Single-index partially functional linear regression model

Abstract

In this paper, we propose a flexible single-index partially functional linear regression model, which combines single-index model with functional linear regression model. All the unknown functions are estimated by B-spline approximation. Under some mild conditions, the convergence rates and asymptotic normality of the estimators are obtained. Finally, simulation studies and a real data analysis are conducted to investigate the performance of the proposed methodologies.

Appendix

The following lemma, which follows easily from Corollary 6.21 of Schumaker (1981), is stated for easy reference.

Lemma 1

If $g_0(u)$ and $\beta _0(t)$ satisfy condition $\text {C}3$, then there exists $\varvec{\eta }_0$ and $\varvec{\gamma }_0$ such that

$$\begin{aligned} \sup _{u\in [a,b]}\mid g_0(u)-\varvec{B}^\tau _1(u)\varvec{\eta }_0 \mid \le c_1k_1^{-r},\quad \sup _{t\in [0,1]}\mid \beta _0(t)-\varvec{B}^\tau _2(t)\varvec{\gamma }_0 \mid \le c_2k_2^{-r}. \end{aligned}$$

where ${{\varvec{\eta }_0}=({\eta }_{01},\ldots ,{\eta }_{0N_1})}$, ${{\varvec{\gamma }_0}=({\gamma }_{01},\ldots ,{\gamma }_{0N_2})}$, and $c_1>0$ and $c_2>0$ depend only on $l_1$ and $l_2$, respectively.

Proof of Theorem 1

Let $\delta =n^{-{\frac{r}{2r+1}}}$, $\varvec{T}_1=\delta ^{-1} \big (\varvec{\phi }-\varvec{\phi }_0\big )$, $\varvec{T}_2=\delta ^{-1}\big (\varvec{\eta }-\varvec{\eta }_0\big )$, $\varvec{T}_3=\delta ^{-1}\big (\varvec{\gamma }-\varvec{\gamma }_0\big )$ and $\varvec{T}=(\varvec{T}_1^\tau ,\varvec{T}_2^\tau ,\varvec{T}_3^\tau )^\tau$. We next show that, for any given $\epsilon >0$, there exists a sufficient large constant $L=L_\epsilon$ such that

$$\begin{aligned} P\bigg \{\inf _{\Vert \varvec{T}\Vert =L}Q(\varvec{\theta }_0+\delta \varvec{T})>Q(\varvec{\theta }_0)\bigg \}\ge 1-\epsilon . \end{aligned}$$

(A1)

This implies with the probability at least $1-\epsilon$ that there exists a local minimizer in the ball $\{\varvec{\theta }_0+\delta \varvec{T}:\Vert \varvec{T}\Vert \le L\}$.

Using Taylor expansion and a simple calculation, we can obtain

$$\begin{aligned} \begin{aligned} \big \{Q(\varvec{\theta }_0+\delta \varvec{T})-Q(\varvec{\theta }_0)\big \}&\ge -2\delta \sum _{i=1}^{n}\big (e_i+R_{1i}+R_{2i}\big )U_i+\delta ^2\sum _{i=1}^{n}U_i^2+o_p(1)\\&\equiv \text {A}_1+\text {A}_2+o_p(1), \end{aligned} \end{aligned}$$

where $R_{1i}=g_0(\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0))-\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0)\big )^\tau \varvec{\eta }_0$, $R_{2i}=\big \langle X_i(t), \beta _0(t)-\varvec{B}_2(t)^\tau \varvec{\gamma }_0\big \rangle$, and $U_i=\varvec{W}_i^\tau \varvec{T}_3+\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0)\big )^\tau \varvec{T}_2+\varvec{B}'_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0)\big )^\tau \varvec{\eta }_0\varvec{T}_1^\tau \varvec{J}^\tau _{\phi _0}\varvec{Z}_i+\delta \varvec{B}'_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0)\big )^\tau \varvec{T}_3\varvec{T}_1^\tau \varvec{J}^\tau _{\phi _0}\varvec{Z}_i$. By condition C1, Lemmas 1 and 8 in Stone (1985), we can obtain that $|R_{1i}|\le ck_1^{-r}$, $|R_{2i}|=O_p(k_2^{-r})$, and

$$\begin{aligned} |g'_0(\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0))-\varvec{B}'_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0)\big )^\tau \varvec{\eta }_0|\le ck_1^{-r+1}. \end{aligned}$$

