Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

Abstract

This paper develops a robust and efficient estimation procedure for quantile partially linear additive models with longitudinal data, where the nonparametric components are approximated by B spline basis functions. The proposed approach can incorporate the correlation structure between repeated measures to improve estimation efficiency. Moreover, the new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations method for non-normal correlated random errors. However, the proposed estimating functions are non-smooth and non-convex. In order to reduce computational burdens, we apply the induced smoothing method for fast and accurate computation of the parameter estimates and its asymptotic covariance. Under some regularity conditions, we establish the asymptotically normal distribution of the estimators for the parametric components and the convergence rate of the estimators for the nonparametric functions. Furthermore, a variable selection procedure based on smooth-threshold estimating equations is developed to simultaneously identify non-zero parametric and nonparametric components. Finally, simulation studies have been conducted to evaluate the finite sample performance of the proposed method, and a real data example is analyzed to illustrate the application of the proposed method.

Appendix

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

• (C1) $E{g_l}({z_l}) = 0$ and ${g _l} \in {\mathscr {H}_r}$, $l=1,\ldots ,d$ for some $r>1/2$, where $\mathscr {H}_r$ is the collection of all functions on [0, 1] whose $\rho$th order derivative satisfies the H$\ddot{o}$lder condition of the order v with $r \equiv \rho + v$ and $0<v\le 1$.

• (C2) There exist constants $\delta _1$ and $\delta _2$ such that the marginal density $f_l(z_l)$ of $Z_l$ satisfies $0< \delta _1 \le f_l(z_l) \le \delta _2 < \infty$ on [0, 1] for every $l=1,\ldots ,d$. The joint density $f_{ll'}(z_l,z_{l'})$ of $(Z_{l},Z_{l'})$ satisfies $0< \delta _1 \le f_{ll'}(z_l,z_{l'})\le \delta _2 < \infty$ for all $(z_l,z_{l'})\in [0,1]^2$, $1\le l\ne {l'}\le d$.

• (C3) The dimensions p, d of $\varvec{x}_{ij}$ and $\varvec{z}_{ij}$ are fixed and $\max n_i$ is bounded when $m\rightarrow \infty$. The distribution functions $F_{ij}(t)=p( {y_{ij}} - \varvec{x}_{ij}^T{\varvec{\beta } } - \sum \nolimits _{l = 1}^d {{g_l}({z_{ijl}}) \le t\left| {{\varvec{x}_{ij}},{\varvec{z}_{ij}}} \right. } )$ are absolutely continuous, with continuous densities $f_{ij}$ uniformly bounded, and its first derivative ${f'_{ij}}( \cdot )$ uniformly bounded away from 0 and $\infty$ at the points $0, i=1,\ldots ,m, j=1,\ldots ,n_i$.

• (C4) For any positive definite matrix $\varvec{W}_i$, $\mathrm{{ }}{m^{ - 1}}\sum \nolimits _{i = 1}^m {\varvec{H}_i^T{\varvec{\Lambda }_i}{\varvec{W}_i}} {\varvec{\Lambda }_i}{\varvec{H}_i}$ converges to a positive definite matrix, where $\varvec{\Lambda } _i$ is an $n_i \times n_i$ diagonal matrix with the jth diagonal element $f_{ij}(0)$ and ${\sup _i}\Vert {{\varvec{H}_i}}\Vert < + \infty$.

• (C5) Matrix $\varvec{\Omega }$ is positive definite and $\varvec{\Omega }=O(\frac{1}{m})$.

• (C6) The parameter space $\Theta$ is compact, and the true parameter vector ${\varvec{\theta } _0}$ is an interior point in $\Theta$.

• (C7) The differentiation of ${{\tilde{U}}_\tau }(\varvec{\theta } )$, $\frac{{\partial {{\tilde{U}}_\tau }(\varvec{\theta } )}}{{\partial \varvec{\theta } }}$ is positive definite with probability 1.

• (C8) Let $\lambda _{max}=\mathrm{max}(\lambda _1,\lambda _2)$ and $\lambda _{min}=\mathrm{min}(\lambda _1,\lambda _2)$. The tuning parameters $\lambda _{min}$ and $\lambda _{max}$ satisfy $m^{r/(2r+1)}\lambda _{max} \rightarrow 0$, and $m^{(1+\kappa )r/(2r+1)}\lambda _{min} \rightarrow \infty$, as $m \rightarrow \infty$.

Proof of Theorem 1

Let $\varvec{G}_i^T = \varvec{H}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}$, so ${U_\tau }(\varvec{\theta } ) = - \sum \nolimits _{i = 1}^m {\varvec{G}_i^T} {\varvec{S}_i}$. Let ${{\bar{U}}_\tau }(\varvec{\theta } )= - \sum \nolimits _{i = 1}^m {\varvec{G}_i^T} {\varvec{P}_i}$ with ${\varvec{P}_i} = {(\tau - p({y_{i1}} - \varvec{h}_{i1}^T\varvec{\theta } \le 0),\ldots ,\tau - p({y_{i{n_i}}} - \varvec{h}_{i{n_i}}^T\varvec{\theta } \le 0))^T}$. We can obtain

$$\begin{aligned}&{m^{ - 1}}\left[ {{{\bar{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \right] = {m^{ - 1}}\sum \limits _{i = 1}^m {\varvec{G}_i^T\left( {{\varvec{S}_i} - {\varvec{P}_i}} \right) } \\&\quad = {m^{ - 1}}\sum \limits _{i = 1}^m {\varvec{G}_i^T\left( {\begin{array}{*{20}{c}} {p({y_{i1}} - \varvec{h}_{i1}^T\varvec{\theta } \le 0) - I({y_{i1}} - \varvec{h}_{i1}^T\varvec{\theta } \le 0)} \\ \vdots \\ {p({y_{i{n_i}}} - \varvec{h}_{i{n_i}}^T\varvec{\theta } \le 0) - I({y_{i{n_i}}} - \varvec{h}_{i{n_i}}^T\varvec{\theta } \le 0)} \\ \end{array}} \right) } \\&\quad = {m^{ - 1}}\sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {{\varvec{g}_{ij}}\left[ {p({y_{ij}} - \varvec{h}_{ij}^T\varvec{\theta } \le 0) - I({y_{ij}} - \varvec{h}_{ij}^T\varvec{\theta } \le 0)} \right] } } , \end{aligned}$$

where $\varvec{g}_{ij}$ is a $(p+dK)\times 1$ vector and $\varvec{G}_i^T = ({\varvec{g}_{i1}},\ldots ,{\varvec{g}_{i{n_i}}})$. Under condition (C4) and from the uniform strong law of large numbers (Pollard 1990), we have

$$\begin{aligned}&\mathop {\sup }\limits _{\varvec{\theta } \in \Theta } \left| {{m^{ - 1}}\sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {{\varvec{g}_{ij}}\left[ {p({y_{ij}} - \varvec{h}_{ij}^T\varvec{\theta } \le 0) - I({y_{ij}} - \varvec{h}_{ij}^T \varvec{\theta } \le 0)} \right] } } } \right| \nonumber \\&\quad = o(m^{-1/2}) ~a.s.. \end{aligned}$$

