Model detection and variable selection for mode varying coefficient model

Abstract

Varying coefficient model is often used in statistical modeling since it is more flexible than the parametric model. However, model detection and variable selection of varying coefficient model are poorly understood in mode regression. Existing methods in the literature for these problems are often based on mean regression and quantile regression. In this paper, we propose a novel method to solve these problems for mode varying coefficient model based on the B-spline approximation and SCAD penalty. Moreover, we present a new algorithm to estimate the parameters of interest, and discuss the parameters selection for the tuning parameters and bandwidth. We also establish the asymptotic properties of estimated coefficients under some regular conditions. Finally, we illustrate the proposed method by some simulation studies and an empirical example.

Appendix

Lemma A.1. Suppose conditions (C1)-(C4) hold, $\varvec{\gamma }^{best}$ is the best approximation of $\varvec{\gamma }$, and $\epsilon _1$, $a_1$, $a_2$ are positive constants, such that

• (1) $\vert \vert \varvec{\gamma }_{j*}^{best} \vert \vert _{L_2} >\epsilon _1,j=1,\ldots ,v$, $\varvec{\gamma }_j^{best}=(\gamma _{j1},\varvec{0}_{q-1}^{\top })^{\top },j=v+1,\ldots ,s$, $\varvec{\gamma }_j^{best}=\varvec{0},j=s+1,\ldots ,p$.

• (2) $\sup \vert \alpha _j(U) - \varvec{B}(U)^{\top } \varvec{\gamma }_j^{best} \vert \le a_1 k_n^{-t}, j=1,\ldots ,v$

• (3) $\sup \vert \varvec{\Pi }^{\top } \varvec{\gamma }^{best} -\varvec{X}\varvec{\alpha }(U) \vert \le a_2k_n^{-t}$

Lemma A.2. Let $\delta =O\,(n^{-t/(2t+1)})$. Define $\varvec{\gamma }=\varvec{\gamma }^{best}+\delta \varvec{v}$, given $\rho >0$, there exists a large C such that ( Zhao et al. (2014))

$$\begin{aligned} P\{\sup _{\vert \vert \varvec{v} \vert \vert =C} l(\varvec{\gamma }) < l(\varvec{\gamma }^{best})\} \le 1-\rho \end{aligned}$$

Proof 1. Proof of Theorem 1 Similar to the proof of Theorem 1 in Zhao et al. (2014), let

$$\begin{aligned} a_n= & {} \max _{j} \{\vert p^{'}_{\lambda _{1j}}(\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2}) \vert , \vert p^{'}_{\lambda _{2j}}(\vert \gamma _{j1} \vert ) \vert :\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2} \ne 0, \vert \gamma _{j1} \vert \ne 0\},\\ b_n= & {} \max _{j} \{\vert p^{''}_{\lambda _{1j}}(\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2})\vert , \vert p^{''}_{\lambda _{2j}}(\vert \gamma _{j1} \vert ) \vert :\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2} \ne 0, \vert \gamma _{j1} \vert \ne 0 \}. \end{aligned}$$

As $b_n \rightarrow 0$, we have $\vert \vert \hat{\varvec{\gamma }} -\varvec{\gamma }^{best} \vert \vert = O_p(n^{-t/(2t+1)} + a_n)$ by the Lemma A.2.. Therefore

$$\begin{aligned} \begin{aligned}&\vert \vert {\hat{\alpha }}_j(U) -\alpha _j(U) \vert \vert _{L_2}^2 \\&\quad = \sum _{i=1}^{n}({\hat{\alpha }}_j(U_i) - \alpha _j(U_i))^2\\&\quad =\int _{0}^{1} ({\hat{\alpha }}_j(U) -\alpha _j(U))^2 dU \\&\quad =\int _{0}^{1} (\varvec{B}(U)^{\top } \hat{\varvec{\gamma }}_j - \varvec{B}(U)^{\top } \varvec{\gamma }^{best}_j + \varvec{B}(U)^{\top } \varvec{\gamma }^{best}_j - \alpha _j(U))^2 dU \\&\quad \le 2 \int _{0}^{1} ((\varvec{B}(U)^{\top } \hat{\varvec{\gamma }}_j - \varvec{B}(U)^{\top } \varvec{\gamma }^{best}_j)^2 dU + 2 \int _{0}^{1} (\varvec{B}(U)^{\top } \varvec{\gamma }^{best}_j - \alpha _j(U))^2 dU\\&\quad = 2(\hat{\varvec{\gamma }}_j - \varvec{\gamma }^{best}_j)^{\top } \int _{0}^{1}{\varvec{B}(U) \varvec{B}(U)^{\top }}dU (\hat{\varvec{\gamma }}_j - \varvec{\gamma }^{best}_j) + 2 \int _{0}^{1} (\varvec{B}(U)^{\top } \varvec{\gamma }^{best}_j - \alpha _j(U))^2 dU \end{aligned} \end{aligned}$$

