Appendix
Throughout the appendix, C is used to represent a generic positive constant that can change from line to line. For a vector x, ||x|| denotes its Euclidean norm and for a matrix \({\mathbf {A}}\), \(\left\| {\mathbf {A}} \right\| = \sqrt{{\lambda _{\max }}({\mathbf {A}}'{\mathbf {A}})}\) denotes its spectral norm.
1.1 Lemmas
For a given covariate \(u_i\), we let \(\varvec{{\tilde{\iota }}(u_i)}=({{\iota }_0(u_i)},\ldots ,{{\iota }_N(u_i)})^T\) denote the corresponding vector of B-spline basis functions of order \(h+1\). A property of B-spline is that \(\sum \nolimits _{s = 1}^N {{{\iota }_s}({u_i})} = 1\), thus to avoid collinearity when fitting models only \({{\varvec{\iota } }(u_i)}=({{\iota }_1(u_i)},\ldots ,\) \({{\iota }_N(u_i))}^T\) is used. For ease of proof the B-spline basis functions are theoretically centered similar to the one defined in Xue and Yang (2006). Specifically, \({{\iota }_s}\) in the above is transformed by defining \({w_s}({u_i}) = \sqrt{{k_n}} \left[ {{{\iota }_s}({u_i}) - \frac{{E({{\iota }_s}({u_i}))}}{{E({{\iota }_0}({u_i}))}}{{\iota }_0}({u_i})} \right] .\) For a given covariate \(u_i\), let \({\mathbf {w}}(u_i) = (w_1(u_i),\ldots , w_N(u_i))^{T}\) be the vector of basis functions, and \({\mathbf {W}}(u_i)\) denote the \(J_n \times 1\) vector \(({\mathbf {w}}(u_i)^{T}X_{i1}, . . . , {\mathbf {w}}(u_i)^{T}X_{ip})^{T}\), where \(J_n = p\times N\) and \({\mathbf {W}} = ({\mathbf {W}}(u_1),\ldots , {\mathbf {W}}(u_n))^{T}\in R^{n\times J_n}\). Let
$$\begin{aligned} (\hat{\beta } ,{\hat{a}}) = \mathop {\arg \min }\limits _{(\beta ,a, b_k)} \sum \limits _{k = 1}^K {\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{\hat{\pi } }}} } {\rho _{{\tau _k}}}({Y_i} - {\varvec{\beta ^T}}{{\mathbf {Z}}_i} - {\mathbf {W}}{({u_i})^T}{\textit{\textbf{a}}} - {b_k}), \end{aligned}$$
then \(\varvec{{\hat{\beta }}}=\varvec{{\hat{\beta }}}^{WBCQR_{{\hat{\pi }}}}\) and \(\varvec{{\hat{\alpha }}}^{WBCQR_{{\hat{\pi }}}}(u)={\mathbf {W}}{({u_i})^T}\varvec{\hat{a}}^{WBCQR_{{\hat{\pi }}}}\). To help analyze the asymptotic behavior of \(\varvec{{\hat{\beta }}}\) , while accounting for the estimation of \({\textit{\textbf{a}}}\), following the techniques of He and Shi (1996), we define:
$$\begin{aligned} {{\mathbf {D}}_n}= & {} \mathrm {diag}\left\{ {\sum \nolimits _{k = 1}^K {{f_i}({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}){\delta _i}\pi _{i0}^{ - 1}} } \right\} _{i=1}^n \in {R^{n \times n}},\\ {{\varvec{{\hat{D}}}}_n}= & {} \mathrm {diag}\left\{ {\sum \nolimits _{k = 1}^K {{f_i}({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}){\delta _i}\hat{\pi } _{i}^{ - 1}} } \right\} _{i=1}^n \in {R^{n \times n}},\\ {{\mathbf {Z}}^*}= & {} {(Z_1^*, \ldots ,Z_n^*)^T} = ({{\mathbf {I}}_n} - {\mathbf {P}}){\mathbf {Z}} \in {R^{n \times q}}, {\mathbf {W}}_D^2 = {{\mathbf {W}}^T}{{{\hat{\mathbf {D}}}}_n}{\mathbf {W}} \in {R^{J_n \times J_n}},\\ {\varvec{\eta } _1}= & {} \sqrt{n} (\varvec{\beta } - {{\varvec{\beta }_0}}) \in {R^q}, {\varvec{\eta } _2} = {{\mathbf {W}}_D}({\textit{\textbf{a}}} - {{\textit{\textbf{a}}}_0}) + {{\textit{\textbf{W}}}_D}^{ - 1}{W^T}{{\varvec{D_n}}}{\mathbf {Z}}({\varvec{\beta }} - {{\varvec{\beta }_0}}) \in {R^{{J_n}}},\\ {{{\tilde{\mathbf {Z}}}}_i}= & {} {n^{{{ - 1} / 2}}}{\mathbf {Z}}_i^* \in {R^{{q}}}, \tilde{ {\mathbf {W}}}({u_i}) = {{\textit{\textbf{W}}}_D}^{ - 1}{\mathbf {W}}({u_i}) \in {R^{{J_n}}}, {{{\tilde{\mathbf {s}}}}_i} = {({\tilde{\mathbf {Z}}}_i^T,\tilde{{\mathbf {W}}}({u_i}))^T} \in {R^{q + {J_n}}}\\ {\phi _{ni}}= & {} ({\textit{\textbf{W}}}{({u_i})^T}{{\textit{\textbf{a}}}_0} - \varvec{\alpha }(u_i))^T {\mathbf {X}}_i,\\ {Q_i}({a_n})= & {} \sum \nolimits _{k = 1}^K {{\rho _{{\tau _k}}}} \left\{ {{\varepsilon _i} - {a_n}{{{\tilde{\mathbf {Z}}}}_i}^T{{\varvec{\eta } _1}} - {a_n}\tilde{{\mathbf {W}}}{{({u_i})}^T}{\varvec{\eta } _2} - {\phi _{ni}}} \right\} , {E_s}({Q_i}) = E({Q_i}|{{\mathbf {Z}}_i},{{U}_i},{{\mathbf {X}}_i}). \end{aligned}$$
We observe that
