Robust and efficient estimator for simultaneous model structure identification and variable selection in generalized partial linear varying coefficient models with longitudinal data

Abstract

This paper proposes a new robust and efficient estimator for the generalized partial linear varying coefficient models with longitudinal data, which can construct variable selection and partial linear structure identification simultaneously. The new method is built upon a newly proposed smooth-threshold robust and efficient generalized estimating equations, which can use the within subject correlation structure, and achieves robustness against outliers by using bounded exponential score function and leverage-based weights. By introducing an additional tuning parameter, it has balance between robustness and efficiency. Under mild conditions, we prove that, with probability tending to one, it can select the relevant variables and identify the partial linear structure correctly. Furthermore, the varying and nonzero constant coefficients can be estimated accurately, just as the true model structure and relevant variables were known in advance. Simulation studies and real data analysis also confirm our method.

Appendix

Lemma 4.1

Suppose that (A1)–(A7) hold. There exist a vector $\varvec{\gamma }_0=\left( \varvec{\gamma }_{01}^T,\ldots ,\varvec{\gamma }_{0p}^T\right) ^T$ satisfying

• (i) $\Vert \varvec{\gamma }_{0k}\Vert _{1}\ne 0,k\in \mathcal {A}_v;\Vert \varvec{\gamma }_{0k}\Vert _{1}=0,k\in \mathcal {A}_c\bigcup \mathcal {A}_z;$

• (ii) $\sup _{u\in [0,1]}|\eta _{k}(u)-\varvec{B}(u)^T\varvec{\gamma }_{0k}|=O(K_{n}^{-r}),k=1,\ldots ,p.$

Let $R_{nijk}=\eta _{k}(t_{ij})-\varvec{B}(t_{ij})^T\varvec{\gamma }_{0k}$, so by Lemma 4.1, $\max _{i,j,k}|R_{nijk}|=O(K_{n}^{-r})$ holds. Lemma 4.1 follows directly from Corollary 6.21 of Schumaker (1981, Chap. 6).

Proof of Proposition 2.1. If, $\alpha _k(t)=\beta _k+\eta _k(t)=a_k+b_k(t)$ and $Eb_k(t)=0$. Then we have $\beta _k+E\eta _k(t)=a_k+Eb_k(t)$. Note that $Eb_k(t)=E\eta _k(t)=0$, this results that $\beta _k=a_k$ and $\eta _k(t)=b_k(t)$. Thus the Proposition 2.1 is proved.

Proof (a) of Theorem 2.2. Let $\delta _n=n^{-r/(2r+1)}$, $\varvec{\beta }=\varvec{\beta }_0+\delta _n\varvec{T}_1$, $\varvec{\gamma }=\varvec{\gamma }_0+\delta _n\varvec{T}_2$ and $\varvec{T}=(\varvec{T}_1^T,\varvec{T}_2^T)^T$, where $\varvec{\gamma }_0$ is the true value of $\varvec{\gamma }$ in Proposition 2.1. Let $\varvec{S}_n(\varvec{\beta },\varvec{\gamma })=\left( \varvec{I}_s-\hat{\varvec{\Lambda }}\right) \varvec{U}\left( \varvec{\beta },\varvec{\gamma },\hat{\delta }\left( \varvec{\beta },\varvec{\gamma },\hat{\phi }(\varvec{\beta },\varvec{\gamma })\right) \right) + \hat{\varvec{\Lambda }}\left( \varvec{\beta }^T,\varvec{\gamma }^T\right) ^T$. Our aim is to show that for $\varepsilon >0$, there exists a constant $C>0$, such that

$$\begin{aligned} \Pr \left( \sup _{\Vert \varvec{T}\Vert =C}\delta _n\varvec{T}^T\varvec{S}_n(\varvec{\beta }_0+\varvec{T}_1,\varvec{\gamma }_0+\varvec{T}_2)>0\right) \ge 1-\varepsilon , \end{aligned}$$