Because $\vert \vert \int _{0}^{1}{\varvec{B}(U) \varvec{B}(U)^{\top }} dU \vert \vert =O(1)$, we have

$$\begin{aligned} (\hat{\varvec{\gamma }}_j - \varvec{\gamma }^{best}_j)^{\top } \int _{0}^{1}{\varvec{B}(U) \varvec{B}(U)^{\top }} dU (\hat{\varvec{\gamma }}_j - \varvec{\gamma }^{best}_j) =O_p(n^{-2t/(2t+1)} + a_n^2) \end{aligned}$$

And according to the Lemma A.1., we have

$$\begin{aligned} \int _{0}^{1} (\varvec{B}(U)^{\top } \varvec{\gamma }^{best}_j - \alpha _j(U))^2 dU =O_p(n^{-2t/(2t+1)}) \end{aligned}$$

Consequently, the proof has been completed. $\square$

Proof 2. Proof of Theorem 2 By the property of SCAD, we know that $\max \{\lambda _{1j},\lambda _{2j}\} \rightarrow 0$ as $n \rightarrow \infty$, then $a_n=0$, then by Theorem 1, we have $\vert \vert \varvec{\gamma } -\varvec{\gamma }^{best} \vert \vert =O_p(n^{-t/2t+1})$

Firstly, if $\varvec{\gamma }_{j*}=0$, it is clear that $\alpha _j(U)$ is a constant. If $\varvec{\gamma }_{j*} \ne 0$, we have

$$\begin{aligned} \begin{aligned} \frac{\partial l_1(\varvec{\gamma })}{\partial {\varvec{\gamma }_{j*}}} =&\sum _{i=1}^{n}K_{h}^{'}(Y_i-\varvec{\Pi }_i^{\top } \varvec{\gamma }) X_i^{(j)}\bar{\varvec{B}}(U_i) + np^{'}_{\lambda _{1j}} (\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2})\\&(sign\,(\gamma _{j,2})|\gamma _{j,2}|,\ldots ,sign(\gamma _{j,q})|\gamma _{j,q}|)^{\top }\\ =&\sum _{i=1}^{n}\Big \{K_{h}^{'}(\epsilon _i + D_{ni})X_i^{(j)}\bar{\varvec{B}}(U_i) + K_{h}^{''}(\epsilon _i+ D_{ni})X_i^{(j)}\bar{\varvec{B}}(U_i)[\varvec{\Pi }_i^{\top }( \varvec{\gamma } - \varvec{\gamma }^{best} )] \\&+ K_{h}^{'''}(\eta _i)X_i^{(j)}\bar{\varvec{B}}(U_i) [\varvec{\Pi }_i^{\top }(\varvec{\gamma } -\varvec{\gamma }^{best})]^2 \Big \} + np^{'}_{\lambda _{1j}}(\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2})\\&(sign(\gamma _{j,2})|\gamma _{j,2}|,\ldots ,sign(\gamma _{j,q})|\gamma _{j,q}|)^{\top }\\ =&n\lambda _{1j}\Big [O_p(\lambda _{1j}^{-1} \bar{\varvec{B}}(U_i) n^{-\frac{t}{2t+1}}) +\lambda _{1j}^{-1}np^{'}_{\lambda _{1j}}(\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2})\\&(sign(\gamma _{j,2})|\gamma _{j,2}|,\ldots , sign(\gamma _{j,q})|\gamma _{j,q}|)^{\top }\Big ] \end{aligned} \end{aligned}$$

where $\eta _i$ is between $Y_i-\varvec{\Pi }_i^{\top } \varvec{\gamma }$ and $\epsilon _i + D_{ni}$, $\epsilon _i=Y_i-\varvec{X}_i^{\top } \varvec{\alpha }(U_i)$, $D_{ni}= \varvec{X}_i^{\top } \varvec{\alpha }(U_i)- \varvec{\Pi }_i^{\top }\varvec{\gamma }^{best}$.

