Model pursuit and variable selection in the additive accelerated failure time model

Abstract

In this paper, we propose a new semiparametric method to simultaneously select important variables, identify the model structure and estimate covariate effects in the additive AFT model, for which the dimension of covariates is allowed to increase with sample size. Instead of directly approximating the non-parametric effects as in most existing studies, we take a linear effect out to weak the condition required for model identifiability. To compute the proposed estimates numerically, we use an alternating direction method of multipliers algorithm so that it can be implemented easily and achieve fast convergence rate. Our method is proved to be selection consistent and possess an asymptotic oracle property. The performance of the proposed methods is illustrated through simulations and the real data analysis.

Appendix: Proofs

Proof of Proposition 1

First, the fact that $\phi _j(x)=\beta _j+\tilde{\phi }_j(x)$ with $\beta _j=\int _{\alpha _1}^{\alpha _2} \phi _j(x)dx$ and $\tilde{\phi }_j(x)=\phi _j(x)-\beta _j$ for $j=1,\ldots , p$ implies that the decomposition (4) holds. To show the uniqueness of the decomposition, we assume that there exist $(\beta _1^{(l)},\ldots ,\beta _{d_n}^{(l)})'\in {\mathbb {R}}^{d_n}$ and $(\tilde{\phi }_1^{(l)},\ldots ,\tilde{\phi }_{d_n}^{(l)})'\in \tilde{{\mathcal {H}}}^{d_n}$, $l=1,2$ such that

$$\begin{aligned} \sum \limits _{j=1}^{d_n} x_j[\beta _j^{(1)}+\tilde{\phi }_j^{(1)}(x_j)]\equiv \sum \limits _{j=1}^{d_n} x_j[\beta _j^{(2)}+\tilde{\phi }_j^{(2)}(x_j)]. \end{aligned}$$

(10)

It suffices to prove that $\beta _j^{(1)}=\beta _j^{(2)}$ and $\tilde{\phi }_j^{(1)}(x)\equiv \tilde{\phi }_j^{(2)}(x)$ for each $j=1,\ldots , d_n$. To the end, we note that (10) implies that

$$\begin{aligned} \sum \limits _{j=1}^{d_n} x_j\Big (\left[ \beta _j^{(1)}-\beta _j^{(2)}\right] +\left[ \tilde{\phi }_j^{(1)}(x_j)-\tilde{\phi }_j^{(2)}(x_j)\right] \Big )\equiv 0. \end{aligned}$$

When the covariates are not linearly dependent, by the Fubini’s theorem, there exists $(x_1^0,\ldots ,x_{j-1}^0,x_{j+1}^0,\ldots ,x_{d_n}^0)\in [\alpha _1,\alpha _2]^{d_n-1}$ such that

$$\begin{aligned} x_j\Big ([\beta _j^{(1)}-\beta _j^{(2)}]+[\tilde{\phi }_j^{(1)}(x_j)-\tilde{\phi }_j^{(2)}(x_j)]\Big )\equiv -\sum \limits _{i\ne j}x_i^0\Big ([\beta _i^{(1)}-\beta _i^{(2)}]+[\tilde{\phi }_i^{(1)}(x_i^0)-\tilde{\phi }_i^{(2)}(x_i^0)]\Big ). \end{aligned}$$

Writing $-\sum \nolimits _{i\ne j}x_i^0\Big ([\beta _i^{(1)}-\beta _i^{(2)}]+[\tilde{\phi }_i^{(1)}(x_i^0)-\tilde{\phi }_i^{(2)}(x_i^0)]\Big )$ as $C_j$ and using the condition that $E(\beta _j^{(l)} X_j+X_j\tilde{\phi }_j^{(l)}(X_j))=E(X_j\phi (X_j))$ for $l=1,2$, we have

$$\begin{aligned} (\beta _j^{(1)}-\beta _j^{(2)})+[\tilde{\phi }_j^{(1)}(x)-\tilde{\phi }_j^{(2)}(x)]\equiv 0 \end{aligned}$$

(11)

for each $j=1,\ldots , d_n$. Noting that $\tilde{\phi }_j^{(l)}(x)\in \tilde{{\mathcal {H}}}$, integrating two sides of (11) on variable x from $\alpha _1$ to $\alpha _2$ gives that

$$\begin{aligned} \beta _j^{(1)}=\beta _j^{(2)}. \end{aligned}$$

Combining with (11), we get that

$$\begin{aligned} \tilde{\phi }_j^{(1)}(x)\equiv \tilde{\phi }_j^{(2)}(x). \end{aligned}$$