$$\begin{aligned}&\sum \limits _{k = 1}^K {\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{\hat{\pi }_i }}} } {\rho _{{\tau _k}}}({Y_i} - {\varvec{\beta ^T}}{Z_i} - {\mathbf {W}}{({u_i})^T}{\textit{\textbf{a}}} - {b_k}) \\&\quad = \sum \limits _{k = 1}^K {\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{\hat{\pi }_i }}} } {\rho _{{\tau _k}}}({\varepsilon _i} - {\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}} - \tilde{{\mathbf {W}}}{({u_i})^T}{\varvec{\eta } _2} - {b_k} - {\phi _{ni}}) \end{aligned}$$
Defining
$$\begin{aligned} ({{\varvec{\hat{\eta }} }_1},{\varvec{\hat{\eta }}_2}) = \mathop {\arg \min }\limits _{({{\varvec{\eta } _1}},{\varvec{\eta } _2})} \sum \limits _{k = 1}^K {\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{\hat{\pi } }}} } {\rho _{{\tau _k}}}({\varepsilon _i} - {\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}} - {\tilde{W}}{({u_i})^T}{\varvec{\eta } _2} - {b_k} - {\phi _{ni}}), \end{aligned}$$
we then have \({{\varvec{\hat{\eta }} }_1} = \sqrt{n} ({{\varvec{\hat{\beta } }}^{WBCQ{R_{\hat{\pi } }}}} - {{\varvec{\beta }_0}})\) and
$$\begin{aligned} {{{\varvec{\hat{\eta }}}}_2} = {{{\textit{\textbf{W}}}_D}}({\varvec{{\hat{a}}}^{WBCQ{R_{\hat{\pi } }}}} - {{\textit{\textbf{a}}}_0}) + {{\textit{\textbf{W}}}}_D^{ - 1}{{{\textit{\textbf{W}}}^T}}{{\varvec{{\hat{D}}}}_n}{\mathbf {Z}}({{\varvec{\hat{\beta }} }^{WBCQ{R_{\hat{\pi } }}}} - {{\varvec{\beta }_0}}). \end{aligned}$$
Lemma 1
Under Condition 3, there exists a constant \(C>0\) such that
$$\begin{aligned} \mathop {\sup }\limits _{u \in U} |{{\alpha }_l}(u) - {\varvec{\iota }^{T} }(u){{\textit{\textbf{a}}} _{0l}}| \le C{k_{n}^{ - r}}, \end{aligned}$$
(A.1)
for \(l=1,\ldots ,p\), where \(k_n\) is the number of knots. r is defined in Condition 3.
The idea to prove this lemma is similar to Schumaker (1981). Therefore, we omit the proof.
Lemma 2
If Conditions 1–5 are satisfied, then \({n^{{{ - 1} / 2}}}{{\mathbf {Z}}^*} = {n^{{{ - 1} / 2}}}{\varvec{\Delta } _n} + {o_p}(1)\). Furthermore, \({n^{ - 1}}{{\mathbf {Z}}^{*T}}{{{\hat{\mathbf {D}}}}_n}{Z^*} = {{\varvec{\Sigma }} _1} + {o_p}(1)\), where \({{\varvec{\Sigma }} _1}\) is defined in Sect. 2.
Proof
By the definition of \(Z^{*}\),
$$\begin{aligned} {n^{{{ - 1} / 2}}}{{\mathbf {Z}}^*} = {n^{{{ - 1} / 2}}}({\mathbf {Z}} - {\mathbf {P}}{\mathbf {Z}}) = {n^{{{ - 1} / 2}}}{\varvec{\Delta } _n} + {n^{{{ - 1} /2}}}({\mathbf {H}} - {\mathbf {P}}{\mathbf {Z}}). \end{aligned}$$
Consider the following weighted least squares problem. Let \(\varvec{\gamma }^{*}_{j}\in R^{J_n}\) be defined as \(\varvec{\gamma } _j^* = \mathop {\arg \min }\limits _{\gamma \in {R^{{J_n}}}} \sum \nolimits _{k = 1}^K {\sum \nolimits _{i = 1}^n {({{{\delta _i}} / {{{\hat{\pi }}_i}}})} } {f_i}({b_k}|{{\mathbf {Z}}_i},{U_i},{{\mathbf {X}}_i}){\left\{ {{Z_{ij}} - {\mathbf {W}}{{({U_i})}^T}\varvec{\gamma } } \right\} ^2}\). Let \({{{\hat{h}}}_j}({U_i}) = {\mathbf {W}}{({U_i})^T}{\varvec{\gamma } ^*}\) and notice that \({({\mathbf {P}}{\mathbf {X}})_{ij}} = {{{\hat{h}}}_j}({U_i})\). Adapting the results from Stone (1985), for the basis function, and from Wang et al. (1997), we obtain that
$$\begin{aligned} {n^{ - 1}}{\left\| {{\mathbf {H}}-\mathbf {PA}} \right\| ^2}= & {} {n^{ - 1}}{\lambda _{\max }}\left\{ {{{(\mathbf {H} - \mathbf {PZ})}^T}(\mathbf {H} - \mathbf{PZ})} \right\} \\\le & {} {n^{ - 1}}\mathrm{trace}\left\{ {{{(\mathbf {H} - \mathbf {PZ})}^T}(\mathbf {H}- \mathbf {PZ})} \right\} \\= & {} {n^{ - 1}}\sum \limits _{i = 1}^n {\sum \limits _{j = 1}^p {{{\left\{ {h_j^*({U_i}) - {{{\hat{h}}}_j}({U_i})} \right\} }^2}} } \\= & {} {o_p}(1) \end{aligned}$$
\(\square \)
Lemma 3
If the Conditions 1–5 hold then for any \(\omega >0\)
$$\begin{aligned} \Pr \left[ {\mathop {\inf }\limits _{\left\| \eta \right\| = L} \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}J_n^{ - 1}\left\{ {{Q_i}(\sqrt{{J_n}} ) - {Q_i}(0)} \right\} > 0} } \right] \ge 1 - \omega \end{aligned}$$
The idea to prove this lemma is similar to Lemma 4 of Sherwood (2015), where L is arbitrary finite positive constant. Therefore, we omit the proof.