As we all know, $\sup _{u} \vert \vert \bar{\varvec{B}}(U) \vert \vert =O(1)$, and by condition (C4) $n^{t/(2t+1)}\min \{\lambda _{1j},\lambda _{2j}\} \rightarrow \infty$ and ${\lim \inf }_{n \rightarrow \infty } {\lim \inf }_{\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2} \rightarrow 0^{+}} \frac{p_{\lambda _{1j}}^{'}(\vert \vert \varvec{\gamma }_{j*} \vert \vert _{L_2})}{\lambda _{1j}} >0 ,j=s+1, \ldots ,p$, we prove that the sign of the derivation is completely determined by the second part of the derivation. Hence, with the Lemma A.2. $l_1(\varvec{\gamma })$ gets its minimizer at $\hat{\varvec{\gamma }}_{j*}^{VC} =0$, and ${\hat{\alpha }}_j(U)\approx {\hat{\gamma }}_{j1}^{VC} + \bar{\varvec{B}}^{\top }(U) \hat{\varvec{\gamma }}_{j*}^{VC} = {\hat{\gamma }}_{j1}^{VC}$, i.e. ${\hat{\alpha }}_j(U), j=v+1,\ldots ,p$ are constants.

Secondly, since we have proved ${\hat{\alpha }}_j,j=v+1,\ldots ,p$ are constant, we only need to prove ${\hat{\gamma }}_{j1}^{CZ}=0$ to obtain ${\hat{\alpha }}_j=0,$ for $j=s+1,\ldots ,p$.

$$\begin{aligned} \begin{aligned} \frac{\partial l_2(\varvec{\gamma })}{\partial {\gamma _{j1}}}&=\sum _{i=1}^{n}K_h^{'}(Y_i-\varvec{\Pi }_i^{\top } \varvec{\gamma })X_i^{(j)}B_1(U_i) + np^{'}_{\lambda _{2j}}(\vert \gamma _{j1} \vert )sgn(\gamma _{j1})\\ =&\sum _{i=1}^{n}\Big \{K_h^{'}(\epsilon _i + D_{ni})X_i^{(j)}B_1(U_i) + K_h^{''}(\epsilon _i+ D_{ni})X_i^{(j)}B_1(U_i)[\varvec{\Pi }_i^{\top }(\varvec{\gamma } -\varvec{\gamma }^{best})] \\&+ K_h^{'''}(\eta _i)X_i^{(j)}B_1(U_i)[\varvec{\Pi }_i^{\top }(\varvec{\gamma } -\varvec{\gamma }^{best})]^2 \Big \} + np^{'}_{\lambda _{2j}}(\vert \gamma _{j1}\vert )sgn(\gamma _{j1})\\ =&n\lambda _{2j}[O_p(\lambda _{2j}^{-1} n^{-\frac{t}{2t+1}}) +\lambda _{2j}^{-1}p^{'}_{\lambda _{2j}}(\vert \gamma _{j1}\vert )sgn(\gamma _{j1}) ] \end{aligned} \end{aligned}$$

where $\eta _i$ is between $Y_i-\varvec{\Pi }_i^{\top } \varvec{\gamma }$ and $\epsilon _i + D_{ni}$, $\epsilon _i=Y_i-\varvec{X}_i^{\top } \varvec{\alpha }(U_i)$, $D_{ni}= \varvec{X}_i^{\top } \varvec{\alpha }(U_i)- \varvec{\Pi }_i^{\top }\varvec{\gamma }^{best}$.