(A2)

Then, invoking conditions C1, C4 and C5 and Eq. (A2), a simple calculation yields

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{n}R_{1i}U_i&=\sum _{i=1}^{n}R_{1i}\Big \{\varvec{W}_i^\tau \varvec{T}_3 +\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0)\big )^\tau \varvec{T}_2+g'_0(\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0))\varvec{T}_1^\tau \varvec{J}^\tau _{\phi _0}\varvec{Z}_i\\&\quad +\big [\varvec{B}'_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0)\big )^\tau \varvec{\eta }_0-g'_0(\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0))\big ]\varvec{T}_1^\tau \varvec{J}^\tau _{\phi _0}\varvec{Z}_i\\&\quad +\delta \varvec{B}'_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\phi _0)\big )^\tau \varvec{T}_3\varvec{T}_1^\tau \varvec{J}^\tau _{\phi _0}\varvec{Z}_i\Big \}\\&\quad =O_p(nk^{-r})\Vert \varvec{T}\Vert . \end{aligned} \end{aligned}$$

Similarly, we can obtain $\sum _{i=1}^{n}e_iU_i=O_p(\sqrt{n})\Vert \varvec{T}\Vert$, $\sum _{i=1}^{n}R_{2i}U_i=O_p(nk^{-r})\Vert \varvec{T}\Vert$, $\sum _{i=1}^{n}U_i^2=O_p(n)\Vert \varvec{T}\Vert ^2$. Thus, it is easy to show that $\text {A}_1=O_p\big (n\delta ^2\big )\Vert \varvec{T}\Vert$, $\text {A}_2=O_p\big (n\delta ^2\big )\Vert \varvec{T}\Vert ^2$, and the order of $\text {A}_2$ is non-degenerate. By choosing a sufficiently large L, $\text {A}_2$ dominates $\text {A}_1$ uniformly in $\Vert \varvec{T}\Vert =L$. Hence, Eq. (A1) holds, and there exists local minimizers $\hat{\varvec{\phi }}$, $\hat{\varvec{\eta }}$ and $\hat{\varvec{\gamma }}$ such that $\Vert \hat{\varvec{\phi }}-\varvec{\phi }_0\Vert =O_p(\delta )$, $\Vert \hat{\varvec{\eta }}-\varvec{\eta }_0\Vert =O_p(\delta )$ and $\Vert \hat{\varvec{\gamma }}-\varvec{\gamma }_0\Vert =O_p(\delta )$. By a simple calculation, we can get $\Vert \hat{\varvec{\alpha }}-\varvec{\alpha }_0\Vert =O_p(\delta )$. Thus, we complete the proof of first part of (7).

Next we consider $\Vert \hat{{\beta }}(\cdot )-\beta _0(\cdot )\Vert$. Let $R_{2k_2}(t)=\beta _0(t)-\varvec{B}^\tau _2(t)\varvec{\gamma }_0$, we have

$$\begin{aligned} \begin{aligned} \Vert \hat{\beta }(t)-\beta _{0}(t)\Vert ^2&=\int _{0}^{1}\big (\varvec{B}^\tau _2(t)\hat{\varvec{\gamma }}-\beta _{0}(t)\big )^2dt\\&=\int _{0}^{1}\big (\varvec{B}^\tau _2(t)\hat{\gamma }-\varvec{B}^\tau _2(t)\varvec{\gamma }_0+R_{2k_2}(t)\big )^2dt\\&\le 2 \int _{0}^{1}\big \{\varvec{B}^\tau _2(t)(\hat{\varvec{\gamma }}-\varvec{\gamma }_0)\big \}^2+2\int _{0}^{1}(R_{2k_2}(t))^2dt\\&=2(\hat{\varvec{\gamma }}-\varvec{\gamma }_0)^\tau \varvec{H}(\hat{\varvec{\gamma }}-\varvec{\gamma }_0)+2\int _{0}^{1}(R_{2k_2}(t))^2dt. \end{aligned} \end{aligned}$$