Therefore, $\mathop {\sup }\limits _{\varvec{\theta } \in \Theta } \Vert {m^{ - 1}}\{ {{{\bar{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \}\Vert = o(m^{-1/2})$ a.s.. Now focusing on ${{\bar{U}}_\tau }(\varvec{\theta } )$,

$$\begin{aligned} m^{-1}{\varvec{D}_\tau }(\varvec{\theta }_0 ) =m^{-1} \frac{{\partial {{\bar{U}}_\tau }(\varvec{\theta } )}}{{\partial \varvec{\theta } }}\left| {_{\varvec{\theta } = {\varvec{\theta } _0}}} \right. = m^{-1}\sum \nolimits _{i = 1}^m {\varvec{H}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{\Lambda } _i}} {\varvec{H}_i}. \end{aligned}$$

is positive definite. Since $p({y_{ij}} - \varvec{h}_{ij}^T{\varvec{\theta } _0} \le 0) = \tau$ by condition (C3), ${\varvec{\theta } _0}$ is the unique solution of the equation ${{\bar{U}}_\tau }(\varvec{\theta } )=\varvec{0}$. Due to $\varvec{\hat{\theta }}$ is the solution of the equation ${U_\tau }(\varvec{\theta } )=\varvec{0}$ together with condition (C4), hence $\varvec{\hat{\theta }} \rightarrow {\varvec{\theta } _0}$ as $m\rightarrow +\infty$. For any $\varvec{\theta }$ satisfying $\Vert {\varvec{\theta } - {\varvec{\theta } _0}}\Vert$ $<$ cm$^{-1/3}$,

$$\begin{aligned} {U_\tau }(\varvec{\theta } ) - {U_\tau }({\varvec{\theta } _0})= & {} \sum \limits _{i = 1}^m {\varvec{G}_i^T(\varvec{\theta } )} {\varvec{S}_i}(\varvec{\theta } ) - \sum \nolimits _{i = 1}^m {\varvec{G}_i^T({\varvec{\theta } _0})} {\varvec{S}_i}({\varvec{\theta } _0}) \\= & {} \sum \limits _{i = 1}^m {\varvec{G}_i^T(\varvec{\theta } )} \{ {\varvec{S}_i}(\varvec{\theta } ) - {\varvec{S}_i}({\varvec{\theta } _0})\}\\&+\, \sum \limits _{i = 1}^m {{{\{ {\varvec{G}_i}(\varvec{\theta } ) - {\varvec{G}_i}({\varvec{\theta } _0})\} }^T}} {\varvec{S}_i}({\varvec{\theta } _0}). \end{aligned}$$

The first term can be written as

$$\begin{aligned}&\sum \limits _{i = 1}^m {\varvec{G}_i^T(\varvec{\theta } )} \{ {\varvec{S}_i}(\varvec{\theta } ) - {\varvec{S}_i}({\varvec{\theta } _0})\} \\&\quad = \sum \limits _{i = 1}^m {\varvec{G}_i^T(\varvec{\theta } )} {\varvec{P}_i}(\varvec{\theta } ) + \sum \nolimits _{i = 1}^m {\varvec{G}_i^T(\varvec{\theta } )} \{ {\varvec{S}_i}(\varvec{\theta } ) - {\varvec{S}_i}({\varvec{\theta } _0}) - {\varvec{P}_i}(\varvec{\theta } )\} \\&\quad = \sum \limits _{i = 1}^m {\varvec{G}_i^T(\varvec{\theta } )} {\varvec{P}_i}(\varvec{\theta } ) + \sum \nolimits _{i = 1}^m {\sum \nolimits _{j = 1}^{{n_i}} {{\varvec{g}_{ij}}\left\{ {p({y_{ij}} - \varvec{h}_{ij}^T\varvec{\theta } \le 0) - I({y_{ij}} - \varvec{h}_{ij}^T\varvec{\theta } \le 0)} \right. } } \\&\qquad \left. { + I({y_{ij}} - \varvec{h}_{ij}^T{\varvec{\theta } _0} \le 0) - \tau } \right\} . \end{aligned}$$

According to Lemma 3 of Jung (1996), we have

$$\begin{aligned}&\sup \left| \sum \limits _{i = 1}^m \sum \limits _{j = 1}^{{n_i}} {\varvec{g}_{ij}}\left\{ p({y_{ij}} - \varvec{h}_{ij}^T\varvec{\theta } \le 0) - I({y_{ij}} - \varvec{h}_{ij}^T\varvec{\theta } \le 0)\right. \right. \\&\quad \qquad \left. \left. +\,I({y_{ij}} - \varvec{h}_{ij}^T{\varvec{\theta } _0} \le 0) - \tau \right\} \right| \\&\qquad = {o_p}(\sqrt{m} ). \end{aligned}$$

Then the first term

$$\begin{aligned} \sum \limits _{i = 1}^m {\varvec{G}_i^T( \varvec{\theta } )} \{ {\varvec{S}_i}(\varvec{\theta } ) - {\varvec{S}_i}({\varvec{\theta } _0})\}= & {} \sum \limits _{i = 1}^m {\varvec{G}_i^T(\varvec{\theta } )} {\varvec{P}_i}(\varvec{\theta } ) + {o_p}(\sqrt{m} ) \\= & {} {{\bar{U}}_\tau }(\varvec{\theta } ) + {o_p}(\sqrt{m} ). \end{aligned}$$

From the law of large numbers (Pollard 1990) the second term

$$\begin{aligned}&\sum \limits _{i = 1}^m {{{\{ {\varvec{G}_i}(\varvec{\theta } ) - {\varvec{G}_i}({\varvec{\theta } _0})\} }^T}} {\varvec{S}_i}({\varvec{\theta } _0}) \\&\quad = \sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {\left\{ {{\varvec{g}_{ij}}(\varvec{\theta } ) - {\varvec{g}_{ij}}({\varvec{\theta } _0})} \right\} \left\{ {p({y_{ij}} - \varvec{h}_{ij}^T{\varvec{\theta } _0} \le 0) - I({y_{ij}} - \varvec{h}_{ij}^T{\varvec{\theta } _0} \le 0)} \right\} } } \\&\quad = {o_p}(\sqrt{m} ). \end{aligned}$$

Therefore, ${U_\tau }(\varvec{\theta } ) - {U_\tau }({\varvec{\theta } _0}) = {{\bar{U}}_\tau }(\varvec{\theta } ) + {o_p}(\sqrt{m} )$. By Taylor’s expansion of ${{\bar{U}}_\tau }(\varvec{\theta } )$ together with ${{\bar{U}}_\tau }(\varvec{\theta }_0 )=0$, we can obtain

$$\begin{aligned} {U_\tau }(\varvec{\theta } ) - {U_\tau }({\varvec{\theta } _0}) = \frac{{\partial {{\bar{U}}_\tau }(\varvec{\theta } )}}{{\partial \varvec{\theta } }}\left| {_{\varvec{\theta } = {\varvec{\theta } _0}}} \right. (\varvec{\theta } - {\varvec{\theta } _0}) + {o_p}(\sqrt{m} ). \end{aligned}$$

Because $\varvec{\hat{\theta }}$ is in the $m^{-1/3}$ neighborhood of $\varvec{\theta }_0$ and ${U_\tau }(\varvec{\hat{\theta }} ) = \varvec{0}$, we have

$$\begin{aligned} \varvec{\hat{\theta }} - {\varvec{\theta } _0} = -\varvec{D}_\tau ^{ - 1}({\varvec{\theta } _0}){U_\tau }({\varvec{\theta } _0}) + {o_p}({m^{{{ - 1} / 2}}}). \end{aligned}$$