(4.1)

for n large enough. This will imply with probability at least $1-\varepsilon$ that there exists a local minimum value of the equation $\varvec{S}_n(\varvec{\beta },\varvec{\gamma })=\varvec{0}$ such that $\Vert \left( \hat{\varvec{\beta }}^T,\hat{\varvec{\gamma }}^T\right) ^T-\left( \varvec{\beta }_0^T,\varvec{\gamma }_0^T\right) ^T\Vert =O_{p}(\delta _n)$. The proof follows that of Theorem 3.6 in Wang (2011), we will evaluate the sign of $\delta _n\varvec{T}^T\varvec{S}_n(\varvec{\beta }_0+\varvec{T}_1,\varvec{\gamma }_0+\varvec{T}_2)$ in the ball $\{\varvec{\beta }_0+\varvec{T}_1,\varvec{\gamma }_0+\varvec{T}_2:\Vert \varvec{T}\Vert =C\}$. By the Taylor approximation, we have that

$$\begin{aligned} \delta _n\varvec{T}^T\varvec{S}_n(\varvec{\beta }_0+\varvec{T}_1,\varvec{\gamma }_0+\varvec{T}_2)&=\delta _n\varvec{T}^T\varvec{S}_n(\varvec{\beta }_0,\varvec{\gamma }_0)+ \delta _n^2\varvec{T}^T\frac{\partial \varvec{S}_n(\tilde{\varvec{\beta }},\tilde{\varvec{\gamma }})}{\partial \left( \varvec{\beta }^T,\varvec{\gamma }^T\right) ^T}\varvec{T}\nonumber \\&=I_{n1}+I_{n2} \end{aligned}$$

(4.2)

where $(\tilde{\varvec{\beta }}^T,\tilde{\varvec{\gamma }}^T)^T$ lies between $\left( \varvec{\beta }_0^T,\varvec{\gamma }_0^T\right) ^T$ and $\left( \varvec{\beta }_0^T,\varvec{\gamma }_0^T\right) ^T+\delta _n\varvec{T}$. Next we will consider $I_{n1}$ and $I_{n2}$ respectively. For $I_{n1}$, by some elementary calculations, we have

$$\begin{aligned} I_{n1}&=\delta _n\varvec{T}^T\left( \varvec{I}_s-\hat{\varvec{\Lambda }}\right) \varvec{U}(\varvec{\beta }_0,\varvec{\gamma }_0,\delta )\\&~~~~+\delta _n\varvec{T}^T\left( \varvec{I}_s- \hat{\varvec{\Lambda }}\right) \left[ \varvec{U}\left( \varvec{\beta },\varvec{\gamma },\hat{\delta }\left( \varvec{\beta }_0,\varvec{\gamma }_0,\hat{\phi }(\varvec{\beta }_0,\varvec{\gamma }_0) \right) \right) -\varvec{U}(\varvec{\beta }_0,\varvec{\gamma }_0,\delta )\right] \\&~~~~+\delta _n\varvec{T}^T\hat{\varvec{\Lambda }}\left( \varvec{\beta }_0^T,\varvec{\gamma }_0^T\right) ^T\\&=I_{n11}+I_{n12}+I_{n13}. \end{aligned}$$

By Cauchy–Schwarz inequality, we can derive that

$$\begin{aligned} |I_{n11}|&\le \delta _n\left\| \varvec{T}^T\left( \varvec{I}_s-\hat{\varvec{\Lambda }}\right) \right\| \Vert \varvec{U}(\varvec{\beta }_0,\varvec{\gamma }_0,\delta )\Vert \\&\le \delta _n\left( 1-\min \left[ \min _{k\in \mathcal {A}_1}\hat{\delta }_{1,k},\min _{k\in \mathcal {A}_2}\hat{\delta }_{2,k}\right] \right) \Vert \varvec{T}\Vert \Vert \varvec{U}(\varvec{\beta }_0,\varvec{\gamma }_0,\delta )\Vert . \end{aligned}$$

Since $\min _{k\in \mathcal {A}_1}\hat{\delta }_{1,k}\le \min _{k\in \mathcal {A}_{1,0}}\hat{\delta }_{1,k}$ and $\min _{k\in \mathcal {A}_2}\hat{\delta }_{2,k}\le \min _{k\in \mathcal {A}_{v}}\hat{\delta }_{2,k}$, where $\mathcal {A}_{1,0}=\mathcal {A}_{c}\bigcup \{k:E\alpha _{0k}(t)\ne 0,k\in \mathcal {A}_{v}\}$. We only need to obtain the convergence rate of $\min _{k\in \mathcal {A}_{1,0}}\hat{\delta }_{1,k}$ and $\min _{k\in \mathcal {A}_{v}}\hat{\delta }_{2,k}$. Assume that $(\hat{\varvec{\beta }}^{(0)},\hat{\varvec{\gamma }}^{(0)})$ is the initial estimator, and satisfied $\Vert (\hat{\varvec{\beta }}^{(0)},\hat{\varvec{\gamma }}^{(0)})-(\varvec{\beta }_0,\varvec{\gamma }_0)\Vert =O_{p}(n^{-r/(2r+1)})$. By using the condition $n^{\frac{r}{1+r}}\lambda \rightarrow 0$, for any $\varepsilon >0$ and $k\in \mathcal {A}_{1,0}$, we have

