Robust variable selection in high-dimensional varying coefficient models based on weighted composite quantile regression

Abstract

In this paper, a new variable selection procedure based on weighted composite quantile regression is proposed for varying coefficient models with a diverging number of parameters. The proposed method is based on basis function approximation and the group SCAD penalty. The new estimation method can achieve both robustness and efficiency. Furthermore, the theoretical properties of our procedure, including consistency in variable selection and the oracle property in estimation are established under some suitable assumptions. Finally, the finite sample behavior of the estimator is evaluated by simulation studies. In addition, some interesting extensions are made to separate constant coefficients from varying coefficients.

Appendix

Let C denote a generic constant that might assume different values at different places.

Lemma 1

Suppose ${\varvec{\pi }^{(j)}}{(u)^T}\varvec{\gamma }_j^0$ is the best approximating spline function for ${\beta _j}(u)$ and ${\varvec{\gamma } ^0} = {(\varvec{\gamma } _1^{0T},\ldots ,\varvec{\gamma } _{{p_n}}^{0T})^T}$. Under the conditions (C1)–(C5) together with two constants $C_1$ and $C_2$, we have

• (a) $\mathop {\sup }\limits _{u \in [0,1]} \left| {{\beta _j}(u) - {\varvec{\pi }^{(j)}} {{(u)}^T}\varvec{\gamma } _j^0} \right| \le C_1 K_n^{ - r}$,

• (b) $\mathop {\sup }\limits _{(u,{\varvec{X}}) \in [0,1] \times {R^{{p_n}}}} \left| {{{\varvec{X}}^T}\varvec{\beta } (u) - \varvec{\Pi }^T{\varvec{\gamma } ^0}} \right| \le C_2 K_n^{ - r}\sqrt{{p_n}}$.

  Let ${R_{ni}} = \varvec{\Pi } _{i}^T{\varvec{\gamma } ^0} - \mathbf{X }_i^T\varvec{\beta } ({U_i}).$ By (b) of Lemma 1, it is easy to see that $\mathop {\max }\limits _{i} \left| {{R_{ni}}} \right| \le C_2 K_n^{ - r}\sqrt{{p_n}}$.

Proof of Lemma 1

Since ${\varvec{\pi }^{(j)}}{(u)^T}\varvec{\gamma }_j^0$ is the best approximating spline function for ${\beta _j}(u)$. According to the result on page 149 of de Boor (2001), for ${\beta _j}(u)$ satisfying condition (C1), we have $\mathop {\sup }\limits _{u \in [0,1]} \left| {{\beta _j}(u) - {\varvec{\pi }^{(j)}} {{(u)}^T}\varvec{\gamma } _j^0} \right| \le C_1 K_n^{ - r}$. We complete the proof of Lemma 1 (a). Now, we show Lemma 1 (b). Let

$$\begin{aligned} \varvec{B}(u) = \left( \begin{array}{l} {B_{11}}(u)~ \cdots ~{B_{1{K_1}}}(u)~~0 ~\cdots 0~~~~0 ~~~~~~~\cdots ~~~~~~0 \\ ~~~\vdots ~~~~~~ \ddots ~~~~~\vdots ~~~~~~~~\vdots ~ \ddots ~ \vdots ~~~~\vdots ~~~~~~~~\ddots ~~~~~\vdots \\ Z~~~0 ~~~~~\cdots ~~~~~0~~~~~~~0 ~\cdots ~0~~{B_{{p_n},1}}(u) ~\cdots ~ {B_{{p_n},{K_{{p_n}}}}}(u) \\ \end{array} \right) . \end{aligned}$$

So $\varvec{\Pi }= {{\varvec{B}(u)}^T}\varvec{X}$. Using the condition (C3), we have

$$\begin{aligned} \begin{array}{l} \mathop {\sup }\limits _{(u,{\varvec{X}}) \in [0,1] \times {R^{{p_n}}}}{\left| {{\varvec{X}^T}\varvec{\beta }(u) - {\varvec{\Pi }^T}{\varvec{\gamma }^0}} \right| ^2}\\ = \mathop {\sup }\limits _{(u,{\varvec{X}}) \in [0,1] \times {R^{{p_n}}}}{\left| {{\varvec{X}^T}\left( {\varvec{\beta }(u) - \varvec{B}(u){\varvec{\gamma }^0}} \right) } \right| ^2} \\ = {\sup _{(u,\varvec{X}) \in [0,1] \times {R^{{p_n}}}}}{\left( {\varvec{\beta }(u) -\varvec{B}(u){\varvec{\gamma }^0}} \right) ^T}\varvec{X}{\varvec{X}^T}\left( {\varvec{\beta }(u) - \varvec{B}(u){\varvec{\gamma }^0}} \right) \\ \le \mathop {\sup }\limits _{(u,X) \in [0,1] \times {R^{{p_n}}}} {\lambda _{\max }}(\varvec{X}{\varvec{X}^T}){\left( {\varvec{\beta }(u) - \varvec{B}\left( u \right) {\varvec{\gamma }^0}} \right) ^T}\left( {\varvec{\beta }(u) - \varvec{B}\left( u \right) {\varvec{\gamma }^0}} \right) \\ \rightarrow \mathop {\sup }\limits _{u \in [0,1]} {\lambda _{\max }}\left\{ {E(\varvec{X}{\varvec{X}^T}\left| {U = u} \right. )} \right\} \sum \nolimits _{j = 1}^{{p_n}} {{{\left( {{\beta _j}(u) - {\pi ^{(j)}}{{(u)}^T}\gamma _j^0} \right) }^2}} \\ \le \mathop {\sup }\limits _{u \in [0,1]} {\lambda _{\max }}\left\{ {E(\varvec{X}{\varvec{X}^T}\left| {U = u} \right. )} \right\} \sum \nolimits _{j = 1}^{{p_n}} {\mathop {\sup }\limits _{u \in [0,1]} {{\left( {{\beta _j}(u) - {\pi ^{(j)}}{{(u)}^T}\gamma _j^0} \right) }^2}} \\ \le {\lambda ^{\max }}C_1^2{p_n}K_n^{ - 2r} \\ \end{array} \end{aligned}$$

where ${\lambda ^{\max }} = \mathop {\sup }\limits _{u \in [0,1]} {\lambda _{\max }}\left\{ {E(\varvec{X}{\varvec{X}^T}\left| {U = u} \right. )} \right\}$ and ${\lambda _{\max }}(\varvec{A})$ denotes the maximum eigenvalues of a positive definite matrix $\varvec{A}$.