1.2 Proof of asymptotic normality from Theorem 1 and Theorem 2
Throughout this section we will use Knight’s identity [presented in Koenker (2005) as a generalization of an identity presented in Knight (1998)] that
$$\begin{aligned} {\rho _\tau }(u - v)-\rho _\tau (u) = - v{\psi _\tau }(u) + \int _o^v {I(u \le s)} - I(u \le 0){d_s}. \end{aligned}$$
Another useful equality is \({\rho _\tau }({\varepsilon _i} - b - c) - {\rho _\tau }({\varepsilon _i} - c) = \int _{ - c}^{ - b - c} {{\psi _\tau }} ({\varepsilon _i} + s){d_s}\).
Define
$$\begin{aligned} {{\mathbf {B}}_n}= & {} \mathrm {diag}(\sum \limits _{k = 1}^K {{f_1}} ({b_k}|{{\mathbf {Z}}_1},{{\mathbf {X}}_1},{U_1}), \ldots ,\sum \limits _{k = 1}^K {{f_n}} ({b_k}|{{\mathbf {Z}}_n},{{\mathbf {X}}_n},{U_n})),\\ \varvec{\hat{\delta }}= & {} \mathrm {diag}({\delta _1}\hat{\pi } _1^{ - 1}, \ldots ,{\delta _n}\hat{\pi } _n^{ - 1}), \varvec{\psi }^{*} (\varepsilon ) = {(\psi ({\varepsilon _1}), \ldots ,\psi ({\varepsilon _n}))^T},\\ {{\varvec{{{\tilde{\eta }}}} }_1}= & {} \sqrt{n} {({{\varvec{Z^{*T}}}}{{\mathbf {B}}_n}{{\mathbf {Z}}^*})^{ - 1}}{{\varvec{Z^{*T}}}}\varvec{\hat{\delta }} \varvec{\psi }^{*} (\varepsilon ), \Delta {(B)_n} = {n^{ - 1}}\Delta _n^T{{\mathbf {B}}_n}{\Delta _n}. \end{aligned}$$
The idea is to show that \(\varvec{{\hat{\eta }}}_{1}\) is asymptotically equivalent to \(\varvec{{\tilde{\eta }}}_{1}\). The following definition is used for ease of notation
$$\begin{aligned} Q_i^*\left( {{{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2}} \right)= & {} \sum \limits _{k = 1}^K \left\{ {\rho _{{\tau _k}}}\left\{ {{\varepsilon _i} - {\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}} - \tilde{{\mathbf {W}}}{{\left( {{U_i}} \right) }^T}{\varvec{\eta } _2} - {\phi _{ni}}-b_k} \right\} \right. \nonumber \\&\left. - {\rho _{{\tau _k}}}\left\{ {{\varepsilon _i} - {\tilde{\mathbf {Z}}}_i^T{\varvec{{{\tilde{\eta }}} }_1} - \tilde{{\mathbf {W}}}{{\left( {{U_i}} \right) }^T}{\varvec{\eta } _2} - {\phi _{ni}}-b_{k}} \right\} \right\} . \end{aligned}$$
(A.2)
For positive constants M, L and C
$$\begin{aligned} \Pr \left[ {\mathop {\inf }\limits _{\begin{array}{c} \left\| {{{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}} \right\| \ge M\\ \scriptstyle \left\| {{\varvec{\eta } _2}} \right\| \le C\sqrt{{k_n}} \end{array}} \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}Q_i^*({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2}) > 0} } \right] \rightarrow 1, \end{aligned}$$
(A.3)
By Lemma B.4 from supplementary material of Sherwood and Wang (2016) and Condition 2, we have
$$\begin{aligned} \mathop {\sup }\limits _{\begin{array}{c} \scriptstyle \left\| {{{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}} \right\| \le M\\ \scriptstyle \left\| {{\varvec{\eta } _2}} \right\| \le C\sqrt{{k_n}} \end{array}} \left| {\displaystyle \sum \limits _{i = 1}^n {\displaystyle \frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}\left[ {Q_i^*\big ({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2}\big ) - {E_s}\left\{ {Q_i^*\big ({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2}\big )} \right\} +{{\tilde{\mathbf {Z}}}_i^T\big ({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}\big )\psi ({\varepsilon _i})} } \right] } } \right| = {o_p}(1), \end{aligned}$$
where
$$\begin{aligned}&\displaystyle \sum \limits _{i = 1}^n {\displaystyle \frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}{E_s}} \left\{ {Q_i^*\big ({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2}\big )} \right\} \\&\quad = \displaystyle \sum \limits _{i = 1}^n {{E_s}} \left[ {\int _{ - \left\{ {{\tilde{\mathbf {Z}}}_i^T{\varvec{{{\tilde{\eta }}} }_1} + {\tilde{\mathbf {W}}}{{({Z_i})}^T}{\varvec{\eta } _2} + {\phi _{ni}}} \right\} }^{ - \left\{ {{\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}} + {\tilde{\mathbf {W}}}{{({Z_i})}^T}{\varvec{\eta } _2} + {\phi _{ni}}} \right\} } {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}}\sum \limits _{k = 1}^K{\psi _{\tau _k} ({\varepsilon _i} + s){d_s}} } \right] \\&\quad = -\displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}} \int _{ - \left\{ {{\tilde{\mathbf {Z}}}_i^T{\varvec{{{\tilde{\eta }}} }_1} +\tilde{ {\mathbf {W}}}{{({Z_i})}^T}{\varvec{\eta } _2} + {\phi _{ni}}} \right\} }^{ - \left\{ {{\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}} +{\tilde{\mathbf {W}}}{{({Z_i})}^T}{\varvec{\eta } _2} + {\phi _{ni}}} \right\} }\sum \limits _{k = 1}^K {\left\{ {{F_i}\big ( - s|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}\big ) - \displaystyle {{F_i}({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i})} } \right\} {d_s}} \\&\quad = \displaystyle \frac{1}{2}\displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}{f_i}\big ({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}\big )\big (1 + {o_p}(1)\big )} } \\&\qquad { \times \left[ {{{\left\{ {{{\tilde{\mathbf{Z}}}}_i^T{{\varvec{\eta } _1}} +{{\tilde{\mathbf {W}}}}{{({Z_i})}^T}{\varvec{\eta } _2} + {\phi _{ni}}} \right\} }^2} - {{\left\{ {{\tilde{\mathbf {Z}}}_i^T{{{{\tilde{\varvec{\eta }}}} }_1} + {\tilde{\mathbf {W}}}{{\big ({{\mathbf {Z}}_i}\big )}^T}{\varvec{\eta } _2} + {\phi _{ni}}} \right\} }^2}} \right] }\\&\quad = \displaystyle \frac{1}{2}\displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}{f_i}\big ({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}\big )} } (1 + {o_p}(1))\\&\qquad { \times \left[ {{{\big ({\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}}\big )}^2} - {{\big ({\tilde{\mathbf {Z}}}_i^T{\varvec{{{\tilde{\eta }}} }_1}\big )}^2} + 2\left\{ {{\tilde{\mathbf {W}}}{{({{{\mathbf {Z}}_i}})}^T}{\varvec{\eta } _2} + {\phi _{ni}}} \right\} \big ({\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}} - {\tilde{\mathbf {Z}}}_i^T{\varvec{{{\tilde{\eta }}} }_1}\big )} \right] }\\&\quad = \displaystyle \frac{1}{2}\displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {\left( \frac{{{\delta _i}}}{{{\pi _{i0}}}} + \frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}} - \frac{{{\delta _i}}}{{{\pi _{i0}}}}\right) {f_i}\big ({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}\big )} } \\&\qquad \times {\left[ {{{({\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}})}^2} - {{\big ({\tilde{\mathbf {Z}}}_i^T{\varvec{{{\tilde{\eta }}} }_1}\big )}^2} + 2\left\{ {{\tilde{\mathbf {W}}}{{({{{\mathbf {Z}}_i}})}^T}{\varvec{\eta } _2} + {\phi _{ni}}}\right\} \big ({\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}} - {\tilde{\mathbf {Z}}}_i^T{\varvec{{{\tilde{\eta }}} }_1}\big )} \right] \times (1 + {o_p}(1))}\\&\quad =\displaystyle \frac{1}{2}\left\{ {{\varvec{\eta } _1}^T\varvec{\Delta } {{(B)}_n}{{\varvec{\eta } _1}} - \tilde{\varvec{\eta } _1}^T\varvec{\Delta } {{(B)}_n}{{\varvec{\eta } _1}}} \right\} \left\{ {1 + {o_p}(1)} \right\} \\&\qquad { + {n^{{{ - 1} / 2}}}{{\big ({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}\big )}^T}\displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{\pi _{i0}}}}{f_i}\big ({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}\big ){{\mathbf {R}}_i}{\phi _{ni}}\left\{ {1 + {o_p}(1)} \right\} } } }\\&\qquad + \displaystyle \frac{1}{2}\displaystyle \sum \limits _{i = 1}^n {\left( \frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}} - \frac{{{\delta _i}}}{{{\pi _{i0}}}}\right) \left\{ {{{\big ({\tilde{\mathbf {Z}}}_i^T{{\varvec{\eta } _1}}\big )}^2} - {{\big ({\tilde{\mathbf {Z}}}_i^T{\varvec{{{\tilde{\eta }}} }_1}\big )}^2}} \right\} } \\&\qquad { + {n^{{{ - 1} / 2}}}} {\big ({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}\big )^T}\displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {\left( \frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}} - \frac{{{\delta _i}}}{{{\pi _{i0}}}}\right) } } {f_i}\big ({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}\big ){{\mathbf {R}}_i}{\phi _{ni}}\left\{ {1 + {o_p}(1)} \right\} \\&\quad =\displaystyle \frac{1}{2}\left\{ {{\varvec{\eta } _1}^T{\varvec{\Delta }} {{(B)}_n}{{\varvec{\eta } _1}} - \tilde{\varvec{\eta } _1}^T{\varvec{\Delta }} {{(B)}_n}{\varvec{{{\tilde{\eta }}} }_1}} \right\} + {o_p}(1). \end{aligned}$$
Therefore, we get