$$\begin{aligned} \Pr \left( \hat{\delta }_{1,k}>n^{\frac{-r}{1+r}}\varepsilon \right)&=\Pr \left( \frac{\lambda }{|\hat{\beta }_j^{(0)}|^{1+\tau }}>n^{\frac{-r}{1+r}}\varepsilon \right) \\&=\Pr \left( (\lambda n^{\frac{r}{1+r}}/\varepsilon )^{1/(1+\tau )}>|\hat{\beta }_k^{(0)}|\right) \\&\le \Pr \left( (\lambda n^{\frac{r}{1+r}}/\varepsilon )^{1/(1+\tau )}>\min _{k\in \mathcal {A}_{1,0}}|\beta _{0k}|-O_{p}(n^{-r/(2r+1)})\right) \\&\rightarrow 0, \end{aligned}$$

which implies that for each $k\in \mathcal {A}_{1,0}$, $\hat{\delta }_{1,k}=o_{p}(n^{-r/(2r+1)})$. Similarly, we can prove that $\hat{\delta }_{2,k}=o_{p}(n^{-r/(2r+1)})$, for each $k\in \mathcal {A}_{v}$. Therefore, we have that

$$\begin{aligned} \min \left[ \min _{k\in \mathcal {A}_1}\hat{\delta }_{1,k},\min _{k\in \mathcal {A}_2}\hat{\delta }_{2,k}\right] =o_{p}(n^{-r/(2r+1)}). \end{aligned}$$

Thus, we can obtain that

$$\begin{aligned} |I_{n11}|=O_{p}(\sqrt{n}\delta _n)\Vert \varvec{T}\Vert -o_{p}(\delta _n)\Vert \varvec{T}\Vert , \end{aligned}$$

which is similar to the proof of Theorem 3.6 in Wang (2011). Furthermore, for the $I_{n12}$, by the regularity conditions and Taylor approximation, we have that

$$\begin{aligned}&\varvec{U}\left( \varvec{\beta },\varvec{\gamma },\hat{\delta }\left( \varvec{\beta }_0,\varvec{\gamma }_0,\hat{\phi }(\varvec{\beta }_0,\varvec{\gamma }_0) \right) \right) -\varvec{U}(\varvec{\beta }_0,\varvec{\gamma }_0,\delta )\\&\quad =\frac{\partial }{\partial \alpha }\varvec{U}(\varvec{\beta }_0,\varvec{\gamma }_0,\delta )\left( \hat{\delta }\left( \varvec{\beta }_0,\varvec{\gamma }_0,\hat{\phi }(\varvec{\beta }_0,\varvec{\gamma }_0) \right) -\delta \right) +o_{p}(1)\\&\quad =\frac{\partial }{\partial \alpha }\varvec{U}(\varvec{\beta }_0,\varvec{\gamma }_0,\delta )\left( \hat{\delta }\left( \varvec{\beta }_0,\varvec{\gamma }_0,\hat{\phi }(\varvec{\beta }_0,\varvec{\gamma }_0) \right) -\hat{\delta }\left( \varvec{\beta }_0,\varvec{\gamma }_0,\phi \right) +\hat{\delta }\left( \varvec{\beta }_0,\varvec{\gamma }_0,\phi \right) -\delta \right) \\&\qquad +o_{p}(1)\\&\quad =\frac{\partial }{\partial \alpha }\varvec{U}(\varvec{\beta }_0,\varvec{\gamma }_0,\delta )\left( \frac{\partial \hat{\delta }\left( \varvec{\beta }_0,\varvec{\gamma }_0,\tilde{\phi } \right) }{\partial \phi }\left( \hat{\phi }(\varvec{\beta }_0,\varvec{\gamma }_0)-\phi \right) +\hat{\delta }\left( \varvec{\beta }_0,\varvec{\gamma }_0,\phi \right) -\delta \right) \\&\qquad +o_{p}(1)\\&\quad =o_{p}(1) \end{aligned}$$