Thus, $\mathop {\sup }\limits _{(u,{\varvec{X}}) \in [0,1] \times {R^{{p_n}}}} \left| {{{\varvec{X}}^T}\varvec{\beta } (u) - \varvec{\Pi }^T{\varvec{\gamma } ^0}} \right| \le {C_2}\sqrt{{p_n}} K_n^{ - r}$, where ${C_2} = \sqrt{{\lambda ^{\max }}C_1^2}$. This complete the proof. $\square$

Proof of Theorem 1

Let ${\alpha _n} = \sqrt{{p_n}} \left( {{n^{ - r/(2r+1)}} + {a_n}} \right) ,{\varvec{u}_n} = \alpha _n^{ - 1}\left( {\varvec{\hat{\gamma }} - {\varvec{\gamma }^0}} \right)$ with $\varvec{u}_{nj}=\alpha _n^{ - 1}\left( {\varvec{\hat{\gamma }}_j - {\varvec{\gamma }_j^0}} \right)$, ${v_k} {=} \alpha _n^{ - 1}\left( {{\hat{c}_{\tau _k}} - c_{\tau _k}} \right)$ and $\mathscr {F}_n{=}\left\{ {\left( {{\varvec{u}_n},\varvec{v}} \right) : {{\left\| {{{\left( {\varvec{u}_n^T,\mathbf{v ^T}} \right) }^T}} \right\| }_2} = C} \right\}$, where C is a large enough constant and $\varvec{c} = {({c_{{\tau _1}}},\ldots ,{c_{{\tau _q}}})^T}, \varvec{v} = {({v_1},\ldots ,{v_q})^T}$. Our aim is to show that for any given $\eta >0$, there is a large constant C such that, for large n we have

$$\begin{aligned} P\left\{ {\mathop {\inf }\limits _{\left( {{\varvec{u}_n},\varvec{v}} \right) \in \mathscr {F}_n} {PL_n}({\varvec{\gamma } ^0} + {\alpha _n}{\varvec{u}_n},\varvec{c} + {\alpha _n}{\varvec{v}},\varvec{\omega } ) > {PL_n}({\varvec{\gamma } ^0},\varvec{c},\varvec{\omega })} \right\} \ge 1 - \eta . \end{aligned}$$

(16)

This implies that, with probability tending to one, there is local minimum $\varvec{\hat{\gamma }}$ in the ball $\left\{ {\left( {{\varvec{\gamma } ^0} + {\alpha _n}{\varvec{u}_n},\varvec{c} + {\alpha _n}{\varvec{v}}} \right) {{: }}{{\left\| {{{\left( {\varvec{u}_n^T,{\varvec{v}^T}} \right) }^T}} \right\| }_2} \le C} \right\}$ such that ${\left\| {\varvec{\hat{\gamma }} - {\varvec{\gamma } ^0}} \right\| _2} = {O_p}({\alpha _n})$. Let ${D_n}({\varvec{u}_n},\varvec{v}) = {{PL_n}({\varvec{\gamma } ^0} + {\alpha _n}{\varvec{u}_n},\varvec{c} + {\alpha _n}{\varvec{v}},\varvec{\omega }) - {PL_n}({\varvec{\gamma } ^0},\varvec{c},\varvec{\omega })}$ and ${S_n}({\varvec{u}_n},\varvec{v})= {{L_n}({\varvec{\gamma } ^0} + {\alpha _n}{\varvec{u}_n},\varvec{c} + {\alpha _n}{\varvec{v}},\varvec{\omega }) - {L_n}({\varvec{\gamma }^0},\varvec{c},\varvec{\omega })}$. Then

$$\begin{aligned} {D_n}({\varvec{u}_n},\varvec{v}) = {S_n}({\varvec{u}_n},\varvec{v}) + {P_{{\lambda _n}}}({\varvec{u}_n}), \end{aligned}$$

(17)

where ${P_{{\lambda _n}}}({\varvec{u}_n})=n\sum \limits _{j = 1}^{{p_n}} {\left[ {{p_{{\lambda _n}}}\left( {{\left\| {\varvec{\gamma }_j^0 + {\alpha _n}{\varvec{u}_{nj}}} \right\| }_2}\right) - {p_{{\lambda _n}}}\left( {{\left\| {\varvec{\gamma }_j^0} \right\| }_2}\right) } \right] }$.

By the identity (Knight 1998),

$$\begin{aligned} \left| {z - y} \right| - \left| z \right| = - y{\mathrm{sgn}} (z) + 2(y - z)\left\{ {I(0 < z < y) - I(y < z < 0)} \right\} , \end{aligned}$$

we have

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

Then we can rewrite ${S_n}({\varvec{u}_n},\varvec{v})$ as

$$\begin{aligned} {S_n}({\varvec{u}_n},\varvec{v})= & {} \sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\alpha _n}(\varvec{\Pi } _i^T{\varvec{u}_n} + {v_k})} } [I({\varepsilon _i} < {R_{ni}} + c_{\tau _k}) - {\tau _k}]\nonumber \\&+\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {\int _0^{{\alpha _n}(\varvec{\Pi } _i^T{\varvec{u}_n} + {v_k})}{[I({\varepsilon _i} \le x + {R_{ni}} + c_{\tau _k}) - I({\varepsilon _i} \le {R_{ni}} +c_{\tau _k})]} } } dx\nonumber \\= & {} \sqrt{n} \alpha _n \left( {\varvec{Z}_n^T{\varvec{u}_n} + \varvec{z} _n^T\varvec{v}} \right) + \sum \limits _{k = 1}^q {{\omega _k}B_n^{(k)}}, \end{aligned}$$

(18)

where

$$\begin{aligned} B_n^{(k)}= & {} \sum \limits _{i = 1}^n {\int _0^{{\alpha _n}(\varvec{\Pi } _i^T{\varvec{u}_n} + {v_k})} {[I({\varepsilon _i} \le x + {R_{ni}} + c_{\tau _k}) - I({\varepsilon _i} \le {R_{ni}} + c_{\tau _k})]} }dx,\\ {\varvec{Z}_n}= & {} n^{-1/2} \sum \limits _{i = 1}^n {{\varvec{\Pi } _i}\sum \limits _{k = 1}^q {{\omega _k}} [I({\varepsilon _i} < {R_{ni}} + c_{\tau _k}) - {\tau _k}],}\\ {z _{n,k}}= & {} n^{-1/2} \sum \limits _{i = 1}^n {{\omega _k}\left[ {I({\varepsilon _i} < {R_{ni}} +c_{\tau _k}) - {\tau _k}} \right] }\;\, \mathrm{and}\;\, {\varvec{z} _n} = {\left( {{z _{n,1}},\ldots ,{z _{n,q}}} \right) ^T}. \end{aligned}$$

Note that, $\Vert {\varvec{\Gamma }_n}\Vert =O(1)$ and $\varepsilon _i$ is independent of $\varvec{X}_i$ and $U_i$, it follow that $E(\varvec{Z}_n^T{\varvec{u}_n}) =0$ and $E\{ {(\varvec{Z}_n^T{\varvec{u}_n})^2}\} = \varvec{u}_n^TE({\varvec{Z}_n}\varvec{Z}_n^T){\varvec{u}_n} =O(\left\| {{\varvec{u}_n}} \right\| _2^2)$. Hence, $\varvec{Z}_n^T{\varvec{u}_n} = {O}(\left\| {{\varvec{u}_n}} \right\| _2)$. This combined with (18) leads to

$$\begin{aligned} {S_n}({\varvec{u}_n},\varvec{v}) = \sum \limits _{k = 1}^q {{\omega _k}B_n^{(k)}} + {o_p}\left( n\alpha _n^2 \right) \left\| {{\varvec{u}_n}} \right\| _2, \end{aligned}$$

(19)