To obtain the closed form expression of $\varvec{\hat{\beta }}$, similar to Ma et al. (2013), we write the inverse of $\varvec{D}_\tau ({\varvec{\theta } _0})=\sum \nolimits _{i = 1}^m {\varvec{H}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{\Lambda } _i}} {\varvec{H}_i}$ as the following block form:

$$\begin{aligned} \varvec{D}_\tau ^{ - 1}({\varvec{\theta } _0})= & {} {\left( {\begin{array}{*{20}{c}} {\sum \nolimits _{i = 1}^m {\varvec{X}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{\Lambda } _i}} {\varvec{X}_i}} &{}\quad {\sum \nolimits _{i = 1}^m {\varvec{X}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{\Lambda } _i}} {\varvec{\Psi }_i}} \\ {\sum \nolimits _{i = 1}^m {\varvec{\Psi }_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{\Lambda } _i}} {\varvec{X}_i}} &{}\quad {\sum \nolimits _{i = 1}^m {\varvec{\Psi } _i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{\Lambda } _i}} {\varvec{\Psi }_i}} \\ \end{array}} \right) ^{ - 1}} \\= & {} {\left( {\begin{array}{*{20}{c}} {\varvec{D_{XX}}} &{}\quad \varvec{{D_{X\Psi }}} \\ {\varvec{D_{\Psi X}}} &{}\quad {\varvec{D_{\Psi \Psi }}} \\ \end{array}} \right) ^{ - 1}} = \left( {\begin{array}{*{20}{c}} {{\varvec{D}^{11}}} &{}\quad {{\varvec{D}^{12}}} \\ {{\varvec{D}^{21}}} &{}\quad {{\varvec{D}^{22}}} \\ \end{array}} \right) , \end{aligned}$$

where ${\varvec{D}^{11}} = {(\varvec{D_{XX}} - \varvec{D_{X\Psi }}\varvec{D}_{\Psi \Psi }^{ - 1} \varvec{D_{\Psi X}})^{ - 1}},{\varvec{D}^{22}} = {(\varvec{D_{\Psi \Psi }} - \varvec{D_{\Psi X}}\varvec{D_{XX}}^{ - 1}\varvec{D_{X\Psi }})^{ - 1}}, {\varvec{D}^{12}} = - {\varvec{D}^{11}}\varvec{D_{X\Psi }}\varvec{D_{\Psi \Psi } }^{ - 1}$ and ${\varvec{D}^{21}} = - {\varvec{D}^{22}}\varvec{D_{\Psi X}}\varvec{D_{XX}}^{ - 1}.$ Furthermore, let ${U_\tau }({\varvec{\theta } _0}) = {(\varvec{U}_1^T,\varvec{U}_2^T)^T},{\varvec{U}_1} = \sum \nolimits _{i = 1}^m {-\varvec{X}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0})} ,{\varvec{U}_2} = \sum \nolimits _{i = 1}^m -\varvec{\Psi } _i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0})$. Then

$$\begin{aligned} \varvec{\hat{\beta }} - {\varvec{\beta } _0}= & {} - \left[ {{\varvec{D}^{11}}{\varvec{U}_1} + {\varvec{D}^{12}}{\varvec{U}_2}} \right] + {o_p}({m^{{{ - 1} / 2}}}) \\= & {} {\varvec{D}^{11}}{\sum \limits _{i = 1}^m {\left[ {{\varvec{X}_i} - {\varvec{\Psi } _i}\varvec{D_{\Psi \Psi } }^{ - 1}\varvec{D_{\Psi X}}} \right] } ^T}{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0}) + {o_p}({m^{{{ - 1} / 2}}}), \\ \varvec{\hat{\gamma }} - {\varvec{\gamma } _0}= & {} - \left[ {{\varvec{D}^{21}}{\varvec{U}_1} + {\varvec{D}^{22}}{\varvec{U}_2}} \right] + {o_p}({m^{{{ - 1} / 2}}}) \\= & {} {\varvec{D}^{22}}{\sum \limits _{i = 1}^m {\left[ {{\varvec{\Psi } _i} - {\varvec{X}_i}\varvec{D_{XX} }^{ - 1}\varvec{D_{ X \Psi }}} \right] } ^T}{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0}) + {o_p}({m^{{{ - 1} / 2}}}). \end{aligned}$$

That is,

$$\begin{aligned}&\sqrt{m} \left( {\varvec{\hat{\beta } } - {\varvec{\beta } _0}} \right) = \left( {m{\varvec{D}^{11}}} \right) {m^{{{ - 1} / 2}}}{\sum \limits _{i = 1}^m {\left[ {{\varvec{X}_i} - {\varvec{\Psi } _i}\varvec{D_{\Psi \Psi } }^{ - 1}\varvec{D_{\Psi X}}} \right] } ^T}\\&\quad \times {\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0}) + {o_p}(1),\\&\sqrt{m} \left( {\varvec{\hat{\gamma }} - {\varvec{\gamma } _0}} \right) = \left( {m{\varvec{D}^{22}}} \right) {m^{{{ - 1} / 2}}}{\sum \limits _{i = 1}^m {\left[ {{\varvec{\Psi } _i} - {\varvec{X}_i}\varvec{D_{XX} }^{ - 1}\varvec{D_{ X \Psi }}} \right] } ^T}\\&\quad \times {\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0}) + {o_p}(1). \end{aligned}$$

Let $\varvec{\tilde{P}} = \varvec{X({X^T}}\varvec{\Lambda } {\varvec{V^{ - 1}}\varvec{\Lambda }\varvec{X}})^{-1}{\varvec{X}^T}\varvec{\Lambda }{\varvec{V^{ - 1}}}\varvec{\Lambda }$, $\varvec{\tilde{\Psi }} = (\varvec{I - \tilde{P}})\varvec{\Psi }$ with $\varvec{\tilde{\Psi } }= {(\varvec{\tilde{\Psi }_1}^T,\ldots ,\varvec{\tilde{\Psi }_m}^T)^T}$, ${\varvec{D_\Psi }} = \mathop {\lim }\nolimits _{m \rightarrow \infty } \frac{1}{m}\sum \nolimits _{i = 1}^m {\varvec{\tilde{\Psi }_i}^T{\varvec{\Lambda _i}}\varvec{V_i^{ - 1}}{\varvec{\Lambda _i}}} {{\varvec{\tilde{\Psi }}_i}}$, ${\varvec{\Sigma _\Psi }} = \mathop {\lim }\nolimits _{m \rightarrow \infty } \frac{1}{m}\sum \nolimits _{i = 1}^m \varvec{\tilde{\Psi }_i}^T{\varvec{\Lambda _i}}\varvec{V_i^{ - 1}}\varvec{\Sigma }_i \varvec{V_i^{ - 1}}{\varvec{\Lambda _i}} {{\varvec{\tilde{\Psi }}_i}}$. Because $\varvec{S}_i$ are independent random variables with mean zero, and

$$\begin{aligned} {\mathop {\mathrm{var}}}\left( {m^{{{ - 1} / 2}}}{\sum \limits _{i = 1}^m {\left[ {{\varvec{X}_i} - {\varvec{\Psi } _i}\varvec{D_{\Psi \Psi }}^{ - 1}\varvec{D_{\Psi X}}} \right] } ^T}{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0}) \right)= & {} \varvec{\Sigma _X},\\ {\mathop {\mathrm{var}}}\left( {m^{{{ - 1} / 2}}}{\sum \limits _{i = 1}^m {\left[ {{\varvec{\Psi } _i} - {\varvec{X}_i}\varvec{D_{XX }}^{ - 1}\varvec{D_{ X \Psi }}} \right] } ^T}{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0})\right)= & {} \varvec{\Sigma _\Psi }. \end{aligned}$$

The multivariate central limit theorem implies that

$$\begin{aligned} {m^{{{ - 1} / 2}}}{\sum \limits _{i = 1}^m {\left[ {{\varvec{X}_i} - {\varvec{\Psi }_i}\varvec{D_{\Psi \Psi } }^{ - 1}\varvec{D_{\Psi X}}} \right] } ^T}{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0})\mathop \rightarrow \limits ^d N\left( {\varvec{0},\varvec{\Sigma _X}} \right) ,\\ {m^{{{ - 1} / 2}}}{\sum \limits _{i = 1}^m {\left[ {{\varvec{\Psi }_i} - {\varvec{X}_i}\varvec{D_{X X} }^{ - 1}\varvec{D_{X\Psi }}} \right] } ^T}{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{S}_i}({\varvec{\theta } _0})\mathop \rightarrow \limits ^d N\left( {\varvec{0},\varvec{\Sigma _\Psi }} \right) . \end{aligned}$$