where $\tilde{\phi }$ lies between $\phi$ and $\hat{\phi }(\varvec{\beta }_0,\varvec{\gamma }_0)$. By above result and using the similar argument of $I_{n11}$, we obtain that $|I_{n12}|=o_{p}(\delta _n)\Vert \varvec{T}\Vert$. Since, $\hat{\delta }_{1,j}=\min \left\{ 1,\lambda /|\hat{\beta }_j^{(0)}|^{1+\tau }\right\}$ and $\hat{\delta }_{2,j}=\min \left\{ 1,\lambda /\Vert \hat{\varvec{\gamma }}_j^{(0)}\Vert ^{1+\tau }\right\}$, we have that $|I_{n13}|\le \delta _n\Vert \varvec{T}\Vert \Vert \left( \varvec{\beta }_0^T,\varvec{\gamma }_0^T\right) ^T\Vert =O_{p}(\delta _n)\Vert \varvec{T}\Vert$. Therefore, $|I_{n1}|=O_{p}(\sqrt{n}\delta _n)\Vert \varvec{T}\Vert =o_{p}(n\delta _n^2)\Vert \varvec{T}\Vert$. Now consider $I_{n2}$, we can derive that

$$\begin{aligned} I_{n2}&=\delta _n^2\varvec{T}^T\frac{\partial \varvec{S}_n(\tilde{\varvec{\beta }},\tilde{\varvec{\gamma }})}{\partial \left( \varvec{\beta }^T,\varvec{\gamma }^T\right) ^T}\varvec{T}\\&=n\delta _n^2\varvec{T}^T\left( \varvec{I}_s-\hat{\varvec{\Lambda }}\right) \left[ \frac{\sum _{i=1}^n\varvec{D}_{i}^T\varvec{\Sigma }_{i}(\varvec{\mu }_i(\varvec{\beta }_0,\varvec{\gamma }_0))\varvec{D}_{i}}{n}\right] \varvec{T}\\&~~~~+\delta _n^2\varvec{T}^T\frac{\partial }{\partial \left( \varvec{\beta }^T,\varvec{\gamma }^T\right) ^T}\left[ \varvec{U}\left( \tilde{\varvec{\beta }},\tilde{\varvec{\gamma }},\hat{\delta } \left( \tilde{\varvec{\beta }},\tilde{\varvec{\gamma }},\hat{\phi }(\tilde{\varvec{\beta }},\tilde{\varvec{\gamma }}) \right) \right) -\varvec{U}(\tilde{\varvec{\beta }},\tilde{\varvec{\gamma }},\delta )\right] \varvec{T}\\&~~~~+\delta _n^2\varvec{T}^T\hat{\varvec{\Lambda }}\varvec{T}\\&= I_{n21}+I_{n22}+I_{n23}. \end{aligned}$$

With the same argument, it is not difficult to prove that $I_{n22}=O_{p}(\sqrt{n}\delta _n)\Vert \varvec{T}\Vert ^2$ and $I_{n23}=O_{p}(\delta _n^2)\Vert \varvec{T}\Vert ^2$. Thus, for sufficiently large n, $\delta _n\varvec{T}^T\varvec{S}_n(\varvec{\beta }_0+\varvec{T}_1,\varvec{\gamma }_0+\varvec{T}_2)$ is asymptotically dominated in probability by $I_{n21}$ on $\{\varvec{\beta }_0+\varvec{T}_1,\varvec{\gamma }_0+\varvec{T}_2:\Vert \varvec{T}\Vert =C\}$, which is positive for the sufficiently large C. This implies, with probability at least $1-\varepsilon$, that there exists a local minimizer $\left( \hat{\varvec{\beta }}^T,\hat{\varvec{\gamma }}^T\right) ^T$ such that $\Vert \left( \hat{\varvec{\beta }}^T,\hat{\varvec{\gamma }}^T\right) ^T-\left( \varvec{\beta }_0^T,\varvec{\gamma }_0^T\right) ^T\Vert =O_{p}(\delta _n)$. Furthermore, note that