Applying the Markov inequality and condition (C3), for constant M, we have

$$\begin{aligned} P\left( {\mathop {\max }\limits _i \left( {{\alpha _n}\left| {\varvec{\Pi }_i^T{u_n}} \right| } \right) > \eta } \right)&= P\left( {\bigcup \limits _{i = 1}^n {\left( {{\alpha _n}\left| {\varvec{\Pi }_i^T{\varvec{u}_n}} \right| > \eta } \right) } } \right) \\&\le nP\left( {{\alpha _n}\left| {\varvec{\Pi }_1^T{\varvec{u}_n}} \right| > \eta } \right) \\&\le n\frac{{\alpha _n^8}}{{{\eta ^8}}}E{\left( {\varvec{\Pi }_1^T{\varvec{u}_n}} \right) ^8} \\&=n\frac{{\alpha _n^8}}{{{\eta ^8}}}E\left\{ {E\left[ {{{\left( {\varvec{\Pi }_1^T{\varvec{u}_n}} \right) }^8}\left| U \right. } \right] } \right\} \\&\le n\frac{{\alpha _n^8}}{{{\eta ^8}}}{\left\| {{\varvec{u}_n}} \right\| ^8}E\left\{ {E\left( {{{\left\| {{\varvec{\Pi }_1}} \right\| }^8}\left| U \right. } \right) } \right\} \\&\le n\frac{{\alpha _n^8}}{{{\eta ^8}}}{\left\| {{\varvec{u}_n}} \right\| ^8}E\left\{ {E\left( {{{\left\| {{\varvec{X} _1}} \right\| }^8}\left| U \right. } \right) } \right\} \\&\le n\frac{{\alpha _n^8}}{{{\eta ^8}}}{C^8}{M^8} \\&\rightarrow 0 \\ \end{aligned}$$

So $\mathop {\max }\limits _i \left( {{\alpha _n}\left| {\varvec{\Pi }_i^T{\varvec{u}_n}} \right| } \right) = o_p\left( 1 \right)$.

Thus, it is easy to show that $\mathop {\max }\limits _i \left( {{\alpha _n}\left| {\varvec{\Pi }_i^T{\varvec{u}_n} + {v_k}} \right| } \right) = o_p\left( 1 \right)$. By condition (C4) and the Lebesgue’s dominated convergence theorem, we have

$$\begin{aligned}&E({B_n}^{(k)}\left| \mathscr {H} \right. ) \nonumber \\&\quad =E\left\{ \sum \limits _{i = 1}^n {\int _0^{{\alpha _n}(\varvec{\Pi } _i^T{\varvec{u}_n} + {v_k})} {[I({\varepsilon _i} \le x + {R_{ni}} + c_{\tau _k}) - I({\varepsilon _i} \le {R_{ni}} + c_{\tau _k})]} } dx \left| \mathscr {H} \right. \right\} \nonumber \\&\quad = \sum \limits _{i = 1}^n {\int _0^{{\alpha _n}(\varvec{\Pi } _i^T{\varvec{u}_n} + {v_k})} {[F(x + {R_{ni}} + {c_{{\tau _k}}}) - F({R_{ni}} + {c_{{\tau _k}}})]} } dx\nonumber \\&\quad = \sum \limits _{i = 1}^n {\int _0^{{\alpha _n}(\varvec{\Pi }_i^T{\varvec{u}_n} + {v_k})} {[f({R_{ni}} + {c_{{\tau _k}}})x(1 + o_p(1))]} } dx\nonumber \\&\quad =\frac{{f{{(}}{c_{{\tau _k}}}{{)}}}}{2}\alpha _n^2\sum \limits _{i = 1}^n {(v_k^2 + \varvec{u}_n^T{\varvec{\Pi } _i}\varvec{\Pi } _i^T{\varvec{u}_n} + 2{v_k}\varvec{\Pi } _i^T{\varvec{u}_n})} {{(1 + }}{o_p}{{(1))}}\nonumber \\&\quad =\frac{{f{{(}}{c_{{\tau _k}}}{{)}}}}{2}n\alpha _n^2 {(v_k^2 + \varvec{u}_n^T {\varvec{\Gamma }_n}{\varvec{u}_n} + 2{v_k} {\varvec{\mu }_n^T}{\varvec{u}_n})} {{(1 + }}{o_p}{{(1))}}. \end{aligned}$$

(20)

Here we use the fact that $\mathop {\max }\limits _i \left( {{\alpha _n}\left| {\varvec{\Pi }_i^T{\varvec{u}_n} + {v_k}} \right| } \right) = o_p\left( 1 \right)$ in the third step.

Moreover,

$$\begin{aligned}&{\mathrm{Var}} ({B_n}^{(k)}\left| \mathscr {H} \right. )\nonumber \\&\quad =\mathrm{Var} \left\{ \sum \limits _{i = 1}^n {\int _0^{{\alpha _n}(\varvec{\Pi } _i^T{\varvec{u}_n} + {v_k})} {[I({\varepsilon _i} \le x + {R_{ni}} + c_{\tau _k}) - I({\varepsilon _i} \le {R_{ni}} + c_{\tau _k})]} } dx \left| \mathscr {H} \right. \right\} \nonumber \\&\quad \le \sum \limits _{i = 1}^n E \left[ {{{\left( {\int _0^{{\alpha _n}(\varvec{\Pi }_i^T{\varvec{u}_n} + {v_k})} {\left\{ {I({\varepsilon _i} < x + {R_{ni}} + {c_{{\tau _k}}}) - I({\varepsilon _i} < {R_{ni}} + {c_{{\tau _k}}})} \right\} dx} } \right) }^2}\left| \mathscr {H} \right. } \right] \nonumber \\&\quad \le \sum \limits _{i = 1}^n {\int _0^{\left| {{\alpha _n}(\varvec{\Pi }_i^T{\varvec{u}_n} + {v_k})} \right| } {\int _0^{\left| {{\alpha _n}(\varvec{\Pi }_i^T{\varvec{u}_n} + {v_k})} \right| } {\Bigg \{ {F\left( {R_{ni}} + {c_{{\tau _k}}} + \left| {{\alpha _n}(\varvec{\Pi }_i^T{\varvec{u}_n} + {v_k})} \right| \right) }} } } \nonumber \\&\qquad { - F({R_{ni}} + {c_{{\tau _k}}})} \Bigg \}d{x_1}d{x_2}\nonumber \\&\quad \le o\left( \sum \limits _{i = 1}^n {{{\left| {{\alpha _n}(\varvec{\Pi }_i^T{\varvec{u}_n} + {v_k})} \right| }^2}}\right) \nonumber \\&\quad = {o_p}( n\alpha _n^2 )\left\| {{\varvec{u}_n}} \right\| _2^2 . \end{aligned}$$

(21)

Hence

$$\begin{aligned} B_n^{(k)}=\frac{{f({c_{{\tau _k}}}\mathrm{{)}}}}{2}n\alpha _n^2 {(v_k^2 + \varvec{u}_n^T {\varvec{\Gamma }_n}{\varvec{u}_n} + 2{v_k} {\varvec{\mu }_n^T}{\varvec{u}_n})} \mathrm{{(1 + }}{o_p}\mathrm{{(1))}}. \end{aligned}$$

This combined with (19), yields that

$$\begin{aligned} {S_n}({\varvec{u}_n},\varvec{v})= & {} \frac{1}{2}n\alpha _n^2\sum \limits _{k = 1}^q {{\omega _k} f({c_{{\tau _k}}}) {(v_k^2 + \varvec{u}_n^T {\varvec{\Gamma }_n}{\varvec{u}_n} + 2{v_k} {\varvec{\mu }_n^T}{\varvec{u}_n})} \mathrm{{(1 + }}{o_p}\mathrm{{(1))}} } \nonumber \\&+ \,{o_p}\left( n\alpha _n^2 \right) \left\| {{\varvec{u}_n}} \right\| _2. \end{aligned}$$

(22)