$$\begin{aligned} \mathop {\sup }\limits _{\begin{array}{c} \scriptstyle \left\| {{{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}} \right\| \le M\\ \scriptstyle \left\| {{\varvec{\eta } _2}} \right\| \le C\sqrt{{k_n}} \end{array}} \left| {\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}{E_s}\left\{ {Q_i^*({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2})} \right\} - \frac{1}{2}\left\{ {{\varvec{\eta } _1}^T{\varvec{\Delta }} {{(B)}_n}{{\varvec{\eta } _1}} - \tilde{\varvec{\eta } _1}^T{\varvec{\Delta }} {{(B)}_n}{\varvec{{{\tilde{\eta }}} }_1}} \right\} } } \right| = {o_p}(1).\nonumber \\ \end{aligned}$$
(A.4)
Then by (A.4) and Condition 5, we have
$$\begin{aligned}&\mathop {\sup }\limits _{\begin{array}{c} \scriptstyle \left\| {{{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}} \right\| \le M\\ \scriptstyle \left\| {{\varvec{\eta } _2}} \right\| \le C\sqrt{{k_n}} \end{array}} \left| \sum \limits _{i = 1}^n \frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}\left[ \left\{ {Q_i^*({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2}) + {\tilde{\mathbf {Z}}}_i^T({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1})\psi ({\varepsilon _i})} \right\} \right. \right. \nonumber \\&\quad \left. \left. - \frac{1}{2}\left\{ {{\varvec{\eta } _1}^T{\varvec{\Delta }} {{(B)}_n}{{\varvec{\eta } _1}} - \tilde{\varvec{\eta } _1}^T{\varvec{\Delta }} {{(B)}_n}{\varvec{{{\tilde{\eta }}} }_1}} \right\} \right] \right| = {o_p}(1). \end{aligned}$$
(A.5)
Note that
$$\begin{aligned} {\big ({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}\big )^T}\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}{{{\tilde{\mathbf {Z}}}}_i}} \psi ({\varepsilon _i}) = {\big ({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}\big )^T}{\varvec{\Delta }} {(B)_n}{\varvec{{{\tilde{\eta }}} }_1} + {o_p}(1), \end{aligned}$$
(A.6)
By combining (A.5) and (A.6) we have
$$\begin{aligned}&\mathop {\sup }\limits _{\begin{array}{c} \scriptstyle \left\| {{{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}} \right\| \le M\\ \scriptstyle \left\| {{\varvec{\eta } _2}} \right\| \le C\sqrt{{k_n}} \end{array}} \left| \displaystyle \sum \limits _{i = 1}^n \displaystyle \frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}\left[ Q_i^*({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2}) + {{({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1})}^T}{\varvec{\Delta }} {{(B)}_n}{\varvec{{{\tilde{\eta }}} }_1}\right. \right. \\&\qquad \left. \left. - \displaystyle \frac{1}{2}\left\{ {{\varvec{\eta } _1}^T{\varvec{\Delta }} {{(B)}_n}{{\varvec{\eta } _1}} - \tilde{\varvec{\eta } _1}^T{\varvec{\Delta }} {{(B)}_n}{\varvec{{{\tilde{\eta }}} }_1}} \right\} \right] \right| = {o_p}(1)\\&\quad \Rightarrow \mathop {\sup }\limits _{\begin{array}{c} \scriptstyle \left\| {{{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}} \right\| \le M\\ \scriptstyle \left\| {{\varvec{\eta } _2}} \right\| \le C\sqrt{{k_n}} \end{array}} \left| {\displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}\left[ {Q_i^*({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2}) - \displaystyle \frac{1}{2}{{({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1})}^T}{\varvec{\Delta }} {{(B)}_n}({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1})} \right] } } \right| = {o_p}(1). \end{aligned}$$
By Conditions 1 and 2 for any \(M>0\) and \(C>0\)
$$\begin{aligned} \frac{1}{2}{\big ({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}\big )^T}{\varvec{\Delta }} {(B)_n}\big ({{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}\big )> 0. \end{aligned}$$
So, we have
$$\begin{aligned} \mathop {\lim }\limits _{n \rightarrow \infty } \mathop {\inf }\limits _{\begin{array}{c} \scriptstyle \left\| {{{\varvec{\eta } _1}} - {\varvec{{{\tilde{\eta }}} }_1}} \right\| \ge M\\ \scriptstyle \left\| {{\varvec{\eta } _2}} \right\| \le C\sqrt{{k_n}} \end{array}} {\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{{\hat{\pi } }_i}}}Q_i^*({{\varvec{\eta } _1}},{\varvec{{{\tilde{\eta }}} }_1},{\varvec{\eta } _2})} }> 0. \end{aligned}$$
Then by the convexity of \(Q^{*}_{i}\) the proof is complete.