Moreover, we have

$$\begin{aligned} m{\varvec{D}^{11}}= & {} {\left[ {m^{-1}\left( {\varvec{D_{XX}} - \varvec{D_{X\Psi }}\varvec{D_{\Psi \Psi } }^{ - 1}\varvec{D_{\Psi X}}} \right) } \right] ^{ - 1}}\mathop \rightarrow \limits ^p \varvec{D_X}^{ - 1},\\ m{\varvec{D}^{22}}= & {} {\left[ {m^{-1}\left( {\varvec{D_{\Psi \Psi }} - \varvec{D_{\Psi X}}\varvec{D_{XX}}^{ - 1}\varvec{D_{X\Psi }}} \right) } \right] ^{ - 1}}\mathop \rightarrow \limits ^p \varvec{D_\Psi } ^{ - 1} \end{aligned}$$

by the law of large numbers. Then, by using Slutskys theorem, it follows that

$$\begin{aligned}&\sqrt{m} \left( {\varvec{\hat{\beta } } - {\varvec{\beta } _0}} \right) \mathop \rightarrow \limits ^d N\left( {\varvec{0},\varvec{D_X}^{ - 1}{\varvec{\Sigma _X}}{{\{ \varvec{D_X}^{ - 1}\} }^T}} \right) , \sqrt{m} \left( \varvec{{\hat{\gamma }} - {\varvec{\gamma } _0}} \right) \mathop \rightarrow \limits ^d N\\&\quad \left( {\varvec{0},\varvec{D_\Psi }^{ - 1}{\varvec{\Sigma _\Psi }}{{\{ \varvec{D_\Psi }^{ - 1}\} }^T}} \right) . \end{aligned}$$

This complete the proof of part (i). Now, we prove part (ii).

In order to obtain the convergence rate of $\hat{g}_l(\cdot )$, let $u_{ij} = \sum \nolimits _{l = 1}^d {g _{0l}}(z_{ijl}) -\varvec{\psi }_{ij}^T\varvec{\gamma }_{0}$. By the conditions (C1), (C2) and Corollary 6.21 in Schumaker (1981), we have $|u_{ij}|=O\left( {{K^{ - r}}} \right)$. From the asymptotic normality of $\varvec{\hat{\gamma } }$, we have

$$\begin{aligned} m^{-1}\sum \nolimits _{l = 1}^d {{{\sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {\left[ {\varvec{\psi }_{ijl}^T({\varvec{\hat{\gamma } }_l} - {\varvec{\gamma } _{0l}})} \right] } } }^2}} = {O_p}({m^{ - 1}}K). \end{aligned}$$

The triangular inequality implies that

$$\begin{aligned}&m^{-1}\sum \limits _{l = 1}^d {{{\sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {\left\{ {{{\hat{g}}_l}(z_{ijl}) - {g_{0l}}(z_{ijl})} \right\} } } }^2}} \\&\quad \le 2m^{-1}\sum \limits _{l = 1}^d {{{\sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {\left[ {\varvec{\psi }_{ijl}^T({\varvec{\hat{\gamma } }_l} - {\varvec{\gamma } _{0l}})} \right] } } }^2}} + C{K^{ - 2r}}. \end{aligned}$$

This completes the proof. $\square$

Proof of Theorem 2

Because ${{\tilde{S}}_{ij}} - {S_{ij}} = {\mathop {\mathrm{sgn}}} ( - {d_{ij}})\Phi ( - \left| {{d_{ij}}} \right| ),$ where $sgn(\cdot )$ is the sign function and $d_{ij}=(\varepsilon _{ij}+u_{ij})/r_{ij}$. We can obtain

$$\begin{aligned} -{m^{{{ - 1} / 2}}}\left[ {{{\tilde{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \right]= & {} {m^{{{ - 1} / 2}}}\sum \limits _{i = 1}^m {\varvec{H}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}} \left( \begin{array}{l} {\mathop {\mathrm{sgn}}} ( - {d_{i1}})\Phi ( - \left| {{d_{i1}}} \right| ) \\ \vdots \\ {\mathop {\mathrm{sgn}}} ( - {d_{i{n_i}}})\Phi ( - \left| {{d_{i{n_i}}}} \right| ) \\ \end{array} \right) \\= & {} {m^{{{ - 1} / 2}}}\sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {{\varvec{g}_{ij}}{\mathop {\mathrm{sgn}}} ( - {d_{ij}})\Phi ( - \left| {{d_{ij}}} \right| )} } , \end{aligned}$$

where $\varvec{g}_{ij}$ is the jth column of $\varvec{H}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}$. Because

$$\begin{aligned} E\left( {{{\tilde{S}}_{ij}} - {S_{ij}}} \right)= & {} \int _{ - \infty }^{ + \infty } {{\mathop {\mathrm{sgn}}} ( - {d_{ij}})\Phi ( - \left| {{d_{ij}}} \right| )} {f_{ij}}(\varepsilon )d\varepsilon \\= & {} \int _{ - \infty }^{ + \infty } {\Phi ( - {{\left| \varepsilon +u_{ij} \right| } / {{r _{ij}}}})} \left\{ {2I(\varepsilon +u_{ij} \le 0) - 1} \right\} {f_{{ij}}}(\varepsilon )d\varepsilon \\= & {} {r _{ij}}\int _{ - \infty }^{ + \infty } {\Phi ( - \left| t \right| )} \left\{ {2I(t \le 0) - 1} \right\} \left[ {{f_{{ij}}}(0) + {f'_{ij}}(\zeta (t))({r_{ij}}t-u_{ij})} \right] dt, \end{aligned}$$

where $\zeta (t)$ is between 0 and ${r _{ij}}t-u_{ij}$. Since $\int _{ - \infty }^{ + \infty } {\Phi ( - \left| t \right| )} \{ {2I(t \le 0) - 1}\}dt = 0$, and by condition (C3), there exists a constant C such that ${\sup _{ij}}\left| {{f'_{ij}}(\zeta (t))} \right| \le C$. Then

$$\begin{aligned}&\int _{ - \infty }^{ + \infty } {\Phi ( - \left| t \right| )} \left\{ {2I(t \le 0) - 1} \right\} ({f_{{ij}}}(0)- {f'_{ij}}(\zeta (t))u_{ij}) dt\\&\quad \le \int _{ - \infty }^{ + \infty } {\Phi ( - \left| t \right| )} \left\{ {2I(t \le 0) - 1} \right\} ({f_{{ij}}}(0)+ C |u_{ij}|) dt= 0. \end{aligned}$$

Because $\int _{ - \infty }^{ + \infty } {\Phi ( - \left| t \right| )} \left| t \right| dt = {1 / 2}$, thus

$$\begin{aligned} \left| {E\left( {{{\tilde{S}}_{ij}} - {S_{ij}}} \right) } \right| \le r _{ij}^2\int _{ - \infty }^{ + \infty } {\Phi ( - \left| t \right| )} \left| t \right| {\left| {{{f'}_{ij}}\left( {\zeta \left( t \right) } \right) } \right| }dt \le {{Cr _{ij}^2} / {2.}} \end{aligned}$$

Under conditions (C4) and (C5), as $m \rightarrow \infty$,

$$\begin{aligned} \left\| {{m^{{{ - 1} / 2}}}E\left[ {{{\tilde{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \right] } \right\| \le {m^{{{ - 1} / 2}}}{\sup _{i,j}}\left| {{\varvec{g}_{ij}}} \right| \sum \limits _{i = 1}^m {{{Cr _{ij}^2} / {2 = o(1).}}} \end{aligned}$$

In addition,

$$\begin{aligned} {m^{ - 1}}{\mathop {\mathrm{var}}} \left[ {{{\tilde{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \right] = {m^{ - 1}}\sum \limits _{i = 1}^m {{\mathop {\mathrm{var}}} \left\{ {\sum \limits _{j = 1}^{{n_i}} {{\varvec{g}_{ij}}{\mathop {\mathrm{sgn}}} \left( - {d_{ij}}\right) \Phi ( - \left| {{d_{ij}}} \right| )} } \right\} } . \end{aligned}$$