By condition (C6), we have

$$\begin{aligned} {P_{{\lambda _n}}}({\varvec{u}_n})&\ge \sum \limits _{j = 1}^{{s}} {\left[ {n{\alpha _n}{{p'}_{{\lambda _n}}}\left( {{\left\| {\varvec{\gamma }_j^0} \right\| }_2}\right) \frac{{\varvec{\gamma }_j^{0T}{\varvec{u}_{nj}}}}{{{{\left\| {\varvec{\gamma }_j^0} \right\| }_2}}} + \frac{1}{2}n\alpha _n^2{{p''}_{{\lambda _n}}}\left( {{\left\| {\varvec{\gamma }_j^0} \right\| }_2}\right) \varvec{u}_{nj}^T{\varvec{u}_{nj}}(1 + o(1))} \right] } \nonumber \\&\ge - (n\alpha _n^2{\left\| {{\varvec{u}_n}} \right\| _2} + {o}(n\alpha _n^2)\left\| {{\varvec{u}_n}} \right\| _2^2). \end{aligned}$$

(23)

It follows from ((19)–(23)) that ${D_n}({\varvec{u}_n},\varvec{v})$ in (17) is dominated by the positive quadratic term $\frac{1}{2}n\alpha _n^2\sum \limits _{k = 1}^q {{\omega _k} f({c_{{\tau _k}}}) {(v_k^2 + \varvec{u}_n^T {\varvec{\Gamma }_n}{\varvec{u}_n} + 2{v_k} {\varvec{\mu }_n^T}{\varvec{u}_n})}}$ as long as ${\left\| {{\varvec{u}_n}} \right\| _2}$ and ${\left\| {{\varvec{v}}} \right\| _2}$ are large enough. This proves (16). By Lemma 1, we have

$$\begin{aligned} {n^{ - 1}}\sum \limits _{i = 1}^n {{{{({{\hat{\beta }}_j}({U_i}) - {\beta _j}({U_i}))}^2}} }\le & {} \frac{2}{n}\sum \limits _{i = 1}^n { {{{(\varvec{\pi } _i^{(j)T}({{\varvec{\hat{\gamma }}}_j} - \varvec{\gamma } _j^0))}^2}} } + \frac{2}{n}\sum \limits _{i = 1}^n { {{{(\varvec{\pi } _i^{(j)T}{\varvec{\gamma }_j}^0 - {\beta _j}({U_i}))}^2}} } \\\le & {} \frac{2}{n} {{{({{\varvec{\hat{\gamma }}}_j} - \varvec{\gamma } _j^0)}^T}\sum \limits _{i = 1}^n {\varvec{\pi } _i^{(j)}\varvec{\pi } _i^{(j)T}} ({{\varvec{\hat{\gamma }} }_j} - \varvec{\gamma } _j^0) + } {C^2K _n^{-2r}}\\= & {} {O_p}({n^{{{ - 2r} / {(2r + 1)}}}} ). \end{aligned}$$

This complete the proof. $\square$

Proof of Theorem 2 (a)

We use proof by contradiction. Suppose that there exists a ${s} + 1 \le {j_0} \le {p_n}$ such that the probability of ${\hat{\beta }_{{j_0}}}(u)$ being a zero function does not converge to one. Then, there exists $\eta > 0$ such that, for infinitely many n, $P({\varvec{\hat{\gamma }}_{{j_0}}} \ne 0) = P({\hat{\beta }_{{j_0}}} (u)\ne 0) \ge \eta .$ Let ${\varvec{\hat{\gamma }} ^*}$ be the vector obtained from $\varvec{\hat{\gamma }}$ with ${\varvec{\hat{\gamma }} _{{j_0}}}$ being replaced by $\varvec{0}$. It will be shown that there exists a $\delta > 0$ such that ${PL_n}\left( {\varvec{\hat{\gamma }} ,\varvec{\hat{c}}; \varvec{\omega } } \right) -{PL_n}\left( {\varvec{\hat{\gamma }}^* ,\varvec{\hat{c}}; \varvec{\omega } } \right) > 0$ with probability at least $\delta$ for infinitely many n, which contradicts with the fact that ${PL_n}\left( {\varvec{\hat{\gamma }} ,\varvec{\hat{c}}; \varvec{\omega } } \right) -{PL_n}\left( {\varvec{\hat{\gamma }}^* ,\varvec{\hat{c}}; \varvec{\omega } } \right) \le 0$.

By Theorem 1, we have ${\left\| {{{\varvec{\hat{\gamma }} }_j} -\varvec{ \gamma } _j^0} \right\| _2} = {O_p}({n^{{{ - r} / {(2r + 1)}}}})$. Since $\varvec{\gamma } _j^0 = \varvec{0}$ for $j=s+1,\ldots ,p_n+1$, we have ${\left\| {{{\varvec{\hat{\gamma }} }_j}} \right\| _2} = {O_p}({n^{{{ - r} / {(2r + 1)}}}})$ for $j=s+1,\ldots ,p_n+1$. So ${\left\| {{{\varvec{\hat{\gamma }}}_{j_0}}} \right\| _2} = {O_p}({n^{{{ - r} / {(2r + 1)}}}})$. With probability tending to one, ${{{\left\| {{{\varvec{\hat{\gamma }}}_{{j_0}}}} \right\| }_2} \le {\lambda _n}}$, since ${n^{r / {(2r + 1)}}} {\lambda _n}/ {\sqrt{{p_n}}} \rightarrow \infty$. By the definition of ${p_{\lambda _n} }(.)$, we have $P\left\{ {{p_{{\lambda _n}}}({{\left\| {{{\varvec{\hat{\gamma }} }_{{j_0}}}} \right\| }_2}) = {\lambda _n}{{\left\| {{{\varvec{\hat{\gamma }}}_{{j_0}}}} \right\| }_2}} \right\} \rightarrow 1.$

Since $\left( {{\rho _\tau }(u) - {\rho _\tau }(v)} \right) \ge (\tau - I(v < 0))(u - v)$ for any $u,v \in R$, we have

$$\begin{aligned}&{PL_n}\left( {\varvec{\hat{\gamma }} ,\varvec{\hat{c}}; \varvec{\omega } } \right) -{PL_n}\left( {\varvec{\hat{\gamma }}^* ,\varvec{\hat{c}}; \varvec{\omega } } \right) \nonumber \\&\quad = {L_n}\left( {\varvec{\hat{\gamma }} ,\varvec{\hat{c}}; \varvec{\omega } } \right) -{L_n}\left( {\varvec{\hat{\gamma }}^* ,\varvec{\hat{c}}; \varvec{\omega } } \right) + n\sum \limits _{j = 1}^{{p_n}} {\left( {{p_{{\lambda _n}}}({{\left\| {{{\varvec{\hat{\gamma }}}_j}} \right\| }_2}) - {p_{{\lambda _n}}}({{\left\| {\varvec{\hat{\gamma }} _j^*} \right\| }_2})} \right) } \nonumber \\&\quad {{= }}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {\left\{ {{\rho _{{\tau _k}}}\left( {{Y_i} - \hat{c}_{{\tau _k}} -\varvec{\Pi }_i^T\varvec{\hat{\gamma }} } \right) - {\rho _{{\tau _k}}}\left( {{Y_i} - \hat{c}_{{\tau _k}} - \varvec{\Pi }_i^T{{\varvec{\hat{\gamma }}}^*}} \right) } \right\} } } + n{\lambda _n}{\left\| {{{\varvec{\hat{\gamma }} }_{{j_0}}}} \right\| _2} \nonumber \\&\quad \ge - \sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {({\tau _k} - I({\varepsilon _i} < 0))} } \varvec{\Pi } _i^T(\varvec{\hat{\gamma }} - {{\varvec{\hat{\gamma }} }^*})\nonumber \\&\qquad - \sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {(I({\varepsilon _i} < 0) - I({\varepsilon _i} < {r_{ni}}+\hat{c}_{{\tau _k}}))} } \varvec{\Pi } _i^T(\varvec{\hat{\gamma }} - {{\varvec{\hat{\gamma }} }^*}) + n{\lambda _n}{\left\| {{{\varvec{\hat{\gamma }}}_{{j_0}}}} \right\| _2} \nonumber \\&\quad \ge \left\{ { - {{\left\| {\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {({\tau _k} - I({\varepsilon _i} < 0))\varvec{\Pi } _i^{({j_0})}} } } \right\| }_2} } \right. \nonumber \\&\quad \left. {~~~~~ -{{\left\| {\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {(I({\varepsilon _i} < 0) - I({\varepsilon _i} < {r_{ni}}+\hat{c}_{{\tau _k}}))\varvec{\Pi } _i^{({j_0})}} } } \right\| }_2}+ n{\lambda _n}} \right\} {\left\| {{{\varvec{\hat{\gamma }} }_{{j_0}}}} \right\| _2},\nonumber \\ \end{aligned}$$

