{\theta }}_0)\Vert ) =O_{p}\left( \delta _{n}^{2}\right) \!. \end{aligned}$$

As a result, we have $K_{n1}=O_{p}\left( \delta _{n}^{2}\right)$.

$$\begin{aligned} K_{n2}&\le m\sum _{j=1}^{m}\Vert \hat{\phi }_{j}-\phi _{j}\Vert ^{2}\gamma _{j}^{2}\le n^{-1}m\sum _{j=1}^{m}j^{2}\gamma _{j}^{2}\\&= O_{p}\left( n^{-1}m\sum _{j=1}^{m}j^{2-2b}\right) =O_{p}\left( n^{-\frac{a+3b-1}{a+2b}}\right) =o_{p}\left( \delta _{n}^{2}\right) \!, \end{aligned}$$

$$\begin{aligned} K_{n3}=\sum _{j=m+1}^{\infty }\gamma _{j}^{2}\le C\sum _{j=m+1}^{\infty }j^{-2b}=O(n^{-\frac{2b-1}{a+2b}})=O\left( \delta _{n}^{2}\right) \!. \end{aligned}$$

Therefore, one has

$$\begin{aligned} \Vert \hat{\beta }-\beta \Vert ^{2}=O_{p}\left( \delta _{n}^{2}\right) \!. \end{aligned}$$

Next we will show the asymptotic normality of $\hat{{\varvec{\theta }}}$, let ${\varvec{\xi }}_{1}^{*}=\frac{1}{f(0)}{{\varvec{S}}}_{n}^{-\frac{1}{2}}\sum _{i=0}^{n}{{\varvec{Z}}}_{i}^{*}\psi _{\tau }(\varepsilon _{i})$, according to Lemmas 1 and 2, ${\varvec{\xi }}_{1}^{*}$ is asymptotically normal with variance-covariance $\frac{\tau (1-\tau )}{f^{2}(0)}I_{p}$.

On the other hand, similar to He and Shi (1996), we can proof that $\Vert \hat{{\varvec{\xi }}}_{1}^{*}-\hat{{\varvec{\xi }}}_{1}\Vert =o_{p}(1).$ Thus,

$$\begin{aligned} \hat{{\varvec{\xi }}}_{1}=\hat{{\varvec{\xi }}}_{1}^{*}+o_{p}(1)=\frac{1}{f(0)}{{\varvec{S}}}_{n}^{-\frac{1}{2}}\sum _{i=0}^{n}Z_{i}^{*} \psi _{\tau }(\varepsilon _{i})+o_{p}(1). \end{aligned}$$

Obviously,

$$\begin{aligned} \sqrt{n}(\hat{{\varvec{\theta }}}_{0}-{\varvec{\theta }}_{0})\rightarrow N\left( 0,\frac{\tau (1-\tau )}{f^{2}(0)}\Sigma \right) . \end{aligned}$$

This completes the proof of Theorem 1.