$$\begin{aligned} \left\| \hat{\eta }_k(u)-\eta _{k}(u)\right\| _2^2&=\int _{[0,1]}[\hat{\eta }_k(u)-\eta _{k}(u)]^2du\nonumber \\&=\int _{[0,1]}\left[ \varvec{B}(u)^T\hat{\varvec{\gamma }}_{k}-\varvec{B}(u)^T\varvec{\gamma }_{0k}+R_k(u)\right] ^2du\nonumber \\&\le 2\int _{[0,1]}\left[ \varvec{B}(u)^T\hat{\varvec{\gamma }}_{k}-\varvec{B}(u)^T\varvec{\gamma }_{0k}\right] ^2du+2\int _{[0,1]}[R_k(u)]^2du\nonumber \\&=2(\hat{\varvec{\gamma }}_{k}-\varvec{\gamma }_{0k})^T\varvec{R}(\hat{\varvec{\gamma }}_{k}-\varvec{\gamma }_{0k})+2\int _{[0,1]}[R_k(u)]^2du, \end{aligned}$$

where $R_k(u)=\eta _{k}(u)-\varvec{B}(u)^T\varvec{\gamma }_{0k}, k=1,\ldots ,p$, and $\varvec{R}$ is a matrix with $\varvec{R}_{ij}=\int _0^1B_i(t)B_j(t)dt$. By Lemma 4.1, we know that $R_k(u)=O(K_{n}^{-r})$. Then, invoking $\Vert \varvec{R}\Vert =O(1)$, a simple calculation yields $(\hat{\varvec{\gamma }}_{k}-\varvec{\gamma }_{0k})^T\varvec{R}(\hat{\varvec{\gamma }}_{k}-\varvec{\gamma }_{0k})=O_{p}(n^{-2r/(2r+1)})$ and $\int _{[0,1]}[R_k(u)]^2du=O_{p}(n^{-2r/(2r+1)})$. Thus, $\left\| \hat{\eta }_k(u)-\eta _{k}(u)\right\| _2^2=O_{p}(n^{-2r/(2r+1)})$ and note $\hat{\alpha }_k(t)=\hat{\beta }_k+\varvec{B}(t)^T\hat{\varvec{\gamma }}_{k}$. Then

$$\begin{aligned} \sum _{k\in \mathcal {A}_{v}}\int _{[0,1]}[\hat{\alpha }_k(u)-\alpha _{0k}(u)]^2du&\le \sum _{k\in \mathcal {A}_{v}}\left[ \int _{[0,1]}2[\hat{\eta }_k(u)-\eta _{k}(u)]^2du+2(\hat{\beta }_k-\beta _k)^2\right] \\&=O_{p}(n^{-2r/(2r+1)}). \end{aligned}$$

The proof is completed.$\square$

Lemma 4.2

Under the same conditions used in Theorem 2.1, we have

• (i) $\hat{\delta }_{1,k}=1$ for $k\in \mathcal {A}_z\bigcup \{k:E\alpha _{0k}(t)=0,k\in \mathcal {A}_v\}$ hold with probability tending to one;

• (ii) $\hat{\delta }_{2,k}=1$ for $k\in \mathcal {A}_c\bigcup \mathcal {A}_z$ hold with probability tending to one.

Proof of Lemma 4.2. We first prove the part (i). Using the same method we can prove that $(\hat{\varvec{\beta }}^{(0)},\hat{\varvec{\gamma }}^{(0)})$ is $n^{-r/(2r+1)}$ consistent. Note that $n^{\frac{r(1+\tau )}{1+r}}\lambda \rightarrow \infty$, we can derive that

$$\begin{aligned} \Pr \left( \lambda /|\hat{\beta }_j^{(0)}|^{1+\tau }<1\right)&=\Pr \left( |\hat{\beta }_j^{(0)}|^{1+\tau }>\lambda \right) \\&\le \lambda ^{-1}E\left( |\hat{\beta }_j^{(0)}|^{1+\tau }\right) \\&=\lambda ^{-1}O(n^{\frac{-r(1+\tau )}{1+r}})\rightarrow 0, \end{aligned}$$

where the first inequality applies Markov’s inequality. This implies that (i) holds. With the same argument, we can derive (ii). Then the proof is completed.$\square$