By Cauchy-Schwartz inequality,

$$\begin{aligned} \frac{1}{m}{\mathop {\mathrm{var}}} \left[ {{{\tilde{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \right]\le & {} \frac{1}{m}\sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {{\varvec{g}_{ij}}\varvec{g}_{ij}^T{\mathop {\mathrm{var}}} ({{\tilde{S}}_{ij}} - {S_{ij}})} } \\&+\,\frac{1}{m}\sum \limits _{i = 1}^m \sum \limits _{j = 1}^{{n_i}} \sum \limits _{j' \ne j}^{{n_i}} {\varvec{g}_{ij}}\varvec{g}_{ij'}^T\\&\times \sqrt{\mathop {\mathrm{var}}} ({{\tilde{S}}_{ij}} - {S_{ij}}){\mathop {\mathrm{var}}} ({{\tilde{S}}_{ij'}} - {S_{ij'}}) . \end{aligned}$$

For each $j=1,\ldots ,n_i$,

$$\begin{aligned} {\mathop {\mathrm{var}}} ({{\tilde{S}}_{ij}} - {S_{ij}})\le & {} E{({{\tilde{S}}_{ij}} - {S_{ij}})^2} \\= & {} \int _{ - \infty }^{ + \infty } {{{\{ {\mathop {\mathrm{sgn}}} ( - {d_{ij}})\Phi ( - \left| {{d_{ij}}} \right| )\} }^2}} {f_{ij}}(\varepsilon )d\varepsilon \\= & {} {r _{ij}}\int _{ - \infty }^{ + \infty } {{\Phi ^2}( - \left| t \right| )} {f_{ij}}({r_{ij}}t-u_{ij})dt \\= & {} {r _{ij}}\int _{\left| t \right| > \Delta } {{\Phi ^2}( - \left| t \right| ){f_{ij}}({r_{ij}}t-u_{ij})dt}\\&+\, {r _{ij}}\int _{\left| t \right| \le \Delta } {{\Phi ^2}( - \left| t \right| ){f_{ij}}({r_{ij}}t-u_{ij})dt} \\\le & {} {\Phi ^2}( - \Delta ) + {r _{ij}}\Delta {f_{ij}}(\xi ), \end{aligned}$$

where $\Delta$ is a positive value, and $\xi$ lies between $(-r_{ij}\Delta -u_{ij},r_{ij}\Delta -u_{ij})$. Let $\Delta =m^{1/3}$. Under condition (C5), since $r_{ij}=O(m^{-1/2})$, then $r_{ij}\Delta =O(m^{-1/6})$. As $m\rightarrow \infty$, both ${\Phi ^2}( - \Delta )$ and ${r _{ij}}\Delta {f_{ij}}(\xi )$ go to 0. By conditions (C4) and (C6), it is easy to obtain ${m^{ - 1}}{\mathop {\mathrm{var}}} \left[ {{{\tilde{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \right] =o(1)$. Therefore, we have $m^{ - 1/2}\left[ {{{\tilde{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \right] \rightarrow 0$ as $m\rightarrow \infty$ for any $\theta$. We complete the proof of Theorem 2. $\square$

Proof of Theorem 3

Since ${\sup _{\varvec{\theta } \in \Theta }}\left\| {{m^{ - 1}}\left[ {{{\bar{U}}_\tau }(\varvec{\theta } ) - {U_\tau }(\varvec{\theta } )} \right] } \right\| = o({m^{{{ - 1} / 2}}} )$ a.s. and Theorem 2, we can obtain ${\sup _{\varvec{\theta } \in \Theta }}\left\| {{m^{ - 1}}\left[ {{{\tilde{U}}_\tau }(\varvec{\theta } ) - {{\bar{U}}_\tau }(\varvec{\theta } )} \right] } \right\| = o({m^{{{ - 1} / 2}}} )$ by the triangle inequality. Since $\varvec{\theta }_0$ is the unique solution of equation ${{\bar{U}}_\tau }(\varvec{\theta } )=0$. This together with the definite of $\varvec{\tilde{\theta }}$ implies $\varvec{\tilde{\theta }}\rightarrow \varvec{\theta }_0$ as $m\rightarrow \infty$. In order to prove the normality of $\varvec{\tilde{\theta }}$, we first prove that ${m^{ - 1}}\{ {{\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0}) - {\varvec{D}_\tau }({\varvec{\theta } _0})}\}\mathop \rightarrow \limits ^p 0$, where ${\tilde{\varvec{D}}_\tau }( {{\varvec{\theta } _0}}) = {{\partial {{\tilde{U}}_\tau }( \varvec{\theta })} / {\partial \varvec{\theta } }}{|_{\varvec{\theta } = {\varvec{\theta } _0}}} = \sum \nolimits _{i = 1}^m {\varvec{H}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{\varvec{\tilde{\Lambda } }_i}} {\varvec{H}_i}$. If we denote $\varvec{G}_i^T = \varvec{H}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}=(\varvec{g}_{i1},\ldots ,\varvec{g}_{in_i})$, where $\varvec{g}_{ij}$ is a $(p+dK)\times 1$ vector, we can obtain that

$$\begin{aligned} E\left\{ {{\tilde{\varvec{D}}_\tau }(\varvec{\theta }_0 )} \right\} - {\varvec{D}_\tau }(\varvec{\theta }_0 ) = \sum \limits _{i = 1}^m {\sum \limits _{j = 1}^{{n_i}} {{\varvec{g}_{ij}}\left\{ {r_{ij}^{ - 1}E\phi \left( {\frac{{{\varepsilon _{ij}+u_{ij}}}}{{{r _{ij}}}}} \right) - {f_{ij}}(0)} \right\} } } {\varvec{h}_{ij}}. \end{aligned}$$

Because

$$\begin{aligned} \left| {r _{ij}^{ - 1}E\phi \left( {\frac{{{\varepsilon _{ij}+u_{ij}}}}{{{r _{ij}}}}} \right) - {f_{ij}}(0)} \right|= & {} \left| {r _{ij}^{ - 1}\int _{ - \infty }^{ + \infty } {\phi \left( {\frac{\varepsilon +u_{ij} }{{{r _{ij}}}}} \right) {f_{ij}}(\varepsilon )d\varepsilon } - {f_{ij}}(0)} \right| \\= & {} \left| \int _{ - \infty }^{ +\infty } {\phi \left( t \right) \left\{ {{f_{ij}}(0) +( {r _{ij}}t-u_{ij}){f'_{ij}}({\zeta _t})} \right\} dt}\right. \\&\left. - {f_{ij}}(0) \right| \\= & {} \left| {\int _{ - \infty }^{ + \infty } {\phi \left( t \right) ({r _{ij}}t-u_{ij}){f'_{ij}}({\zeta _t})} dt } \right| \\\le & {} {r_{ij}}\int _{ - \infty }^{ +\infty } {\left| {\phi \left( t \right) t{f'_{ij}}({\zeta _t})} \right| } dt\\&+\int _{ - \infty }^{ + \infty } \phi \left( t \right) |u_{ij}{f'_{ij}}({\zeta _t}) |dt, \end{aligned}$$

where $\zeta _t$ lies between 0 and $r _{ij}t-u_{ij}$. By condition (C3), $f'_{ij}(\cdot )$ is uniformly bounded, hence there exists a constant C satisfying $\left| {{f'_{ij}}({\zeta _t})} \right| \le C$, and by condition (C5), we have

$$\begin{aligned} \left| {r _{ij}^{ - 1}E\phi \left( {\frac{{{\varepsilon _{ij}+u_{ij}}}}{{{r _{ij}}}}} \right) - {f_{ij}}(0)} \right| \le \sqrt{\frac{2}{\pi }} {r _{ij}}C +C|u_{ij}|\rightarrow 0. \end{aligned}$$