where $\varvec{H}=\int _{0}^{1}\varvec{B}_2(t)\varvec{B}^\tau _2(t)dt$. Then, invoking $\Vert \varvec{H}\Vert =O(1)$ (see, e.g., Feng and Xue 2013; Zhu et al. 2015) and $\Vert \varvec{\gamma }-\varvec{\hat{\gamma }}_0\Vert =O_p(\delta )$, a simple calculation yields $(\hat{\varvec{\gamma }}-\varvec{\gamma }_0)^\tau \varvec{H}(\hat{\varvec{\gamma }}-\varvec{\gamma }_0)=O_p(\delta ^2)$. In addition, it is easy to show that $\int _{0}^{1}(R_{2k_2}(t))^2dt=O_p(\delta ^2)$. Then, we complete the proof of second part of (7). Using an argument similar to $\hat{\beta }(t)$, we can obtain the third part of (7).$\square$

Proof of Theorem 2

For convenience sake, we denote $D_i=g'_0\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )\varvec{J}^\tau _{{\phi }_0}\varvec{Z}_i$, $\hat{g}\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )=\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )^\tau \varvec{\hat{\eta }}$, $\hat{g}'\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )=\varvec{B}'_1\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )^\tau \varvec{\hat{\eta }}$. According to Theorem 1, we know that, as $n\rightarrow \infty$, with probability tending to 1, $Q(\varvec{\theta })$ attains the minimal value at $\varvec{\hat{\theta }}=(\hat{\varvec{\phi }}^\tau ,\hat{\varvec{\eta }}^\tau ,\hat{\varvec{\gamma }}^\tau )^\tau$. Then , we have

$$\begin{aligned} \begin{aligned} -\frac{1}{2n}\frac{\partial Q(\varvec{\hat{\theta }})}{\partial \varvec{\phi }}&=\frac{1}{n}\sum _{i=1}^{n}\big [Y_i-\hat{g}\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )-\varvec{W}^\tau _i \hat{\varvec{\gamma }}\big ]\hat{g}'\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )\varvec{J}^\tau _{\hat{\phi }}\varvec{Z}_i=0 \end{aligned} \end{aligned}$$

Furthermore, invoking condition C3, Lemma 1 and Eq. (A2), a simple calculation yields

$$\begin{aligned} -\frac{1}{2n}\frac{\partial Q(\varvec{\hat{\theta }})}{\partial \varvec{\phi }}= & {} \frac{1}{n}\sum _{i=1}^{n}\big [e_i+R_{2i}-\varvec{W}^\tau _i(\hat{\varvec{\gamma }}-{\varvec{\gamma }}_0)+{g}_0\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )\qquad \nonumber \\&-\,\hat{g}\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )\big ]\varvec{D}_i+o_p(\hat{\varvec{\phi }}-\varvec{\phi }_0)=0. \end{aligned}$$

(A3)

Similarly, we have

$$\begin{aligned} -\frac{1}{2n}\frac{\partial Q(\varvec{\hat{\theta }})}{\partial \varvec{\gamma }}= & {} \frac{1}{n}\sum _{i=1}^{n}\big [Y_i-\hat{g}\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )-\varvec{W}^\tau _i \hat{\varvec{\gamma }}\big ]\varvec{W}_i \nonumber \\= & {} \frac{1}{n}\sum _{i=1}^{n}\big [e_i+R_{2i}-\varvec{W}^\tau _i(\hat{\varvec{\gamma }}-{\varvec{\gamma }}_0)+{g}_0\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )\nonumber \\&-\,\hat{g}\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )\big ]\varvec{W}_i=0, \end{aligned}$$

(A4)

where $R_{2i}=\big \langle X_i(t), \beta _0(t)-\varvec{B}_2(t)^\tau \varvec{\gamma }_0\big \rangle$.

Let $\Phi _n=\frac{1}{n}\sum _{i=1}^{n}\varvec{W}_i\varvec{W}^\tau _i$, $\Psi _n=\frac{1}{n}\sum _{i=1}^{n}\varvec{W}_i[g_0(\varvec{Z}^\tau _i\beta (\phi _0))-\hat{g}(\varvec{Z}^\tau _i\beta (\hat{\phi }))]$, $\Gamma _n=\frac{1}{n}\sum _{i=1}^{n}\varvec{W}_i\varvec{D}^\tau _i$, $\Lambda _n=\frac{1}{n}\sum _{i=1}^{n}\varvec{W}_i(e_i+R_{2i})$. Then, by Eq. (A4), we have

$$\begin{aligned} \hat{\varvec{\gamma }}-\varvec{\gamma }_0=\Phi _n^{-1}(\Lambda _n+\Psi _n). \end{aligned}$$