By condition (C4) $n^{t/(2t+1)}\min \{\lambda _{1j},\lambda _{2j}\} \rightarrow \infty$ and ${\lim \inf }_{n \rightarrow \infty } {\lim \inf }_{\gamma _{j1} \rightarrow 0^{+}} \frac{p_{\lambda _{2j}}^{'}(|\alpha _{j1}|)}{\lambda _{2j}} >0 ,j=s+1, \ldots ,p$, we prove

$$\begin{aligned} \frac{\partial l_2(\varvec{\gamma })}{\partial {\gamma _{j1}}}< & {} 0\quad when \quad -\delta<{\hat{\gamma }}_{j1}^{CZ}<0\\ \frac{\partial l_2(\varvec{\gamma })}{\partial {\gamma _{j1}}}> & {} 0\quad when \quad 0< {\hat{\gamma }}_{j1}^{CZ} < \delta \end{aligned}$$

where $\delta =O(n^{-t/(2t+1)})$. Hence, $l_2(\varvec{\gamma })$ gets its minimizer at ${\hat{\gamma }}_{j1}^{CZ} =0$, i.e. ${\hat{\alpha }}_j=0,j=s+1,\ldots ,p$. $\square$

Proof 3. Proof of Theorem 3 Let $\hat{\varvec{\gamma }}=(\hat{\varvec{\gamma }}_{v}^{\top }, \hat{\varvec{\gamma }}_{s-v}^{\top },0)^{\top }$, where $\hat{\varvec{\gamma }}_{v}^{\top }=(\hat{\varvec{\gamma }}_0, \ldots , \hat{\varvec{\gamma }}_{v})^{\top }, \hat{\varvec{\gamma }}_{s-v}^{\top }=({\hat{\gamma }}_{v+1,1},\ldots , {\hat{\gamma }}_{s,1})$, and for simplicity of notations, let $\hat{\varvec{\gamma }}_{s-v}^{\top }=({\hat{\gamma }}_{v+1},\ldots , {\hat{\gamma }}_{s})$. From Theorems 1 and 2, we know that as $n \rightarrow \infty$, with probability tending to 1, $l(\varvec{\gamma })$ attain the minimal value at $\hat{\varvec{\gamma }}$, let $L_1=\frac{\partial l(\varvec{\gamma })}{\partial \varvec{\gamma }_{s-v}}$, $L_2=\frac{\partial l(\varvec{\gamma })}{\partial \varvec{\gamma }_{v}}$, we know that

$$\begin{aligned} \begin{aligned} L_1(\hat{\varvec{\gamma }})&= \sum _{i=1}^{n}\varvec{X}_{ci}B_1(U_i) K_h^{'}(Y_i-\varvec{\Pi }_i^{\top } \hat{\varvec{\gamma }}) + np^{'}_{\lambda _{2}}(\vert \hat{\varvec{\gamma }}_{s-v} \vert ) {\circ } sgn(\hat{\varvec{\gamma }}_{s-v})\\&=\sum _{i=1}^{n}\varvec{X}_{ci} K_h^{'}(Y_i-\varvec{\Pi }_{vi}^{\top } \hat{\varvec{\gamma }}_{v}- \varvec{X}_{ci}^{\top } \hat{\varvec{\gamma }}_{s-v} ) + np^{'}_{\lambda _{2}}(\vert \hat{\varvec{\gamma }}_{s-v} \vert ) {\circ } sgn(\hat{\varvec{\gamma }}_{s-v}) \end{aligned} \end{aligned}$$

(8)

and

$$\begin{aligned} \begin{aligned} L_2(\hat{\varvec{\gamma }})=\sum _{i=1}^{n}\varvec{\Pi }_{vi} K_h^{'}(Y_i-\varvec{\Pi }_{vi}^{\top } \hat{\varvec{\gamma }}_{v}- \varvec{X}_{ci}^{\top } \hat{\varvec{\gamma }}_{s-v} ) + \varvec{\kappa } \end{aligned} \end{aligned}$$