Proof (b) of Theorem 2.2. By Lemma 4.2, we know that $\hat{\beta }_k=0$ for $k\in \mathcal {A}_z$ and $\hat{\varvec{\gamma }}_k=\varvec{0}$ for $k\in \mathcal {A}_c\bigcup \mathcal {A}_z$ hold with probability tending to 1. Let $\varvec{\gamma }_v=((\beta _k,\varvec{\gamma }_k^T),k\in \mathcal {A}_v)^T$, $\varvec{D}_{ij}^{1}=({\varvec{\Pi }_{ij}^v}^T,{\varvec{x}_{ij}^c}^T)^T$, and here, $\mu _{ij}(\varvec{\beta }^c,\varvec{\gamma }_v)= g^{-1}({\varvec{D}_{ij}^{1}}^T({\varvec{\gamma }_v}^T,{\varvec{\beta }^c}^T)^T)$. Thus, with probability tending to 1, $(\hat{\varvec{\beta }}^c,\hat{\varvec{\gamma }}_v)$ satisfies the following robust smooth-threshold estimating equations

$$\begin{aligned} \left( \varvec{I}_{s_0}-\hat{\varvec{\Lambda }}_1\right) \varvec{U}\left( \varvec{\beta }^c,\varvec{\gamma }_v,\hat{\delta }\left( \varvec{\beta }^c,\varvec{\gamma }_v,\hat{\phi }(\varvec{\beta }^c,\varvec{\gamma }_v) \right) \right) + \hat{\varvec{\Lambda }}_1\left( {\varvec{\beta }^c}^T,{\varvec{\gamma }_v}^T\right) ^T=\varvec{0}, \end{aligned}$$

where $\hat{\varvec{\Lambda }}_1$ is the sub-matrix of $\hat{\varvec{\Lambda }}$ corresponding to $({\varvec{\gamma }_v}^T,{\varvec{\beta }^c}^T)^T$ and $\varvec{I}_{s_0}$ is identity matrix having the same dimension with $\hat{\varvec{\Lambda }}_1$. Thus, we have that

$$\begin{aligned} \frac{1}{\sqrt{n}}\varvec{U}\left( \varvec{\beta }^c,\varvec{\gamma }_v,\hat{\delta }\left( \varvec{\beta }^c,\varvec{\gamma }_v,\hat{\phi }(\varvec{\beta }^c,\varvec{\gamma }_v) \right) \right) + \frac{1}{\sqrt{n}}\hat{\varvec{S}}_1\left( {\varvec{\beta }^c}^T,{\varvec{\gamma }_v}^T\right) ^T=\varvec{0}, \end{aligned}$$

where $\hat{\varvec{S}}_1=\left( \varvec{I}_{s_0}-\hat{\varvec{\Lambda }}_1\right) ^{-1}\hat{\varvec{\Lambda }}_1$. On the other hand,

$$\begin{aligned} \left\| \frac{1}{\sqrt{n}}\hat{\varvec{S}}_1\left( {\varvec{\beta }^c}^T,{\varvec{\gamma }_v}^T\right) ^T\right\| ^2&\le \frac{\lambda ^2}{n(1-\max \{\max _{j\in \mathcal {A}_c}\hat{\delta }_{1,j},\max _{j\in \mathcal {A}_v}\hat{\delta }_{2,j}\})^2}\Biggr [\sum _{j\in \mathcal {A}_c}\Big |\hat{\beta }_j^{(0)(-\tau )}\\&\quad +\,(\beta _j-\hat{\beta }_j^{(0)})\hat{\beta }_j^{(0)(-\tau -1)}\Big |^2+\sum _{k\in \mathcal {A}_v}\sum _{j=1}^{K_n}\Big |\hat{\gamma }_{kj}^{(0)(-\tau )}\\&\quad +\,(\gamma _{kj}-\hat{\gamma }_{kj}^{(0)})\hat{\gamma }_{kj}^{(0)(-\tau -1)}\Big |^2\Biggr ]\\&=O\Big (\frac{\lambda ^2}{n}\Big )\Biggr [\sum _{j\in \mathcal {A}_c}\Big |\hat{\beta }_j^{(0)(-\tau )}+(\beta _j-\hat{\beta }_j^{(0)})\hat{\beta }_j^{(0)(-\tau -1)}\Big |^2\\&\quad +\,\sum _{k\in \mathcal {A}_v}\sum _{j=1}^{K_n}\Big |\hat{\gamma }_{kj}^{(0)(-\tau )}+(\gamma _{kj}-\hat{\gamma }_{kj}^{(0)})\hat{\gamma }_{kj}^{(0)(-\tau -1)}\Big |^2\Biggr ]\\&=O_p(n^{\frac{2r}{2r+1}}\lambda ^2n^{\frac{-4r}{2r+1}}\iota ^{-2\tau })(1+O_p(n^{\frac{-4r+1}{2r+1}}\iota ^{-2}))\\&=O_p(n^{\frac{-4r}{2r+1}}), \end{aligned}$$