By the strong law of large number, we know that ${m^{ - 1}}{\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0}) \rightarrow E\{ {{m^{ - 1}}{\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0})}\}$. Using the triangle inequality, we have

$$\begin{aligned} \left| {{m^{ - 1}}\left\{ {{\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0}) - {\varvec{D}_\tau }({\varvec{\theta } _0})} \right\} } \right|\le & {} \left| {{m^{ - 1}}\left\{ {{\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0}) - E{\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0})} \right\} } \right| \\&+ \left| {{m^{ - 1}}\left\{ {{E \tilde{\varvec{D}}_\tau }({\varvec{\theta } _0}) - {\varvec{D}_\tau }({\varvec{\theta } _0})} \right\} } \right| \rightarrow o(1), \end{aligned}$$

which is equivalent to ${{m^{ - 1}}\{ {{\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0}) - {\varvec{D}_\tau }({\varvec{\theta } _0})}\}}\mathop \rightarrow \limits ^p 0.$ By Taylor series expansion of ${{{\tilde{U}}_\tau }({\varvec{\theta } })}$ around $\varvec{\theta }_0$, we have

$$\begin{aligned} {{\tilde{U}}_\tau }(\varvec{\theta } ) = {{\tilde{U}}_\tau }({\varvec{\theta } _0}) + {\tilde{\varvec{D}}_\tau }({\varvec{\theta } ^*})\left( {\varvec{\theta } - {\varvec{\theta } _0}} \right) , \end{aligned}$$

where $\varvec{\theta } ^*$ lies between $\varvec{\theta }$ and $\varvec{\theta } _0$. Let $\varvec{\theta }=\varvec{\tilde{\theta }}$. Because ${{\tilde{U}}_\tau }(\varvec{\tilde{\theta } }) = 0$ and $\varvec{\tilde{\theta } } \rightarrow {\varvec{\theta } _0}$, we therefore obtain $\varvec{\theta } ^* \rightarrow {\varvec{\theta } _0}$ and ${\tilde{\varvec{D}}_\tau }({\varvec{\theta } ^*}) \rightarrow {\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0})$. By Theorem 2 and ${m^{ - 1}}\{ {{\tilde{\varvec{D}}_\tau }({\varvec{\theta } _0}) - {\varvec{D}_\tau }({\varvec{\theta } _0})}\} = o_p(1)$, we have

$$\begin{aligned} \sqrt{m} \left( {\varvec{\tilde{\theta }} - {\varvec{\theta } _0}} \right) = - m \varvec{D}_\tau ^{ - 1}({\varvec{\theta } _0}){m^{{{ - 1} / 2}}}{{ U}_\tau }({\varvec{\theta } _0}) + {o_p}(1). \end{aligned}$$

Applying Theorem 1, we can easily complete the proof Theorem 3. $\square$

Proof of Theorem 4

Let ${\delta _m}=m^{-r/(2r+1)}$, $\varvec{\theta }={{\varvec{\theta }}_0}+{\delta _m}{\varvec{T}}$. Define ${ S_m}(\varvec{\theta } ) =({\varvec{I}_{p+dK}} - \varvec{\hat{\Delta }} ){\tilde{U}_\tau }(\varvec{\theta } )+\varvec{\hat{\Delta }}\varvec{\theta }$. We want to prove that $\forall \varepsilon > 0$, there exists a constant $C > 0$, such that

$$\begin{aligned} p\left( {\mathop {\inf }\limits _{\left\| \varvec{ T} \right\| = C} {\delta _m}{\varvec{T}^T}{S_m}({\varvec{\theta } _0} + {\delta _m}\varvec{T}) > 0} \right) \ge 1 - \varepsilon \end{aligned}$$

for m large enough. This will imply with probability at least $1 - \varepsilon$ that there exists a solution of the equation ${S_m}(\varvec{\theta } )=\varvec{0}$ such that $\Vert \varvec{\bar{\theta }} - {\varvec{\theta } _0}\Vert = {O_p}({\delta _m})$. The key of the proof is to evaluate the sign of ${\delta _m}{\varvec{T}^T}{S_m}({\varvec{\theta } _0} + {\delta _m}\varvec{T})$ on $\{ {{\varvec{\theta } _0} + {\delta _m}\varvec{T}:\Vert \varvec{T}\Vert = C}\}$. Note that

$$\begin{aligned} {\delta _m}{\varvec{T}^T}{S_m}({\varvec{\theta } _0} + {\delta _m}\varvec{T}) = {\delta _m}{\varvec{T}^T}{S_m}({\varvec{\theta } _0}) + {\delta _m^2}{\varvec{T}^T}\frac{\partial }{{\partial \varvec{\theta } }}{S_m}(\varvec{\tilde{\theta }} )\varvec{T} \buildrel \Delta \over = {I_{m1}} + {I_{m2}}, \end{aligned}$$

where ${\varvec{\tilde{\theta }} }$ is between $\varvec{\theta }_0$ and ${{\varvec{\theta } _0} +{\delta _ m}\varvec{T}}$. Next we will consider $I_{m1}$ and $I_{m2}$ respectively. For $I_{m1}$, by some basic calculations, we have

$$\begin{aligned} {I_{m1}} = {\delta _n}{\varvec{T}^T}({\varvec{I}_{p+dK}} - \varvec{\hat{\Delta }} ){\tilde{U}_\tau }({\varvec{\theta } _0})+ {\delta _m}{\varvec{T}^T} \varvec{\hat{\Delta }}\varvec{\theta }_0 \buildrel \Delta \over = {I_{m11}} + {I_{m12}}. \end{aligned}$$

Using the Cauchy-Schwarz inequality, we have

$$\begin{aligned} {|I_{m11}|}\le & {} {\delta _m}\left\| \varvec{T}^T({\varvec{I}_{p+dK}} - \varvec{\hat{\Delta }} ) \right\| \left\| {{\tilde{U}_\tau }({\varvec{\theta } _0} )} \right\| \\\le & {} {\delta _m}\left\{ {1 - \min \left( {\mathop {\min }\limits _{j \in \mathscr {A}} {{\hat{\delta } }_j}({\lambda _1},\kappa ),\mathop {\min }\limits _{j \in {\mathscr {B}}} {{\hat{\delta } }_j}({\lambda _2},\kappa )} \right) } \right\} \left\| \varvec{T} \right\| \left\| {{\tilde{U}_\tau }({\varvec{\theta } _0} )} \right\| . \end{aligned}$$

Since $\mathop {\min {{\hat{\delta } }}}\nolimits _{j \in \mathscr {A}} (\lambda _1 ,\kappa ) \le \mathop {\min {{\hat{\delta } }}}\nolimits _{j \in {\mathscr {A}_0}} (\lambda _1 ,\kappa )$ and $\mathop {\min {{\hat{\delta } }}}\nolimits _{j \in {\mathscr {B}}} (\lambda _2 ,\kappa ) \le \mathop {\min {{\hat{\delta } }}}\nolimits _{j \in {{\mathscr {B}}_0}} (\lambda _2 ,\kappa )$, we only need to gain the convergence rate of $\mathop {\min {{\hat{\delta } }}}\nolimits _{j \in {\mathscr {A}_0}} (\lambda _1 ,\kappa )$ and $\mathop {\min {{\hat{\delta } }}}\nolimits _{j \in {{\mathscr {B}}_0}} (\lambda _2 ,\kappa )$. By Theorem 3, we know that the initial estimator ${\varvec{{\tilde{\gamma }} }}$ is $m^{r/(2r+1)}$-consistent. By using the condition $m^{r/(2r+1)}\lambda _{max} \rightarrow 0$, for $\varepsilon >0$ and ${j \in {{\mathscr {B}}_0}}$, we can derive that