(9)

where “${\circ }$” denotes the Hadamard (componentwise) product and the k-th component of $p^{'}_{\lambda _{2}}(\vert \hat{\varvec{\gamma }}_{s-v} \vert )$ is $p^{'}_{\lambda _{2k}}(\vert {\hat{\gamma }}_{k} \vert )$, $v+1 \le k \le s$. The j-th block subvector of $\varvec{\kappa }$ is $p^{'}_{\lambda _{1j}}(\vert \vert \varvec{\gamma }_{j*}\vert \vert _{L_2})$

$\Big (\frac{p^{'}_{\lambda _{2j}}(\vert {\hat{\gamma }}_{j} \vert )}{p^{'}_{\lambda _{1j}}(\vert \vert \varvec{\gamma }_{j*}\vert \vert _{L_2})}sign(\gamma _{j,1}), sign(\gamma _{j,2})\vert \gamma _{j,2}\vert ,\ldots ,sign(\gamma _{j,q})\vert \gamma _{j,q} \vert \Big )^{\top }$, $0 \le\, j \,\le v$. Applying the Taylor expansion to $p^{'}_{\lambda _{2k}}(\vert {\hat{\gamma }}_{k} \vert )$,

$$\begin{aligned} p^{'}_{\lambda _{2k}}(\vert {\hat{\gamma }}_{k} \vert )=p^{'}_{\lambda _{2k}}(\vert \gamma _{k}^{best} \vert ) +[p^{''}_{\lambda _{2k}}(\vert \gamma _{k}^{best} \vert )+o_p(1)]({\hat{\gamma }}_{k}-\gamma _{k}^{best}), v+1 \le k \le s \end{aligned}$$

Note that $p^{'}_{\lambda _{2k}}(\vert {\hat{\gamma }}_{k}^{best} \vert )=0$ as $\lambda _{max} \rightarrow 0$ and $b_n \rightarrow 0$, as for (8) we have

$$\begin{aligned} \begin{aligned} \frac{1}{n}&\sum _{i=1}^{n}\varvec{X}_{ci}\Big \{ K_h^{'}(\epsilon _i)+K_h^{''}(\epsilon _i)\{\varvec{X}_i^{\top }\varvec{R}(U_i)- [\varvec{X}_{ci}^{\top }(\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}^{best})+\varvec{\Pi }_{vi}(\hat{\varvec{\gamma }}_{v}- \varvec{\gamma }_{v}^{best})]\}\\&+K_h^{'''}(\xi _i)\{\varvec{X}_i^{\top }\varvec{R}(U_i)- [\varvec{X}_{ci}^{\top }(\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}^{best})+\varvec{\Pi }_{vi}(\hat{\varvec{\gamma }}_{v}- \varvec{\gamma }_{v}^{best})]\}^2 \Big \}+o_p(\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}^{best}) \end{aligned} \end{aligned}$$

(10)

where $\varvec{R}(U_i)=(R_0(U_i),\ldots , R_v(U_i))^{\top }$, $R_j(U_i)=\alpha _j(U_i) - \varvec{B}(U_i)^{\top } \varvec{\gamma }_j^{best}, 0\le j \le v$, and $\xi _i$ between $\epsilon _i$ and $Y_i-\varvec{\Pi }_i^{\top } \hat{\varvec{\gamma }}$.

From Theorem 3 in Zhao et al. (2014), we know that

$$\begin{aligned} \begin{aligned} \hat{\varvec{\gamma }}_{v}- \varvec{\gamma }_{v}^{best}=-(\Phi +O_p(1))^{-1}\Psi (\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}^{best})+ (\Phi +o_p(1))^{-1}\Gamma _n \end{aligned} \end{aligned}$$

(11)

where $\Psi =E(K^{''}_h(\epsilon )\varvec{\Pi }_{v}\varvec{X}_{c}^{\top })$, $\Phi =E(K^{''}_h(\epsilon )\varvec{\Pi }_{v}\varvec{\Pi }_{v}^{\top })$ and $\Gamma _n=\frac{1}{n}\sum _{i=1}^{n}\varvec{\Pi }_{vi}\{K_h^{'}(\epsilon _i) + K_h^{''}(\epsilon _i)\varvec{X}_{vi}^{\top }\varvec{R}(U_i)\}$.