(24)

where ${r_{ni}} = {R_{ni}} + \varvec{\Pi } _{i}^T({\varvec{\hat{\gamma }} ^*} - {\varvec{\gamma } ^0}).$

Let $\mathbf{T _n} = {\sum \limits _{i = 1}^n {(I({\varepsilon _i} < 0) - I({\varepsilon _i} < {r_{ni}} + \hat{c}_{{\tau _k}}))\varvec{\Pi } _i^{({j_0})}} }$. From conditions (C3) and (C4), we obtain that for any $L > 0$ and $\Delta ={n^{{{ - r} / {(2r + 1)}}}}\sqrt{{p_n}}$,

$$\begin{aligned}&E(\varvec{T}_n^T{\varvec{T}_n}) \\&\quad = E\left\{ {\sum \limits _{i = 1}^n {(I({\varepsilon _i} < 0) - I({\varepsilon _i} < L\Delta ))\varvec{\Pi } _i^{({j_0})T}\sum \limits _{k = 1}^n {{{(I({\varepsilon _k} < 0) - I({\varepsilon _k} < L\Delta ))}}\varvec{\Pi } _k^{({j_0})}} } } \right\} \\&\quad \le nE{\left\{ {(I(\varepsilon < 0) - I(\varepsilon < L\Delta ))\left| {{\varvec{\Pi }^{({j_0})}}} \right| } \right\} ^2} \\&\qquad + n(n - 1){\left[ {E\left\{ {(I(\varepsilon < 0) - I(\varepsilon < L\Delta ))\left| {{\varvec{\Pi }^{({j_0})}}} \right| } \right\} } \right] ^2} \\&\quad \le {M}^2\left\{ {\left( {L\Delta n} \right) + {{\left( {L\Delta n} \right) }^2}} \right\} \\&\quad \le C^2 {n^2}{p_n}{n^{{{ -2 r} / {(2r + 1)}}}}.\\ \end{aligned}$$

Thus

$$\begin{aligned} {{{\left\| {\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {(I({\varepsilon _i} < 0) - I({\varepsilon _i} < {r_{ni}}))\varvec{\Pi } _i^{({j_0})}} } } \right\| }_2}}={O_p}(n\sqrt{{p_n}} {n^{{{ - r} / {(2r + 1)}}}}), \end{aligned}$$

(25)

By simple calculation , we obtain

$$\begin{aligned} {{{\left\| {\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {({\tau _k} - I({\varepsilon _i} < 0))\varvec{\Pi } _i^{({j_0})}} } } \right\| }_2}}={O_p}(\sqrt{n{p_n}}). \end{aligned}$$

(26)

By the fact that ${{{n^{{r / {(2r + 1)}}}}{\lambda _n}} / {\sqrt{{p_n}} }} \rightarrow \infty$, so $n{\lambda _n}$ is of higher order than $O(n\sqrt{{p_n}} {n^{{{ - r} / {(2r + 1)}}}})$. This combined with (25) and (26), we can conclude that (24) is dominated by $n{\lambda _n}{\left\| {{{\varvec{\hat{\gamma }}}_{{j_0}}}} \right\| _2}$, which contradicts to ${PL_n}\left( {\varvec{\hat{\gamma }} ,\varvec{\hat{c}}; \varvec{\omega } } \right) -{PL_n}\left( {\varvec{\hat{\gamma }}^* ,\varvec{\hat{c}}; \varvec{\omega } } \right) \le 0$.   $\square$

Proof of Theorem 2 (b)

Let ${\varvec{u}_n} = \alpha _n^{ - 1}(\varvec{\gamma }- {\varvec{\gamma }^0})$. Partition the vectors ${\varvec{u}_n} = {(\varvec{u}_{na}^T,\varvec{u}_{nb}^T)^T}$ and ${\varvec{\Pi }_i} = {(\varvec{\Pi }_{ia}^T,\varvec{\Pi }_{ib}^T)^T}$ in the same way as $\varvec{\gamma }= {(\varvec{\gamma }_a^T,\varvec{\gamma }_b^T)^T}$. By (17) and $P_{\lambda _n}(0)=0$, we can write

$$\begin{aligned} {D_n}(({\varvec{u}_{na}^T, \varvec{0}^T})^T,\varvec{v}) = {S_n}(({\varvec{u}_{na}^T, \varvec{0}^T})^T,\varvec{v}) + {P_{{\lambda _n}}}({\varvec{u}_{na}}), \end{aligned}$$

(27)

where ${P_{{\lambda _n}}}\left( {\varvec{u}_{na}}\right) =n\sum \limits _{j = 1}^{{s}} {\left[ {{p_{{\lambda _n}}}\left( {{\left\| {\varvec{\gamma }_j^0 + {\alpha _n}{\varvec{u}_{nj}}} \right\| }_2}\right) - {p_{{\lambda _n}}}\left( {{\left\| {\varvec{\gamma }_j^0} \right\| }_2}\right) } \right] }$. By taking Taylor’s expansion for ${P_{{\lambda _n}}}\left( {\varvec{u}_{na}}\right)$ at $\varvec{u}_{na}=0$, we obtain that

$$\begin{aligned} {P_{{\lambda _n}}}({\varvec{u}_{na}})=n{\alpha _n}\varvec{c}_n^T{\varvec{u}_{na}} + \frac{1}{2}n\alpha _n^2\varvec{u}_{na}^T{\varvec{\Sigma }_{{\lambda _n}}}{\varvec{u}_{na}}(1 + o(1)). \end{aligned}$$

Then the minimizer $(\varvec{\hat{u}}_{na}^T,\varvec{\hat{v}}^T)^T$ of ${D_n}(({\varvec{u}_{na}^T, \varvec{0}^T})^T,\varvec{v})$ satisfies the score equations

$$\begin{aligned} \begin{array}{l} {n^{ - 1}}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\varvec{\Pi }_{ia}}{\psi _{{\tau _k}}}({\varepsilon _i} - {c_{{\tau _k}}} -R_{ni}- {\alpha _n}(\varvec{\Pi }_{ia}^T{\varvec{\hat{u}}_{na}} + {{\hat{v}}_k}))} } (1 + o_p(1)) \\ \quad = {\varvec{c}_n} + {\alpha _n}{\varvec{\Sigma }_{{\lambda _n}}}{\varvec{\hat{u}}_{na}}(1 + {o_p}(1)), \\ \end{array} \end{aligned}$$