$$\begin{aligned} P\left( {\frac{{{\lambda _2}}}{{\left\| {{\varvec{\tilde{\gamma } }_j}} \right\| _{\varvec{W}_j}^{1 + \kappa }}} > {m^{\frac{{ - r}}{{2r + 1}}}}\varepsilon } \right)\le & {} P\left( {\frac{{{\lambda _{\max }}}}{{\left\| {{\varvec{\tilde{\gamma } }_j}} \right\| _{\varvec{W}_j}^{1 + \kappa }}} > {m^{\frac{{ - r}}{{2r + 1}}}}\varepsilon } \right) \\= & {} P\left( {{{\left( {{{{m^{\frac{r}{{2r + 1}}}}{\lambda _{\max }}} / \varepsilon }} \right) }^{{1 / {(1 + \kappa )}}}} > {{\left\| {{\varvec{\tilde{\gamma } }_j}} \right\| }_{\varvec{W}_j}}} \right) \\\le & {} P\left( {{\left( {{{{m^{\frac{r}{{2r + 1}}}}{\lambda _{\max }}} / \varepsilon }} \right) }^{{1 / {(1 + \kappa )}}}} > \mathop {\min }\limits _{j \in {\mathscr {B}}_0} \{ {{\left\| {{\varvec{\tilde{\gamma } }_{0j}}} \right\| }_{\varvec{W}_j}}\}\right. \\&\left. - {O_p}\left( {{m^{\frac{{ - r}}{{2r + 1}}}}} \right) \right) \\\rightarrow & {} 0, \end{aligned}$$

which implies that ${{\hat{\delta } }_j}(\lambda _2 ,\kappa ) ={o_p}( {{m^{{{ - r} / {(2r + 1)}}}}})$ for ${j \in {{\mathscr {B}}_0}}$. Therefore, we can get ${\min _{j \in {\mathscr {B}}}}{{\hat{\delta } }_j}(\lambda _2 ,\kappa ) = {o_p}( {{m^{{{ - r} / {(2r + 1)}}}}})$. With the similar argument, we can obtain that ${\min _{j \in \mathscr {A}}}{{\hat{\delta } }_j}(\lambda ,\kappa ) = {o_p}( {{m^{{{ - r} / {(2r + 1)}}}}})$. Thus, $|I_{m11}|= {O_p}(\sqrt{m}{\delta _m})\Vert \varvec{T } \Vert -{o_p}({\sqrt{m} \delta _m^2})\Vert \varvec{T }\Vert$. In addition, ${{\hat{\delta } }_j} = \min ( {1,{\lambda _1 / {{{\left| {{{\tilde{\beta } }_j}} \right| }^{1 + \kappa }}}}})$, $j = 1,\ldots ,p$, and ${{\hat{\delta } }_j} = \min ( {1,{\lambda _2 / {\Vert {{\varvec{\tilde{\gamma } }_j}} \Vert _{\varvec{W}_j}^{1 + \kappa }}}})$, $j= p + 1,\ldots ,p + d$. Thus, we have $\Vert {{I_{m12}}}\Vert \le {\delta _m}\Vert \varvec{T}\Vert \Vert {{\varvec{\theta } _0}}\Vert =O_p(\delta _m\Vert \varvec{T}\Vert )$. Therefore, $| {{I_{m1}}}\Vert = {O_p}(\sqrt{m}{\delta _m})\Vert \varvec{T }\Vert ={o_p}({m}{\delta _m^2})\Vert \varvec{T }\Vert$. For $I_{m2}$, we can derive that

$$\begin{aligned} {I_{m2}}= & {} \delta _m^2{\varvec{T}^T}\frac{\partial }{{\partial \theta }}{S_m}(\varvec{\tilde{\theta }} )\varvec{T} \\= & {} m \delta _m^2 {\varvec{T}^T}({\varvec{I}_{p+dK}} {-} \varvec{\hat{\Delta }} ) \left[ {\frac{1}{m}\sum \nolimits _{i = 1}^m {\varvec{H}_i^T{\varvec{\Lambda } _i}\varvec{V}_i^{ - 1}{{\varvec{\Lambda } }_i}{\varvec{H}_i}} }\right] \varvec{T}{+} \delta _m^2{\varvec{T}^T}\varvec{\hat{\Delta }}\varvec{T} +o_p(m\delta _m^2 ) \\&\buildrel \Delta \over = {I_{m21}} + {I_{m22}}+o_p(m\delta _m^2 ). \end{aligned}$$

With the same argument, it’s easy to prove that ${I_{m22}} = {O_p}( \delta _m^2 ){\Vert \varvec{T}\Vert ^2}=o(m\delta _m^2\Vert \varvec{T} \Vert ^2)$. Thus ${\delta _m}{\varvec{T}^T}{S_m}({\varvec{\theta } _0} + {\delta _m}\varvec{T})$ is asymptotically dominated in probability by $I_{m21}$ on $\{ {{\varvec{\theta } _0} + {\delta _m}\varvec{T}:\Vert \varvec{T}\Vert = C}\}$ for the sufficiently large C, and $I_{m21}>0$ by condition (C7). This implies $\Vert {\varvec{\bar{\theta } }- {\varvec{\theta } _0}}\Vert =O_p({m^{{{ - r} / {(2r + 1)}}}})$, which in turn immediately implies that ${| \sum \nolimits _j{ \bar{g} _{j}}(z_j) - \varvec{\psi }_{ij}^T\varvec{\gamma }_0 |} =O_p({m^{{{ - r} / {(2r + 1)}}}})$ and ${|\sum \nolimits _j{ g _{0j}}(z_j) - \varvec{\psi }_{ij}^T\varvec{\gamma }_0| } = O_p({m^{{{ - r} / {(2r + 1)}}}})$. This complete the proof of part (i).

Now, we prove part (ii). For any given $j \in {{\mathscr {B}}_0^c}$, we have $\Vert {\varvec{\tilde{\gamma }}_j}\Vert _{\varvec{W}_j} = {O_p}(m^{-r/(2r+1)})$, together with $m^{(1+\kappa )r/(2r+1)} \lambda _{min} \rightarrow \infty$, we can derive that

$$\begin{aligned} P\left( {{{{\lambda _2}} / {\left\| {{\varvec{\tilde{\gamma } }_j}} \right\| _{\varvec{W}_j}^{1 + \kappa }}} < 1} \right)\le & {} \lambda _2^{ - 1}E\left( {\left\| {{\varvec{\tilde{\gamma } }_j}} \right\| _{\varvec{W}_j}^{1 + \kappa }} \right) \\= & {} \lambda _2^{ - 1}{m^{{{ - r(1+\kappa )} / {(2r + 1)}}}} \\\le & {} {\lambda _{min} ^{ - 1}}{m^{{{ - r(1+\kappa )} / {(2r + 1)}}}} \\&\rightarrow 0. \end{aligned}$$

This implies that

$$\begin{aligned} P\left\{ {{{\hat{\delta } }_j} = 1~~for~all~j \in {\mathscr {B}}_0^c} \right\} \rightarrow 1. \end{aligned}$$

On the other hand, by the condition $m^{r/(2r+1)}\lambda _{max}\rightarrow 0$, for $\varepsilon >0$ and $j \in {\mathscr {B}}_0$, we have

$$\begin{aligned} P\left( {\frac{{{\lambda _2}}}{{\left\| {{\varvec{\tilde{\gamma } }_j}} \right\| _{\varvec{W}_j}^{1 + \kappa }}} > {m^{\frac{{ - r}}{{2r + 1}}}}\varepsilon } \right)\le & {} P\left( {\frac{{{\lambda _{\max }}}}{{\left\| {{\varvec{\tilde{\gamma } }_j}} \right\| _{\varvec{W}_j}^{1 + \kappa }}} > {m^{\frac{{ - r}}{{2r + 1}}}}\varepsilon } \right) \\= & {} P\left( {{{\left( {{{{m^{\frac{r}{{2r + 1}}}}{\lambda _{\max }}} / \varepsilon }} \right) }^{{1 / {(1 + \kappa )}}}} > {{\left\| {{\varvec{\tilde{\gamma } }_j}} \right\| }_{\varvec{W}_j}}} \right) \\\le & {} P\left( {{\left( {{{{m^{\frac{r}{{2r + 1}}}}{\lambda _{\max }}} / \varepsilon }} \right) }^{{1 / {(1 + \kappa )}}}} > \mathop {\min }\limits _{j \in {\mathscr {B}}_0} \{ {{\left\| {{\varvec{\tilde{\gamma } }_{j0}}} \right\| }_{\varvec{W}_j}}\}\right. \\&\left. - {O_p}\left( {{m^{\frac{{ - r}}{{2r + 1}}}}} \right) \right) \\&\rightarrow 0, \end{aligned}$$

which implies that ${{\hat{\delta } }_j} = {o_p}(m^{-r/(2r+1)})$ for each $j \in {{\mathscr {B}}_0}$. Therefore, we prove that $P\{ {{{\hat{\delta } }_j} < 1~~for~all~j \in {\mathscr {B}}_0}\} \rightarrow 1.$ Thus, we complete the proof of (ii).