Hence, we have
$$\begin{aligned}&\sqrt{n} ({{\varvec{\hat{\beta } }^{WBCQ{R_\pi }}}} - {{\varvec{\beta }_0}}) \\&\quad = {\left\{ {{n^{ - 1}}\sum \limits _{k = 1}^K {\sum \limits _{i = 1}^n {{f_i}({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}){\mathbf {Z}}_i^*{{\mathbf {Z}}_i}^{*T}} } } \right\} ^{ -1/\sqrt{n}}}\sum \limits _{i = 1}^n {({\delta _i}/{\pi _{i0}}){\mathbf {Z}}_i^*\psi ({\varepsilon _i})} + {o_p}(1), \end{aligned}$$
Using Lemma 2 for the first equality, we get
$$\begin{aligned} \sqrt{n} ({{\varvec{\hat{\beta } }^{WBCQ{R_\pi }}}} - {{\varvec{\beta }_0}}) = {\left\{ {{{\varvec{\Sigma }} _1} + {o_p}(1)} \right\} ^{ - 1}}{n^{{{ - 1} / 2}}}\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{\pi _{i0}}}}{{\mathbf {R}}_i}\psi ({\varepsilon _i})} \left\{ {1 + {o_p}(1)} \right\} \nonumber \\ \end{aligned}$$
(A.7)
The expected value of (A.7) is zero. Therefore, it suffices to compute the variance of the sums.
$$\begin{aligned} Var\left\{ {\frac{{{\delta _i}}}{{{\pi _{i0}}}}{{\mathbf {R}}_i}\psi ({\varepsilon _i})} \right\} = E\left\{ {\frac{{{\delta _i}}}{{\pi _{i0}}}{{\mathbf {R}}_i}{\mathbf {R}}_i^T\psi {{({\varepsilon _i})}^2}} \right\} = E\left\{ {\frac{1}{{\pi _{i0}}}{{\mathbf {R}}_i}{\mathbf {R}}_i^T\psi {{({\varepsilon _i})}^2}} \right\} = {{\varvec{\Sigma }} _2}. \end{aligned}$$
Therefore, we have
$$\begin{aligned} \sqrt{n} ({{\varvec{\hat{\beta } }^{WBCQ{R_\pi }}}} - {{\varvec{\beta }_0}}) \rightarrow N(0,{\varvec{\Sigma }} _1^{ - 1}{{\varvec{\Sigma }} _2}{\varvec{\Sigma }} _1^{ - 1}). \end{aligned}$$
This completes the proof of Part (i) for Theorem 1.
$$\begin{aligned}&\sqrt{n} ({\varvec{\hat{\beta } }^{WBCQ{R_{\hat{\pi } }}}} - {{\varvec{\beta }_0}}) \\&\quad = {\left\{ {{n^{ - 1}}\sum \limits _{k = 1}^K {\sum \limits _{i = 1}^n {{f_i}({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}){\mathbf {Z}}_i^*{{\mathbf {Z}}_i}^{*T}} } } \right\} ^{ -1/\sqrt{n}}}\sum \limits _{i = 1}^n {({\delta _i}/{{\hat{\pi } }_i}){\mathbf {Z}}_i^*\psi ({\varepsilon _i})} + {o_p}(1). \end{aligned}$$
Using Lemma 2 for the first equality and Lemma 3, we get
$$\begin{aligned} \begin{aligned}&\sqrt{n}\left( \hat{\varvec{\beta }}^{W B C Q R_{\hat{\pi }}}-\varvec{\beta }_{0}\right) =\left\{ \varvec{\Sigma }_{1}+o_{p}(1)\right\} ^{-1} n^{-1 / 2} \sum _{i=1}^{n} \frac{\delta _{i}}{{\hat{\pi }}_{i}} {\mathbf {R}}_{i} \psi \left( \varepsilon _{i}\right) \left\{ 1+o_{p}(1)\right\} \end{aligned} \end{aligned}$$
Again
$$\begin{aligned} \displaystyle \sum _{i=1}^{n} \displaystyle \frac{\delta _{i}}{{\hat{\pi }}_{i}} {\mathbf {R}}_{i} \psi ={\displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{\pi _{i0}}}}{{\mathbf {R}}_i}\psi ({\varepsilon _i}) - \displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i} - {\pi _{i0}}}}{{{\pi _{i0}}}}E\left\{ {{{\mathbf {R}}_i}\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} } } } . \end{aligned}$$
So
$$\begin{aligned} \sqrt{n} ({\varvec{\hat{\beta } }^{WBCQ{R_{\hat{\pi } }}}} - {{\varvec{\beta }_0}})= & {} {\left\{ {{{\varvec{\Sigma }} _1} + {o_p}(1)} \right\} ^{ - 1}}\left\{ {n^{{{ - 1} / 2}}}\displaystyle \sum \limits _{i = 1}^n \frac{{{\delta _i}}}{{{\pi _{i0}}}}{{\mathbf {R}}_i}\psi ({\varepsilon _i}) \right. \\&\left. - {n^{{{ - 1} / 2}}}\displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i} - {\pi _{i0}}}}{{{\pi _{i0}}}}E\left\{ {{{\mathbf {R}}_i}\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} } \right\} \left\{ {1 + {o_p}(1)} \right\} . \end{aligned}$$
The expected value of each of these two sums is zero. Therefore, it suffices to compute the variance of the two sums and their covariance. The variance of the first sum is \({{{\varvec{\Sigma }} _2}}\). The variance of the second sum is