(28)

$$\begin{aligned} {\omega _k}\sum \limits _{i = 1}^n {{\psi _{{\tau _k}}}({\varepsilon _i} - {c_{{\tau _k}}} - R_{ni}-{\alpha _n}(\varvec{\Pi }_{ia}^T{\varvec{\hat{u}}_{na}} + {{\hat{v}}_k})) = 0}, \end{aligned}$$

(29)

where ${\psi _\tau }(u) ={{\rho '}_{{\tau }}}(u)= \tau - I(u < 0),$ we can write

$$\begin{aligned} \begin{array}{l} {n^{ - 1}}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\varvec{\Pi }_{ia}}{\psi _{{\tau _k}}}({\varepsilon _i} - {c_{{\tau _k}}} - {R_{ni}} - {\alpha _n}(\varvec{\Pi }_{ia}^T{\varvec{\hat{u}}_{na}} + {{\hat{v}}_k}))} } \\ = - {n^{ - {1 / 2}}}{\varvec{H}_{n}} + \sum \limits _{k = 1}^q {{\omega _k}(B_{n21}^{(k)} + B_{n22}^{(k)})} , \\ \end{array} \end{aligned}$$

(30)

where ${\varvec{H}_n} = {n^{ - {1 / 2}}}\sum \limits _{i = 1}^n {{\varvec{\Pi }_{ia}}\sum \limits _{k = 1}^q {{\omega _k}} [I({\varepsilon _i} < {R_{ni}} + {c_{{\tau _k}}}) - {\tau _k}],}$

$$\begin{aligned} \begin{array}{l} B_{n21}^{(k)} = {n^{ - 1}}\sum \limits _{i = 1}^n {{\varvec{\Pi }_{ia}}[F({c_{{\tau _k}}} + {R_{ni}}) - F({R_{ni}} + {c_{{\tau _k}}} + {\alpha _n}(\varvec{\Pi }_{ia}^T{\varvec{\hat{u}}_{na}} + {{\hat{v}}_k}))],} \\ B_{n22}^{(k)} = {n^{ - 1}}\sum \limits _{i = 1}^n {{\varvec{\Pi }_{ia}}\left\{ {[I({\varepsilon _i} < {c_{{\tau _k}}} + {R_{ni}}) - I({\varepsilon _i} < {R_{ni}} + {c_{{\tau _k}}} + {\alpha _n}(\varvec{\Pi }_{ia}^T{\varvec{\hat{u}}_{na}} + {{\hat{v}}_k}))]} \right. } \\ \qquad \qquad \left. -{[F({c_{{\tau _k}}} + {R_{ni}}) - F({R_{ni}} + {c_{{\tau _k}}} + {\alpha _n}(\varvec{\Pi }_{ia}^T{\varvec{\hat{u}}_{na}} + {{\hat{v}}_k}))]} \right\} .\\ \end{array} \end{aligned}$$

Taking Taylor’s explanation for ${F({R_{ni}} + {c_{{\tau _k}}} + {\alpha _n}(\varvec{\Pi }_{ia}^T{\varvec{\hat{u}}_{na}} + {{\hat{v}}_k}))}$ at ${{R_{ni}} + {c_{{\tau _k}}}}$ gives

$$\begin{aligned} \begin{aligned} B_{n21}^{(k)}&= - {n^{ - 1}}\sum \limits _{i = 1}^n {{\varvec{\Pi }_{ia}}[f({c_{{\tau _k}}} + {R_{ni}}){\alpha _n}(\varvec{\Pi }_{ia}^T{\varvec{\hat{u}}_{na}} + {{\hat{v}}_k})](1 + o(1)),} \\&= - {\alpha _n}f({c_{{\tau _k}}})({\varvec{\Gamma }_{na}}{\varvec{\hat{u}}_{na}} + {\varvec{\mu }_{na}}{{\hat{v}}_k})(1 + {o_p}(1)). \\ \end{aligned} \end{aligned}$$

By direct calculation of the mean and variance, we can show, as in Jiang et al. (2001), $B_{n22}^{(k)} = {o_p}({\alpha _n})$. This combined with (28) and (30) lead to

$$\begin{aligned} - ({n^{ - {1 / 2}}}{\varvec{H}_{n}} + {\varvec{c}_n}) = {\alpha _n}\left\{ \sum \limits _{k = 1}^q {{\omega _k}f({c_{{\tau _k}}})({\varvec{\Gamma }_{na}}{\varvec{\hat{u}}_{na}} + {\varvec{\mu }_{na}}{{\hat{v}}_k}) + {\varvec{\Sigma }_{{\lambda _n}}}{\varvec{\hat{u}}_{na}}}\right\} (1 + {o_p}(1)).\nonumber \\ \end{aligned}$$

(31)

Similarly, (29) can be simplified as

$$\begin{aligned} {n^{ - 1/2}}\zeta _{n,k}+ {\alpha _n}{\omega _k}f({c_{{\tau _k}}})({{\hat{v}}_k} + \varvec{\mu }_{na}^T{\varvec{\hat{u}}_{na}}(1 + {o_p}(1))) = 0, \end{aligned}$$

(32)

where $\zeta _{n,k} = {n^{{{ - 1} / 2}}}{\omega _k}\sum \limits _{i = 1}^n {[I({\varepsilon _i} < {c_{{\tau _k}}}+R_{ni}) - {\tau _k}]}$. Solving (31) and (32), we obtain that

$$\begin{aligned} {\alpha _n}\left( {{\varvec{G}_{na}} + \frac{{{\varvec{\Sigma }_{{\lambda _n}}}}}{{{\varvec{\omega }^T}\varvec{f}}}} \right) {\varvec{\hat{u}}_{na}} + \frac{{{\varvec{c}_n}}}{{{\varvec{\omega }^T}\varvec{f}}} = - {n^{ - {1 / 2}}}\left( {\frac{{{\varvec{H}_{n}}}}{{{\varvec{\omega }^T}\varvec{f}}} - {\varvec{\mu }_{na}}\sum \limits _{k = 1}^q {\frac{{{\zeta _{n,k}}}}{{{\varvec{\omega }^T}\varvec{f}}}} } \right) + {o_p}({n^{{{ - 1} / 2}}}). \end{aligned}$$

Let $\varvec{H}_n^* = {n^{ - {1 / 2}}}\sum \limits _{i = 1}^n {{\varvec{\Pi }_{ia}}\sum \limits _{k = 1}^q {{\omega _k}} [I({\varepsilon _i} < {c_{{\tau _k}}}) - {\tau _k}]}$, $\zeta _{n,k}^* = {n^{{{ - 1} / 2}}}{\omega _k}\sum \limits _{i = 1}^n [I({\varepsilon _i} < {c_{{\tau _k}}}) - {\tau _k}]$. Following Jiang et al. (2012), we have

$$\begin{aligned} \varvec{e}^T\varvec{G}_{na}^{{{ - 1} / 2}}{{(\varvec{H}_n^* - {\varvec{\mu }_{na}}\sum \nolimits _{k = 1}^q {\zeta _{n,k}^*} )} / {{\varvec{\omega }^T}\varvec{f}}}\mathop \rightarrow \limits ^d N(0,R(q)). \end{aligned}$$

(33)