(A5)

Substituting Eqs. (A5) into (A3), we can obtain that

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^{n}\big [e_i+R_{2i}-\varvec{W}^\tau _i\Phi _n^{-1}(\Lambda _n+\Psi _n)+{g}_0\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\nonumber \\&\quad -\,\hat{g}\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )\big ]\varvec{D}_i+o_p(n^{-1/2})=0. \end{aligned}$$

(A6)

Note that

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^{n}{\Gamma }_n^\tau \Phi _n^{-1}\varvec{W}_i\big [\varvec{D}_i^\tau -\varvec{W}_i^\tau \Phi _n^{-1}\Gamma _n\big ]=0, \end{aligned}$$

(A7)

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^{n}{\Gamma }_n^\tau \Phi _n^{-1}\varvec{W}_i\big [e_i+R_{2i}-\varvec{W}_i^\tau \Phi _n^{-1}\Lambda _n\big ]=0, \end{aligned}$$

(A8)

and

$$\begin{aligned}&{g}_0\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )-\hat{g}\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )\nonumber \\&\quad ={g}_0\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )-{g}\big (\varvec{Z_i}^\tau \varvec{\alpha ({\hat{\phi }})}\big )+{g}\big (\varvec{Z_i}^\tau \varvec{\alpha ({\hat{\phi }})}\big )-\hat{g}\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )\nonumber \\&\quad =D_i^\tau (\varvec{{\phi }_0}-{\hat{\varvec{\phi }}})+o_p(\varvec{{\phi }_0}-{\hat{\varvec{\phi }}})+O_p(k^{-r}). \end{aligned}$$

(A9)

Let $\widetilde{\varvec{D}}_i={\varvec{D}}_i-\varvec{\Gamma }_n^\tau \Phi _n^{-1}\varvec{W}_i$. By the law of large numbers and the definition of $\Sigma _n$, we have

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\widetilde{\varvec{D}}_i\widetilde{\varvec{D}}_i^\tau \overset{p}{\longrightarrow } \Sigma , \end{aligned}$$

(A10)

where $\overset{p}{\longrightarrow }$ means the convergence in probability. Invoking Eqs. (A6)–(A9), it is easy to show that

$$\begin{aligned} \begin{aligned}&\Big (\frac{1}{n}\sum _{i=1}^{n}\widetilde{\varvec{D}}_i\widetilde{\varvec{D}}_i^\tau \Big )\sqrt{n} (\hat{\varvec{\phi }}-{\varvec{\phi }}_0)\\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\widetilde{\varvec{D}}_ie_i+\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\widetilde{\varvec{D}}_iR_{2i}+ \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\widetilde{\varvec{D}}_i\varvec{W}_i^\tau \Phi _n^{-1}\Lambda _n+o_p(1)\\&\quad \equiv S_1+S_2+S_3+o_p(1). \end{aligned} \end{aligned}$$

Note that $\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\widetilde{\varvec{D}}_i\varvec{W}_i^\tau =0$, then, we have $S_3=0$. Furthermore, a simple calculation yields

$$\begin{aligned} \begin{aligned} S_2=&\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\big [{\varvec{D}}_i-E(\varvec{\Gamma }_n^\tau )E(\varvec{\Phi _n}^{-1})\varvec{W}_i\big ]R_{2i} +\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\big [E(\varvec{\Gamma }_n^\tau )E(\varvec{\Phi _n}^{-1})\\&-\,\varvec{\Gamma }_n^\tau \varvec{\Phi _n}^{-1}\big ]\varvec{W}_iR_{2i}\equiv S_{21}+S_{22}. \end{aligned} \end{aligned}$$

Using an argument similar to Theorem 3 in Zhao and Xue (2010), invoking

$$\begin{aligned} E\big ([{\varvec{D}}_i-E(\varvec{\Gamma }_n^\tau )E(\varvec{\Phi _n}^{-1})\varvec{W}_i]\varvec{W}_i^\tau \big )=0, \end{aligned}$$

we can prove

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\big [{\varvec{D}}_i-E(\varvec{\Gamma }_n^\tau )E(\varvec{\Phi _n}^{-1})\varvec{W}_i\big ]\varvec{W}_i^\tau =O_p(1). \end{aligned}$$