where $\iota =\min \{\min _{j\in \mathcal {A}_c}|\hat{\beta }_j^{(0)}|,\min _{k\in \mathcal {A}_v,j=1,\ldots ,K_n}|\hat{\gamma }_{kj}^{(0)}|\}$. Then by using Taylor approximation and the same arguments used in the proof of Theorem 3 in Tian et al. (2015), we can get that

$$\begin{aligned}&~~{\varvec{x}^c}^T\varvec{H}\varvec{\mu }^0-{\varvec{x}^c}^T\varvec{\Sigma }_0\varvec{\Pi }_v(\varvec{\Pi }_v^T\varvec{\Sigma }_0\varvec{\Pi }_v)^{-1}\varvec{\Pi }_v^T\varvec{H}\varvec{\mu }^0\\&~~~~~~~+({\varvec{x}^c}^T\varvec{\Sigma }_0\varvec{x}^c-{\varvec{x}^c}^T\varvec{\Sigma }_0\varvec{\Pi }_v(\varvec{\Pi }_v^T\varvec{\Sigma }_0\varvec{\Pi }_v)^{-1}\varvec{\Pi }_v^T\varvec{\Sigma }_0 \varvec{x}^c)(\hat{\varvec{\beta }}^c-\varvec{\beta }_0^c)=o_p(\sqrt{n}), \end{aligned}$$

where $\varvec{H}(\varvec{\mu }^0)=(\varvec{h}_{1,0}^h(\varvec{e}_1)^T,\ldots ,\varvec{h}_{n,0}^h(\varvec{e}_n)^T)^T$. Then, by the definition of $\varvec{P}_v$ in Sect. 2.2, we have that $(\varvec{I}-\varvec{P}_v)^2=(\varvec{I}-\varvec{P}_v)$ and $\varvec{P}_v^T\varvec{\Sigma }_0=\varvec{\Sigma }_0\varvec{P}_v=(\varvec{\Sigma }_0\varvec{P}_v)^T$. Thus, we can obtain that

$$\begin{aligned} {\varvec{x}^c}^T\varvec{\Sigma }_0(\varvec{I}-\varvec{P}_v)(\varvec{I}-\varvec{P}_v)\varvec{x}^c(\hat{\varvec{\beta }}^c-\varvec{\beta }_0^c) =-{\varvec{x}^c}^T(\varvec{I}-\varvec{P}_v^T)\varvec{H}\varvec{\mu }^0+o_p(\sqrt{n}). \end{aligned}$$

That is

$$\begin{aligned} {\varvec{x}_{*}^{c}}^T\varvec{\Sigma }_0\varvec{x}_{*}^{c}(\hat{\varvec{\beta }}^c-\varvec{\beta }_0^c)=-\sum _{i=1}^n{\varvec{x}_{*i}^{c}}^T\varvec{h}_{i,0}^h(\varvec{e}_i)+o_p(\sqrt{n}), \end{aligned}$$

moreover, note that $\frac{1}{n}{\varvec{x}_{*}^{c}}^T\varvec{\Sigma }_0\varvec{x}_{*}^{c}\rightarrow _p\varvec{K}_{c}$, by the central limit theorem, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{i=1}^n{\varvec{x}_{*i}^{c}}^T\varvec{h}_{i,0}^h(\varvec{e}_i)\rightarrow _d N(\varvec{0},\varvec{S}_{c}). \end{aligned}$$

Then the proof is completed.$\square$

Proof of Theorem 2.1. The result in Theorem 2.1 can be obtained directly by combining Lemma 4.2, (a) and (b) in Theorem 2.2. The proof is completed.$\square$