Put ${\eta _{i,k}} = I({\varepsilon _i} < {R_{ni}} + {c_{{\tau _k}}}) - {\tau _k},\eta _{i,k}^* = I({\varepsilon _i} < {c_{{\tau _k}}}) - {\tau _k},$ Moreover,

$$\begin{aligned} \begin{array}{l} Var\left( {(\varvec{H}_n - {\varvec{\mu }_{na}}\sum \nolimits _{k = 1}^q {\zeta _{n,k}} ) - (\varvec{H}_n^* - {\varvec{\mu }_{na}}\sum \nolimits _{k = 1}^q {\zeta _{n,k}^*} )\left| \mathscr {H} \right. } \right) \\ \quad = Var\left( {[{n^{ - {1 / 2}}}\sum \limits _{i = 1}^n {\sum \limits _{k = 1}^q {{\varvec{\Pi }_{ia}}{\omega _k}({\eta _{i,k}} - \eta _{i,k}^*) - {n^{ - {1 / 2}}}{\varvec{\mu }_{na}}\sum \limits _{i = 1}^n {\sum \limits _{k = 1}^q {{\omega _k}({\eta _{i,k}} - \eta _{i,k}^*)} } } } ]\left| \mathscr {H} \right. } \right) \\ \quad \le 2\left\{ {\frac{1}{n}\sum \limits _{i = 1}^n ({{\varvec{\Pi }_{ia}}\varvec{\Pi }_{ia}^T + {\varvec{\mu }_{na}}\varvec{\mu }_{na}^T}) } \right\} Var\left( {\sum \limits _{k = 1}^q {{\omega _k}({\eta _{i,k}} - \eta _{i,k}^*)} \left| \mathscr {H} \right. } \right) \\ \quad \le 2{q^2}\left\{ {\frac{1}{n}\sum \limits _{i = 1}^n ({{\varvec{\Pi }_{ia}}\varvec{\Pi }_{ia}^T + {\varvec{\mu }_{na}}\varvec{\mu }_{na}^T})} \right\} \mathop {\max }\limits _k \left| E \{{\omega _k^2[I({\varepsilon _i} < {R_{ni}} + {c_{{\tau _k}}}) - I({\varepsilon _i} < {c_{{\tau _k}}})]\left| \mathscr {H} \right. }\} \right| \\ \quad \le 2{q^2}\left\{ {\frac{1}{n}\sum \limits _{i = 1}^n ({{\varvec{\Pi }_{ia}}\varvec{\Pi }_{ia}^T + {\varvec{\mu }_{na}}\varvec{\mu }_{na}^T}) } \right\} \mathop {\max }\limits _k \omega _k^2|F({R_{ni}} + {c_{{\tau _k}}}) - F({c_{{\tau _k}}})| \\ \quad = {o}(1). \\ \end{array} \end{aligned}$$

Thus, we have

$$\begin{aligned} \varvec{H}_n- {\varvec{\mu }_{na}}\sum \nolimits _{k = 1}^q {\zeta _{n,k}} \mathop \rightarrow \limits ^p \varvec{H}_n^* - {\varvec{\mu }_{na}}\sum \nolimits _{k = 1}^q {\zeta _{n,k}^*} . \end{aligned}$$

(34)

By Slutsky’s theorem, conditioning on $\mathscr {H}$, we have

$$\begin{aligned} \varvec{e}^T\varvec{G}_{na}^{{{ - 1} / 2}}{{(\varvec{H}_n - {\varvec{\mu }_{na}}\sum \nolimits _{k = 1}^q {\zeta _{n,k}} )} / {{\varvec{\omega }^T}\varvec{f}}}\mathop \rightarrow \limits ^d N(0,R(q)). \end{aligned}$$

(35)

Note that ${\varvec{u}_{na}} = \alpha _n^{ - 1}(\varvec{\gamma }_a - {\varvec{\gamma }_ a^0})$. It follows that

$$\begin{aligned} \sqrt{n} \varvec{e}^T\varvec{G}_{na}^{{{ - 1} / 2}}({\varvec{G}_{na}} + \frac{{{\varvec{\Sigma }_{{\lambda _n}}}}}{{{\varvec{\omega }^T}\varvec{f}}})\left[ {({\varvec{\hat{\gamma }}_a} - \varvec{\gamma }_a^0) + {{({\varvec{G}_{na}} + \frac{{{\varvec{\Sigma }_{{\lambda _n}}}}}{{{\varvec{\omega }^T}\varvec{f}}})}^{ - 1}}\frac{{{\varvec{c}_n}}}{{{\varvec{\omega }^T}\varvec{f}}}} \right] \mathop \rightarrow \limits ^d N(0,R(q)). \end{aligned}$$

$\square$

Proof of Theorem 2 (c)

By the proof Theorem 2 (a), we know immediately that ${\varvec{\hat{\gamma }} _{b,{\lambda _n}}} = 0$ with probability tending to one. Consequently, we know that ${\varvec{\hat{\gamma }}_{a,{\lambda _n}}}$ must be the solution of the following normal equation

$$\begin{aligned} \frac{1}{n}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\varvec{\Pi } _{ia}}{\psi _{{\tau _k}}}\left( {{Y_i} - {\hat{c}_{{\tau _k}}} - \varvec{\Pi }_{ia}^T{{\varvec{\hat{\gamma }}}_a}} \right) } } - \varvec{c}_n=\varvec{0}, \end{aligned}$$

On the other hand, the oracle estimator must be the solution of the normal equation

$$\begin{aligned} \frac{1}{n}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\varvec{\Pi }_{ia}}{\psi _{{\tau _k}}}\left( {{Y_i} - {\hat{c}_{{\tau _k}}} - \varvec{\Pi }_{ia}^T{{\varvec{\hat{\gamma }} }_{ora}}} \right) } } = \varvec{0}. \end{aligned}$$

So we have

$$\begin{aligned} \begin{array}{l} \left[ \frac{1}{n}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\varvec{\Pi } _{ia}}{\psi _{{\tau _k}}}\left( {{Y_i} -{\hat{c}_{{\tau _k}}} - \varvec{\Pi }_{ia}^T{{\varvec{\hat{\gamma }} }_{ora}}} \right) } }\right. \\ \left. \quad - \frac{1}{n}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\varvec{\Pi } _{ia}}{\psi _{{\tau _k}}}\left( {{Y_i} - {\hat{c}_{{\tau _k}}}- \varvec{\Pi }_{ia}^T{{\varvec{\hat{\gamma }} }_a}} \right) } } \right] \\ \quad + {\left( {p'_{{\lambda _n}}}\left( {\left\| {{{\varvec{\hat{\gamma }}}_1}} \right\| _2}\right) \frac{{\varvec{\hat{\gamma }}_1^T}}{{{{\left\| {{{\varvec{\hat{\gamma }} }_1}} \right\| }_2}}},\ldots ,{p'_{{\lambda _n}}}\left( {\left\| {{{\varvec{\hat{\gamma }} }_{{s}}}} \right\| _2}\right) \frac{{\varvec{\hat{\gamma }} _{{s}}^T}}{{{{\left\| {{{\varvec{\hat{\gamma }}}_{{s}}}} \right\| }_2}}}\right) ^T} = \varvec{0}.\\ \end{array} \end{aligned}$$

(36)