For part (iii), apply the similar techniques as in part (ii), we have, with probability tending to 1, that ${{\hat{\delta } }_j} = 1$ for $j\in \mathscr {A}_0^c$ and ${{\hat{\delta } }_j} <1$ for $j\in \mathscr {A}_0$. $\square$

Proof of Theorem 5

As shown in (ii) and (iii) in Theorem 4, ${{\bar{\beta } }_j} = 0$ for $j \in {\mathscr {A}_0^c}$ and ${\varvec{\bar{\gamma }}_j} = 0$ for $j \in {{\mathscr {B}}_0^c}$ with probability tending to 1. Meanwhile, with probability tending to 1, $({\varvec{\bar{\beta }}_{\mathscr {A}_0}},\varvec{\bar{\gamma }_{{\mathscr {B}}_0}})$ satisfies the RSGEE

$$\begin{aligned} \tilde{U}_\tau \left( {\varvec{\beta }_{\mathscr {A}_0} , \varvec{\gamma }_{{\mathscr {B}}_0}} \right) +\varvec{\hat{S}}_{{\mathscr {A}_0},{{\mathscr {B}}_0}}\varvec{\theta }_{{\mathscr {A}_0},{{\mathscr {B}}_0}}=0, \end{aligned}$$

where $\varvec{\theta }_{{\mathscr {A}_0},{{\mathscr {B}}_0}}=(\varvec{\beta }_{\mathscr {A}_0}^T,\varvec{\gamma }_{{\mathscr {B}}_0}^T)^T$, $\varvec{\hat{S}}_{{\mathscr {A}_0},{{\mathscr {B}}_0}}=({\varvec{I}_{s+tK}} - {\varvec{\hat{\Delta }}}_{{\mathscr {A}_0},{{\mathscr {B}}_0}} )^{-1}\varvec{\hat{\Delta }}_{{\mathscr {A}_0},{{\mathscr {B}}_0}}$ and $\varvec{\hat{\Delta }}_{{\mathscr {A}_0},{{\mathscr {B}}_0}} = {\mathrm{diag}}\{ {{{\hat{\delta } }_1},\ldots ,{{\hat{\delta } }_s},\underbrace{{{\hat{\delta } }_{p + 1}},\ldots ,{{\hat{\delta } }_{p + 1}}}_{K},\ldots ,\underbrace{{{\hat{\delta } }_{p + t}},\ldots ,{{\hat{\delta } }_{p + t}}}_{K}} \}$. Moreover, let $\varvec{\hat{\Delta }}_{{\mathscr {A}_0}}={\mathrm{diag}}\{ {{{\hat{\delta } }_1},\ldots ,{{\hat{\delta } }_s}}\}$ and $\varvec{\hat{\Delta }}_{{{\mathscr {B}}_0}}={\mathrm{diag}}\{ {\underbrace{{{\hat{\delta } }_{p + 1}},\ldots ,{{\hat{\delta } }_{p + 1}}}_{K},\ldots ,\underbrace{{{\hat{\delta } }_{p + t}},\ldots ,{{\hat{\delta } }_{p + t}}}_{K}} \}$. Define $\varvec{\hat{S}}_{{\mathscr {A}_0}}=({\varvec{I}_{s}} - {\varvec{\hat{\Delta }}}_{{\mathscr {A}_0}} )^{-1}\varvec{\hat{\Delta }}_{{\mathscr {A}_0}}$ and $\varvec{\hat{S}}_{{{\mathscr {B}}_0}}=({\varvec{I}_{tK}} - {\varvec{\hat{\Delta }}}_{{{\mathscr {B}}_0}} )^{-1}\varvec{\hat{\Delta }}_{{{\mathscr {B}}_0}}$. Then ${\Vert {\varvec{\hat{S}}_{{\mathscr {A}_0},{{\mathscr {B}}_0}}\varvec{\theta }_{{\mathscr {A}_0},{{\mathscr {B}}_0}} }\Vert ^2} ={\Vert {\left( \begin{array}{l} \varvec{\hat{S}}_{\mathscr {A}_0}\varvec{\beta }_{\mathscr {A}_0} \\ \varvec{\hat{S}}_{{\mathscr {B}}_0}\varvec{\gamma }_{{\mathscr {B}}_0}\\ \end{array} \right) }\Vert ^2} = {\Vert { {\varvec{\hat{S}}_{\mathscr {A}_0}\varvec{\beta }_{\mathscr {A}_0} } }\Vert ^2}+{\Vert { { \varvec{\hat{S}}_{{\mathscr {B}}_0}\varvec{\gamma }_{{\mathscr {B}}_0}} }\Vert ^2}$, where

$$\begin{aligned} {\left\| { {\varvec{\hat{S}}_{\mathscr {A}_0}\varvec{\beta }_{\mathscr {A}_0} } } \right\| ^2}\le & {} \frac{1}{{{{\left\{ {1 - \mathop {\max }\limits _{j \in {\mathscr {A}_0}} {{\hat{\delta } }_j}(\lambda _1 ,\kappa )} \right\} }^2}}}\sum \limits _{j \in {\mathscr {A}_0}} {\frac{{{{(\lambda _1 {\beta _j})}^2}}}{{\tilde{\beta } _j^{2(1 + \kappa )}}}} \\= & {} \frac{\lambda _1^2}{{{{\left\{ {1 - \mathop {\max }\limits _{j \in {\mathscr {A}_0}} {{\hat{\delta } }_j}(\lambda _1 ,\kappa )} \right\} }^2}}}\sum \limits _{j \in {\mathscr {A}_0}} {{{\left| {\tilde{\beta } _j^{( - \kappa )} + ({\beta _j} - \tilde{\beta } _j)\tilde{\beta } _j^{( - 1 -\kappa )}} \right| }^2}} \\\le & {} {O_p}(\lambda _1^2 )\sum \limits _{j \in {\mathscr {A}_0}} {\left( {2{{\left| {\tilde{\beta } _j} \right| }^{ - 2\kappa }} + 2{{\left| {({\beta _j} - \tilde{\beta } _j)\tilde{\beta } _j^{( - 1 - \kappa )}} \right| }^2}} \right) } \\\le & {} {O_p}({\lambda _{max}^2})\left( {2s\mathop {\min }\limits _{j \in {\mathscr {A}_0}} {{\left| {\tilde{\beta } _j} \right| }^{ - 2\kappa }} + 2\mathop {\min }\limits _{j \in {\mathscr {A}_0}} {{\left| {\tilde{\beta } _j} \right| }^{2( - 1 - \kappa )}}{{\left\| {{\varvec{\beta } _{\mathscr {A}_0}} - \varvec{\tilde{\beta }} _{\mathscr {A}_0}} \right\| }^2}} \right) \\= & {} {O_p}\left\{ {{{((m^{r/(2r+1)} \lambda _{max})}^2}m^{-2r/(2r+1)}{\nu ^{ - 2\kappa }}s)(1 + {O_p}({\nu ^{ - 2}}m^{-2r/(2r+1)}))} \right\} \\= & {} {o_p}(m^{-2r/(2r+1)}), \end{aligned}$$

where $\nu = \mathop {\min }\nolimits _{j \in {\mathscr {A}_0}} \left| {\tilde{\beta } _j} \right|$. Similarly, we have ${\Vert {\varvec{\hat{S}}_{{\mathscr {B}}_0}\varvec{\gamma }_{{\mathscr {B}}_0} } \Vert ^2}={o_p}(m^{-2r/(2r+1)})$. Applying Theorem 3, we can easily complete the proof Theorem 5. $\square$