Taking this together with condition C1, $\sup _{u}\Vert \varvec{B}_2(u)\Vert =O(1)$ and $|R_{2i}|=O_p(k^{-r})$, we can obtain $S_{21}=o_p(1)$. Similarly, we can prove that $S_{22}=o_p(1)$. Hence, we obtain that $S_{2}=o_p(1)$. Thus, we have

$$\begin{aligned} \sqrt{n} (\hat{\varvec{\phi }}-{\varvec{\phi }}_0)=\Sigma ^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\widetilde{\varvec{D}}_ie_i+o_p(1). \end{aligned}$$

(A11)

Invoking Eq. (5), we can prove

$$\begin{aligned} \sqrt{n} (\hat{\varvec{\alpha }}-{\varvec{\alpha }}_0)={\sqrt{n}}\varvec{J}_{\phi _0}(\hat{\varvec{\phi }}-{\varvec{\phi }}_0)+O_p(n^{-1/2}). \end{aligned}$$

Moreover, we have

$$\begin{aligned} \sqrt{n} (\hat{\varvec{\alpha }}-{\varvec{\alpha }}_0)={n}^{-1/2}\varvec{J}_{\phi _0}\Sigma ^{-1}\sum _{i=1}^{n}\widetilde{\varvec{D}}_ie_i+o_p(1). \end{aligned}$$

The claim then follows from the central limiting theorem and Slutsky’s theorem.$\square$

Proof of Theorem 3

Following the same arguments used in Eqs. (A3) and (A4), by Theorems 1 and 2 and a simple calculation, we have

$$\begin{aligned} -\frac{1}{2n}\frac{\partial Q(\varvec{\hat{\theta }})}{\partial \varvec{\eta }}= & {} \frac{1}{n}\sum _{i=1}^{n}\big [Y_i-\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )^\tau \varvec{\hat{\eta }}-\varvec{W}^\tau _i \hat{\varvec{\gamma }}\big ]\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )^\tau \nonumber \\= & {} \frac{1}{n}\sum _{i=1}^{n}\big [\tilde{e}_i-\varvec{W}^\tau _i(\hat{\varvec{\gamma }}-{\varvec{\gamma }}_0)- \varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )^\tau \nonumber \\&\times (\varvec{\hat{\eta }}-\varvec{\eta }_0)\big ]\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )^\tau +o_p(1)=0, \end{aligned}$$

(A12)

where $\tilde{e}_i=e_i+R_{1i}+R_{2i}$, $R_{1i}=g_0(\varvec{Z_i}^\tau \varvec{\alpha }(\varvec{\phi }_0))-\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\varvec{\phi }_0)\big )^\tau \varvec{\eta }_0$ and $R_{2i}=\big \langle X_i(t), \beta _0(t)-\varvec{B}_2(t)^\tau \varvec{\gamma }_0\big \rangle$. The remainder is $o_p(1)$ because Theorem 2 and $\big \Vert \varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )-\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\varvec{\phi }_0)\big )\big \Vert =o_p(1)$. In addition, Eq. (A4) can be rewrite as

$$\begin{aligned} -\frac{1}{2n}\frac{\partial Q(\varvec{\hat{\theta }})}{\partial \varvec{\gamma }}= & {} \frac{1}{n}\sum _{i=1}^{n}\big [\tilde{e}_i-\varvec{W}^\tau _i(\hat{\varvec{\gamma }}-{\varvec{\gamma }}_0)- \varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )^\tau \nonumber \\&\times (\varvec{\hat{\eta }}-\varvec{\eta }_0)\big ]\varvec{W}^\tau _i +o_p(1)=0, \end{aligned}$$