Furthermore, the first term of the left hand side of (36) can be written as

$$\begin{aligned} \begin{array}{l} \frac{1}{n}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\varvec{\Pi } _{ia}}} } \left[ {{\psi _{{\tau _k}}}\left( {{Y_i} - {{\hat{c}}_{{\tau _k}}} - \varvec{\Pi }_{ia}^T{{\varvec{\hat{\gamma }} }_{ora}}} \right) - {\psi _{{\tau _k}}}\left( {{Y_i} - {c_{{\tau _k}}} - \varvec{\Pi }_{ia}^T{\varvec{\gamma } ^0}} \right) } \right] \\ - \frac{1}{n}\sum \limits _{k = 1}^q {{\omega _k}\sum \limits _{i = 1}^n {{\varvec{\Pi } _{ia}}} } \left[ {{\psi _{{\tau _k}}}\left( {{Y_i} - {{\hat{c}}_{{\tau _k}}} - \varvec{\Pi }_{ia}^T{{\varvec{\hat{\gamma }} }_a}} \right) - {\psi _{{\tau _k}}}\left( {{Y_i} - {c_{{\tau _k}}} - \varvec{\Pi }_{ia}^T{\varvec{\gamma } ^0}} \right) } \right] \\ \buildrel \Delta \over = {\varvec{G}_1}{{ - }}{\varvec{G}_2} .\\ \end{array} \end{aligned}$$

For $\varvec{G}_1$ and $\varvec{G}_2$, after some direct calculation, we have

$$\begin{aligned} {\varvec{G}_1} = \sum \limits _{k = 1}^q {{\omega _k}} {\varvec{\hat{S}}_{na}}({\varvec{\gamma } ^0} - {\varvec{\hat{\gamma }} _{ora}}) + {o_p}({n^{{{ - r} / {(2r + 1)}}}}) ,\\ {\varvec{G}_2} = \sum \limits _{k = 1}^q {{\omega _k}} {\varvec{\hat{S}}_{na}}({\varvec{\gamma }^0} - {\varvec{\hat{\gamma }} _a}) + {o_p}({n^{{{ - r} / {(2r + 1)}}}}) , \end{aligned}$$

where ${\varvec{\hat{S}}_{na}} = {n^{ - 1}}f{{(}}{c_{{\tau _k}}}{{)}}\sum \limits _{i = 1}^n {{\varvec{\Pi } _{ia}}\varvec{\Pi } _{ia}^T} .$ So

$$\begin{aligned} \begin{array}{l} \mathop {\sup }\limits _{u \in [0,1]} {\left\| {{{\varvec{\hat{\gamma }}}_{aj}} - {{\varvec{\hat{\gamma }} }_{oraj}}} \right\| _2} \\ = \mathop {\sup }\limits _{u \in [0,1]}{\left\| {{{(\sum \limits _{k = 1}^q {{\omega _k}} {{\varvec{\hat{S}}}_{naj}})}^{ - 1}}{{({{p'}_{{\lambda _n}}}({{\left\| {{{\varvec{\hat{\gamma }}}_j}} \right\| }_2})\frac{{\varvec{\gamma }_j}}{{{{\left\| {{{\varvec{\hat{\gamma }}}_j}} \right\| }_2}}})}}} \right\| _2} + {o_p}({n^{{{ - r} / {(2r + 1)}}}}) \\ \le \hat{\lambda }_{\min j }^{ - 1}{a_n} +{o_p}({n^{{{ - r} / {(2r + 1)}}}}) \\ ={o_p}({n^{{{ - r} / {(2r + 1)}}}}),\\ \end{array} \end{aligned}$$

where ${\hat{\lambda }_{\min ,j }} = {\inf _u}{\lambda _{\min }}({\varvec{\hat{S}}_{na,j}}).$

Thus, we have $\mathop {\sup }\limits _{u \in [0,1]}| {{{{\hat{\beta }}}_{aj}}(u) - {{{\hat{\beta }} }_{ora,j}}(u)}| ^2 = {o_p}({n^{{{ - 2r} / {(2r + 1)}}}})$. This completes the proof. $\square$

Proof of Theorem 3

Suppose for some $s_1 + 1 \le {j_0} \le s$, ${\varvec{\pi } ^{({j_0})T}}{\varvec{\hat{\gamma }} _{{j_0}}}$ does not represent a constant coefficient. Let ${\varvec{\hat{\gamma }} ^*}$ be the vector obtained from $\varvec{\hat{\gamma }}$ with ${\varvec{\hat{\gamma }} _{{j_0}}}$ being replaced by its projection onto the subspace $\left\{ {{\varvec{\gamma } _{{j_0}}}{{:}}{\varvec{\pi } ^{({j_0})T}}{\varvec{\gamma } _{{j_0}}}\mathrm{{ represents ~ a ~constant~ coefficient}}} \right\}$. For $j=s_1+1,\ldots ,s$, ${{{\varvec{\hat{\gamma }} }_j}^T{{{\varvec{F}}}_j}{{\varvec{\hat{\gamma }} }_j} = 0}$. By definition of ${\varvec{\hat{\gamma }}}$ and ${\varvec{\hat{\gamma }}^* }$, we have

(37)

Since ${n^{r / {(2r + 1)}}} {\lambda _n} / {\sqrt{{p_n}} } \rightarrow \infty$, so ${\sqrt{{{\varvec{\hat{\gamma }} }_{{j_0}}}^T{{{\varvec{F}}}_{{j_0}}}{{\varvec{\hat{\gamma }} }_{{j_0}}}} } ={O_p}( {{n^{{{ - r} / {(2r + 1)}}}}}) = o({\lambda _n})$ and $n{p_{{\lambda _n}}}\left\{ {\sqrt{{{\varvec{\hat{\gamma }} }_{{j_0}}}^T{{{\varvec{F}}}_{{j_0}}}{{\varvec{\hat{\gamma }} }_{{j_0}}}} } \right\} =n\lambda _n{\sqrt{{{\varvec{\hat{\gamma }} }_{{j_0}}}^T{{{\varvec{F}}}_{{j_0}}}{{\varvec{\hat{\gamma }} }_{{j_0}}}} }$ with probability tending to 1, by the definition of SCAD penalty function. By the proof of Theorem 2 (a), we have ${I} ={O_p}( {n\sqrt{{p_n}} {n^{{{ - r} / {(2r + 1)}}}}} )\left\| {\varvec{\hat{\gamma }} - {{\varvec{\hat{\gamma }} }^*}} \right\|$. Noting that $\left\| {\varvec{\hat{\gamma }} - {{\varvec{\hat{\gamma }} }^*}} \right\| =\left\| {\varvec{\hat{\gamma }}_{j_0} - {{\varvec{\hat{\gamma }}_{j_0} }^*}} \right\| =O_p( {\sqrt{{{\varvec{\hat{\gamma }} }_{{j_0}}}^T{{{\varvec{F}}}_{{j_0}}}{{\varvec{\hat{\gamma }} }_{{j_0}}}} } )$. We can conclude that $n{p_{{\lambda _n}}}\left\{ {\sqrt{{{\varvec{\hat{\gamma }} }_{{j_0}}}^T{{{\varvec{F}}}_{{j_0}}}{{\varvec{\hat{\gamma }} }_{{j_0}}}} } \right\}$ dominates the other term in (37), which contradicts to $PL(\varvec{\hat{\gamma }},\varvec{\hat{c}},\varvec{\omega }) - PL({\varvec{\hat{\gamma }} ^*},\varvec{\hat{c}},\varvec{\omega }) \le 0.$ $\square$

Proof of Theorem 4

We note that because of Theorem 1–Theorem 3, we only need to consider a correctly specified PLVCM without regularization terms. This reduces the problem to the one studied in Theorem 4.1 of Zou and Yuan (2008) and the results there directly apply, showing the asymptotic normality of the parametric component.

$\square$