$$\begin{aligned}&{\mathrm{Var}} \left\{ {\displaystyle \frac{{{\delta _i} - {\pi _{i0}}}}{{{\pi _{i0}}}}E\left\{ {{{\mathbf {R}}_i}\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} } \right\} \\&\quad = E\left\{ {\displaystyle \frac{{{{({\delta _i} - {\pi _{i0}})}^2}}}{{\pi _{i0}^2}}E\left\{ {{{\mathbf {R}}_i}\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} E\left\{ {R_i^T\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} } \right\} \\&\quad = E\left\{ {\displaystyle \frac{{1 - {\pi _{i0}}}}{{\pi _{i0}}}E\left\{ {{{\mathbf {R}}_i}\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} E\left\{ {R_i^T\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} } \right\} ={\varvec{\Sigma }}_3. \end{aligned}$$
For the covariance of the sums, we can use the assumption that \(\pi _{i0}\) is known given \( {\mathbf {V}}_{i}\) and the law of iterated expectations to get
$$\begin{aligned}&Cov\left\{ {\displaystyle \frac{{{\delta _i}}}{{{\pi _{i0}}}}{{\mathbf {R}}_i}\psi ({\varepsilon _i}),\frac{{{\delta _i} - {\pi _{i0}}}}{{{\pi _{i0}}}}E\left\{ {{{\mathbf {R}}_i}\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} } \right\} \\&\quad = E\left\{ {\displaystyle \frac{{{\delta _i}({\delta _i} - {\pi _{i0}})}}{{{\pi _{i0}}}}{{\mathbf {R}}_i}\psi ({\varepsilon _i})E\left\{ {{{\mathbf {R}}_i}\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} } \right\} \\&\quad = E\left\{ {\displaystyle \frac{{1 - {\pi _{i0}}}}{{{\pi _{i0}}}}E\left\{ {{{\mathbf {R}}_i}\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} E\left\{ {{\mathbf {R}}_i^T\psi ({\varepsilon _i})|{{{{\mathbf {V}}}_i}}} \right\} } \right\} ={\varvec{\Sigma }}_3 \end{aligned}$$
Therefore, we have
$$\begin{aligned} \sqrt{n} ({{\varvec{\hat{\beta }} }^{WBCQ{R_{{\hat{\pi }}} }}} - {{\varvec{\beta }_0}}) \rightarrow N(0,{\varvec{\Sigma }} _1^{ - 1}{{\varvec{\Sigma }} _m}{\varvec{\Sigma }} _1^{ - 1}). \end{aligned}$$
This completes the proof of Part (i) for Theorem 2.
1.3 Proof of coefficient functions convergence rate for Theorem 1 and Theorem 2
It follows Lemma 3 that
$$\begin{aligned} \left\| {{{{\textit{\textbf{W}}}_D}}({{\varvec{{\hat{a}}}}^{WBCQ{R_{\hat{\pi } }}}} - {\varvec{a_0}})} \right\| = {O_p}\left( {\sqrt{{k_n}} } \right) . \end{aligned}$$
By Schumaker (1981) it follows that
$$\begin{aligned} \max \left| {{\phi _{ni}}} \right| = {O_p}(k_n^{ - r}). \end{aligned}$$
Combining this with Condition 3, we have
$$\begin{aligned}&\displaystyle \frac{1}{n}\displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {{f_i}} } ({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}){\{ {{\hat{\alpha } }^{WBCQ{R_{\hat{\pi } }}}_{l}}({U_i}) - {\alpha _{0l}}({U_i})\} ^2}\\&\quad =\displaystyle \frac{1}{n}\displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {{f_i}} } ({b_k}|{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i}){\left\{ {{\mathbf {W}}{{({U_i})}^T}({\varvec{{\hat{a}}_l}^{WBCQ{R_{\hat{\pi } }}}} - \varvec{a_{0l}}) - {\phi _{ni}}} \right\} ^2}\\&\quad \le \displaystyle \frac{1}{n}({{\varvec{{\hat{a}}_l}}^{WBCQ{R_{\hat{\pi } }}}} - {{\textit{\textbf{a}}}_{0l}}){{\textit{\textbf{W}}}_D}^2({\varvec{{\hat{a}}_l}^{WBCQ{R_{\hat{\pi } }}}} - \varvec{a_{0l}}) + {O_p}({n^{{{ - 2r} / {\left( {2r + 1} \right) }}}})\\&\quad = {O_p}({n^{{{ - 2r} / {\left( {2r + 1} \right) }}}}) \end{aligned}$$
The proof is complete by Condition 1, which provides a uniform lower and upper bound for \({f_i}( \cdot |{{\mathbf {Z}}_i},{{\mathbf {X}}_i},{U_i})\). This completes the proof of Part (ii) for Theorem 2. The proof of Part (i) of Theorem 2 is similar to the proof of Part (ii) of Theorem 2, so it is omitted here.