Substituing (11) into (10), we obtain

$$\begin{aligned} \begin{aligned}&\frac{1}{n}\sum _{i=1}^{n}K_h{''}(\epsilon _i)\varvec{X}_{ci}\{ \varvec{X}_{ci}- \Psi ^{\top }\Phi ^{-1}\varvec{\Pi }_{vi}\}^{\top }(\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}^{best})+o_p(\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}^{best})\\&\quad =\frac{1}{n}\sum _{i=1}^{n}\varvec{X}_{ci}\Big \{K_h{'}(\epsilon _i)+K_h{''}(\epsilon _i)\varvec{X}_{vi}^{\top }\varvec{R}(U_i)- K_h{''}(\epsilon _i)\varvec{\Pi }_{vi}^{\top }\frac{1}{n}\sum _{j=1}^{n}\Phi ^{-1}\varvec{\Pi }_{vj}K_h{'}(\epsilon _j)\Big \}\\&\qquad -\frac{1}{n}\sum _{i=1}^{n}\varvec{X}_{ci}K_h{''}(\epsilon _i)\varvec{\Pi }_{vi}^{\top }\frac{1}{n}\sum _{j=1}^{n}\varvec{X}_{vj}^{\top }\varvec{R}(U_j) \end{aligned} \end{aligned}$$

(12)

Note that

$$\begin{aligned}&E\Big \{\frac{1}{n}\sum _{i=1}^{n}K_h{''}(\epsilon _i)\Psi ^{\top }\Phi ^{-1}\varvec{\Pi }_{vi}[\varvec{X}_{ci}^{\top } -\varvec{\Pi }_{vi}^{\top }\Phi ^{-1}\Psi ] \Big \}=0\\&Var \Big \{\frac{1}{n}\sum _{i=1}^{n}K_h{''}(\epsilon _i)\Psi ^{\top }\Phi ^{-1}\varvec{\Pi }_{vi}[\varvec{X}_{ci}^{\top } -\varvec{\Pi }_{vi}^{\top }\Phi ^{-1}\Psi ] \Big \}=o_p(1/n) \end{aligned}$$

Hence

$$\begin{aligned} \begin{aligned}&\Big \{\frac{1}{n}\sum _{i=1}^{n}K_h{''}(\epsilon _i)\tilde{\varvec{X}}_{ci}\tilde{\varvec{X}}_{ci}^{\top } +o_p(1) \Big \}\sqrt{n}(\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}^{best})\\&\quad =\Big \{\frac{1}{n}\sum _{i=1}^{n}K_h{''}(\epsilon _i)\tilde{\varvec{X}}_{ci}\tilde{\varvec{X}}_{ci}^{\top } +o_p(1) \Big \}\sqrt{n}(\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v})\\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\tilde{\varvec{X}}_{ci}K_h{'}(\epsilon _i)+\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\tilde{\varvec{X}}_{ci}K_h{''}(\epsilon _i) \varvec{X}_{vi}^{\top }\,\varvec{R}(U_i) =J_1+J_2 \end{aligned} \end{aligned}$$

(13)

From Theorem 3 in Zhao et al. (2014), we know that $J_2=o_p(1)$, and

$$\begin{aligned}&\frac{1}{n}K_h{''}(\epsilon _i)\tilde{\varvec{X}}_{ci}\tilde{\varvec{X}}_{ci}^{\top } \overset{P}{\rightarrow } \Sigma \nonumber \\&J_1 \overset{d}{\rightarrow } N(0,\Delta ) \end{aligned}$$

(14)

where $\Delta =E(G(\varvec{x},u,h)\tilde{\varvec{X}}_{c} \tilde{\varvec{X}}_{c}^{\top })$. Then combining (13) and (14) and using the Slutsky’s theorem, it follows that

$$\begin{aligned} \begin{aligned} \sqrt{n}(\hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}) \overset{d}{\rightarrow } N(0,\Sigma ^{-1}\Delta \Sigma ^{-1}) \end{aligned} \end{aligned}$$

(15)

It’s clearly that $\hat{\varvec{\alpha }}_c - \varvec{\alpha }_c= \hat{\varvec{\gamma }}_{s-v}- \varvec{\gamma }_{s-v}$, so we complete the proof of Theorem 3. And the proof of (14) can be found in Zhao et al. (2014).