(A13)

where the remainder is $o_p(1)$ because $\big \Vert \varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha (\hat{\phi })}\big )-\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha }(\varvec{\phi }_0)\big )\big \Vert =o_p(1)$ and $\Vert \varvec{W}_i\Vert =O_p(1)$. Invoking Eq. (A12), we have

$$\begin{aligned} \hat{\varvec{\eta }}-{\varvec{\eta }}_0= & {} \bigg \{\frac{1}{n}\sum _{i=1}^{n}\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )^\tau \varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big ) \bigg \}^{-1} \bigg \{\frac{1}{n}\sum _{i=1}^{n}\widetilde{e}_i\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )\nonumber \\&-\,\frac{1}{n}\sum _{i=1}^{n}\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )\varvec{W}^\tau _i(\hat{\varvec{\gamma }}-{\varvec{\gamma }}_0)+o_p(1)\bigg \}. \end{aligned}$$

(A14)

Let

$$\begin{aligned} \bar{\Delta }_n= & {} \bigg \{\frac{1}{n}\sum _{i=1}^{n}\varvec{W}_i\varvec{W}^\tau _i-\bigg [\frac{1}{n}\sum _{i=1}^{n}\varvec{W}_i \varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )^\tau \bigg ]\nonumber \\&\times \bigg [\frac{1}{n}\sum _{j=1}^{n}\varvec{B}_1\big (\varvec{Z_j}^\tau \varvec{\alpha ({\phi }_0)}\big )\varvec{B}_1\big (\varvec{Z_j}^\tau \varvec{\alpha ({\phi }_0)}\big )^\tau \bigg ]^{-1}\bigg [\frac{1}{n}\sum _{i=1}^{n} \varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )\varvec{W}_i^\tau \bigg ]\bigg \}.\nonumber \\ \end{aligned}$$

(A15)

Substituting Eqs. (A14) into (A13), a simple calculation yields

$$\begin{aligned} \hat{\varvec{\gamma }}-{\varvec{\gamma }}_0= & {} (\bar{\Delta }_n)^{-1}\Bigg \{\frac{1}{n}\sum _{i=1}^{n}\tilde{e}_i\bigg [\varvec{W}_i-\bigg (\frac{1}{n}\sum _{k=1}^{n}\varvec{W}_k \varvec{B}_1\big (\varvec{Z_k}^\tau \varvec{\alpha ({\phi }_0)}\big )^\tau \bigg )\\&\times \bigg (\frac{1}{n}\sum _{j=1}^{n}\varvec{B}_1\big (\varvec{Z_j}^\tau \varvec{\alpha ({\phi }_0)}\big )\varvec{B}_1\big (\varvec{Z_j}^\tau \varvec{\alpha ({\phi }_0)}\big )^\tau \bigg )^{-1}\varvec{B}_1\big (\varvec{Z_i}^\tau \varvec{\alpha ({\phi }_0)}\big )\bigg ]+o_p(1)\Bigg \}. \end{aligned}$$

Since that $|R_{1i}| \le ck_1^{-r}$, $|R_{2i}|=O_p(k_2^{-r})$ and $\hat{\beta }(t)-\beta ^*(t)=\varvec{B}^\tau _2(t)(\hat{\varvec{\gamma }}-{\varvec{\gamma }}_0)$, for any fixed point $t\in (0,1)$, as $n\rightarrow \infty$, by the law of large numbers, the Slutsky’s theorem and the property of multivariate normal distribution, a simple calculation yields

$$\begin{aligned} \sqrt{\frac{n}{k_2}}(\hat{\beta }(t)-\beta ^*(t)) \overset{d}{\longrightarrow }N\big (0,\Xi (t)\big ), \end{aligned}$$

where $\Xi (t)=\lim _{n \rightarrow \infty }\frac{\sigma ^2}{k_2}\varvec{B}^\tau _2(t)\Delta _n\varvec{B}_2(t).$

Next, the argument for index function $g(\cdot )$ is the same. Since $\hat{g}(u)-g^*(u)=\varvec{B}^\tau _1(u)(\hat{\varvec{\eta }}-{\varvec{\eta }}_0)$, for any fixed point $u\in (a,b)$, as $n\rightarrow \infty$, we have

$$\begin{aligned} \sqrt{\frac{n}{k_1}}(\hat{g}(u)-g^*(u)) \overset{d}{\longrightarrow }N\big (0,\Pi (u)\big ), \end{aligned}$$

where $\Pi (u)=\lim _{n \rightarrow \infty }\frac{\sigma ^2}{k_1}\varvec{B}^\tau _1(u)\Lambda _n\varvec{B}_1(u).$ This completes the proof of Theorem 3.$\square$