1.4 Proof of Theorem 3 and Theorem 4
Let \(\sqrt{n} ({{\varvec{\hat{\beta }} }^{PWBCQ{R_{ \pi }}}} - {{\varvec{\beta }_0}}) = {{\textit{\textbf{u}}}^*}\), \(\sqrt{n} ({\hat{b}}_k^{PWBCQ{R_\pi }} - {b_k}) = v_k^*\), and \({{\varvec{\theta }} ^*} = ({{\textit{\textbf{u}}}^*},v_k^*)\). \({{\varvec{\theta }} ^*}\) is the minimizer of the following criterion:
$$\begin{aligned} {Q_n}(\pi ,{{\varvec{\theta }} ^*})= & {} \displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {\frac{{{{{{\mathbf {V}}}_i}}}}{{{\pi _{i0}}}}} } \left[ {{\rho _{{\tau _k}}}\left( {{\varepsilon _i} - {b_k} - {\phi _{ni}} - \frac{{v_k^* + {{\mathbf {Z}}_i}^T{{\textit{\textbf{u}}}^*}}}{{\sqrt{n} }}} \right) - {\rho _{{\tau _k}}}({\varepsilon _i} - {b_k})} \right] \\&+\displaystyle \sum \limits _{j = 1}^q {{\lambda _\pi }} \frac{{\left( {\left| {{\beta _j} +\displaystyle \frac{{u_j^*}}{{\sqrt{n} }}} \right| - \left| {{\beta _j}} \right| } \right) }}{{{{\left| {{{\hat{\beta } }_{j}^{WBCQ{R_\pi }}}} \right| }^2}}} \end{aligned}$$
Similar to the proof of Theorem 4.1 in Zou and Yuan (2008), the second term above can be expressed as
$$\begin{aligned} \frac{{{\lambda _\pi }}}{{\sqrt{n} {{\left| {{{\hat{\beta } }_{j}^{WBCQ{R_\pi }}}} \right| }^2}}}\sqrt{n} \left( {\left| {{\beta _j} + \frac{{u_j^*}}{{\sqrt{n} }}} \right| - \left| {{\beta _j}} \right| } \right) {\mathop {\longrightarrow }\limits ^{P}}\left\{ \begin{array}{lllll} 0,&{}\quad if&{} \beta _j \ne 0,\\ 0,&{}\quad if&{}{\beta _j} = 0 &{} and &{} u_j^* = 0,\\ \infty ,&{}\quad if&{}{\beta _j} = 0&{} and &{} u_j^* \ne 0. \end{array} \right. \end{aligned}$$
Let \({{\textit{\textbf{u}}}^*} = {({\textit{\textbf{u}}}_1^{T*},{\textit{\textbf{u}}}_2^{T*})^T}\) where \({\textit{\textbf{u}}}^{*}_{1}\) contains nonzero element of \({\textit{\textbf{u}}}^{*}\). Using the same arguments in Knight (1998) and Koenker (2005), we have \({\textit{\textbf{u}}}^{*}_{2}{\mathop {\rightarrow }\limits ^{P}} 0\) and \({\textit{\textbf{u}}}^{*}_{1}{\mathop {\rightarrow }\limits ^{d}}N(0,{[{{\varvec{\Sigma }} _1^{ - 1}{{\varvec{\Sigma }} _m}{\varvec{\Sigma }} _1^{ - 1}]}_{\Lambda \Lambda }})\). Thus, asymptotic normality is proven.
Next, we prove the consistency part. Let \({{\hat{\Lambda }}_n} = \left\{ {j:\hat{\beta } _j^{PWBCQ{R_\pi }} \ne 0} \right\} \) and \(\Lambda = \left\{ {j:{\beta _j} \ne 0} \right\} \), \(\forall {}j \in \Lambda \), the asymptotic normality indicates \(P(j \in {{\hat{\Lambda }}_n}){\rightarrow } 1\). It suffices to show \(\forall {}j \notin \Lambda \), \(P(j \in {{\hat{\Lambda }}_n}){\rightarrow } 0\). Note that,
$$\begin{aligned}&(b_1^{PWBCQ{R_\pi }}, \ldots ,b_K^{PWBCQ{R_\pi }},{{\varvec{\hat{\beta }} }^{PWBCQ{R_\pi }}},{{\varvec{{\hat{a}}}}^{PWBCQ{R_\pi }}})\\&\quad = \mathop {\arg \min }\limits _{{b_1}, \ldots ,{b_K},\varvec{\beta } ,{\textit{\textbf{a}}}} \displaystyle \sum \limits _{k = 1}^K {\displaystyle \sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{\pi _{i0}}}}{\rho _{{\tau _k}}}({Y_i} - {\varvec{\beta ^T}}{{{\mathbf {Z}}_i}} - \varvec{\Pi } _i^T{\textit{\textbf{a}}} - {b_k})} } + {\lambda _\pi }\sum \limits _{j = 1}^q {\frac{{|{\beta _j}|}}{{|\hat{\beta } _j^{WBCQ{R_\pi }}{|^2}}}} \end{aligned}$$
Using the fact that
$$\begin{aligned} \left| {\frac{{{\rho _\tau }({x_1}) - {\rho _\tau }({x_2})}}{{{x_1} - {x_2}}}} \right| \le \max \left( {\tau ,1 - \tau } \right) < 1, \end{aligned}$$
therefore, we have
$$\begin{aligned} \frac{{{\lambda _\pi }}}{{|\hat{\beta } _j^{WBCQ{R_\pi }}{|^2}}} < \sum \limits _{k = 1}^K {\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{\pi _{i0}}}}\left| {{Z_{ij}}} \right| } } = K\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{\pi _{i0}}}}\left| {{Z_{ij}}} \right| }. \end{aligned}$$
So
$$\begin{aligned} P(j \in {{\hat{\Lambda }}_n}) \le P\left( {\frac{{{\lambda _\pi }}}{{|\hat{\beta } _j^{WBCQ{R_\pi }}{|^2}}} < K\sum \limits _{i = 1}^n {\frac{{{\delta _i}}}{{{\pi _{i0}}}}\left| {{Z_{ij}}} \right| } } \right) {\rightarrow }0. \end{aligned}$$
This completes the proof of Theorem 3. The proof of Theorem 4 is similar to the proof of Theorem 3, so it is omitted here.