$\square$

Let ${\mathbb {P}}_n$ be the empirical measure of $\{(Y_i,\delta _i, \varvec{X}_i):i=1,2,\ldots ,n\}$, and ${\mathbb {P}}$ be the probability measure of $(Y,\delta ,\varvec{X})$. Define $g_{nj}^*(X_j)=g_{j}^*(\phi _{nj},X_j)$ and $g_{0j}^*(X_j)=g_{j}^*(\phi _{0j},X_j)$ for $\phi _{nj}\in {\Omega _n}$. Then denote $g_n(\varvec{X})=\sum _{j=1}^{d_n} X_j\phi _{nj}(X_j)$, $g_n^*(\varvec{X})=\sum _{j=1}^{d_n} g_{nj}^*(X_j)$ and $g_0^*(\varvec{X})=\sum _{j=1}^{d_n} g_{0j}^*(X_j)$. Define

$$\begin{aligned} \varvec{{C}_{\xi }}=\left( \begin{array}{lllll} -\frac{m}{u_{m+1}-u_1} &{}\quad \frac{m}{u_{m+1}-u_1} &{}\quad 0 &{}\quad \cdots \ &{}\quad 0 \\ 0 &{}\quad -\frac{m}{u_{m+2}-u_2} &{} \frac{m}{u_{m+2}-u_2} &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad -\frac{m}{u_{q_n+m-1}-u_{q_n-1}} &{}\quad \frac{m}{u_{q_n+m-1}-u_{q_n-1}} \end{array} \right) _{(q_n-1)\times q_n,} \end{aligned}$$

for $u_0=\cdots =u_m=\xi _0,\ u_{m+1}=\xi _1,\ldots , u_{q_n-1}=\xi _{K_n-1},\ u_{q_n}=\cdots =u_{q_n+m}=\xi _{K_n}$. Let $\xrightarrow {P}$ and $\xrightarrow {d}$ represent convergence in probability and in distribution, respectively, as $n\rightarrow \infty$ unless otherwise stated. Similar to Lemma A5 in Huang (1999), the following lemma can be established first.

Lemma 1

Assume that Conditions (C1)–(C4) hold for any $1\le j\le d_n$. Then there exists a function $\phi _{nj} \in {\Omega _n}$ such that

$$\begin{aligned} \Vert g_n^*-g_0^*\Vert _\infty =\Vert g_n-g_0\Vert _\infty =O_p(d_n(n^{-\nu p}+n^{-(1-\nu )/2})) \end{aligned}$$

with ${\mathbb {P}}_n\delta g_{nj}=0$.

Proof

According to Corollary 6.21 of Schumaker (1981), for any $1\le j\le d_n$, there exists $\phi _{nj}\in \Omega _n$ such that $\Vert \phi _{nj}-\phi _{0j}\Vert _\infty =O(n^{-\nu p})$. We define $\widetilde{g}_{nj}(X_j)=X_j\phi _{nj}(X_j)$ and

$$\begin{aligned} g_{nj}=\widetilde{g}_{nj}-n_{\delta }^{-1}{\mathbb {P}}_n\delta \widetilde{g}_{nj}, \end{aligned}$$

where $n_{\delta }=\sum _{i=1}^n\delta _i/n$. Then it is easy to see that ${\mathbb {P}}_n\delta g_{nj}=0$ for any $1\le j\le d_n$. Furthermore, we note that

$$\begin{aligned} \Vert g_{nj}-g_{0j}\Vert _{\infty }\le \Vert g_{nj}-\widetilde{g}_{nj}\Vert _\infty +\Vert \widetilde{g}_{nj}-g_{0j}\Vert _\infty \triangleq I_{1n}+I_{2n}, \end{aligned}$$

(12)

where

$$\begin{aligned} I_{1n}=\Vert g_{nj}-\widetilde{g}_{nj}\Vert _\infty \le c\Vert {\mathbb {P}}_n\delta \widetilde{g}_{nj}\Vert _{\infty }\le c(\Vert ({\mathbb {P}}_n-{\mathbb {P}})\delta \widetilde{g}_{nj}\Vert _\infty +\Vert {\mathbb {P}}(\delta \widetilde{g}_{nj}-\delta g_{0j})\Vert _\infty ), \end{aligned}$$

with c being a constant independent of n. By Lemma 3.4.2 in van der Vaart and Wellner (1996), we have $({\mathbb {P}}_n-{\mathbb {P}})\delta \widetilde{g}_{nj}=O_p(n^{-1/2}n^{\nu /2})$. And the definition of $\phi _{nj}$ shows that $\Vert {\mathbb {P}}(\delta \widetilde{g}_{nj}-\delta g_{0j})\Vert _\infty \le E(\delta )\Vert \widetilde{g}_{nj}-g_{0j}\Vert _\infty =O(n^{-\nu p})$. Hence we have

$$\begin{aligned} I_{1n}=O_p(n^{-\nu p}+n^{-(1-\nu )/2}). \end{aligned}$$

(13)

In addition,

$$\begin{aligned} I_{2n}=\Vert X_j\phi _{nj}-X_j\phi _{0j}\Vert _{\infty }=O_p(n^{-\nu p}). \end{aligned}$$

(14)

Plugging (13) and (14) into (12), we can get $\Vert g_{nj}-g_{0j}\Vert _\infty =O_p(n^{-\nu p}+n^{-(1-\nu )/2})$. By using the property of Kaplan–Meier weights (Stute 1993) and Lemma 3.4.2 in van der Vaart and Wellner (1996), we have

$$\begin{aligned} \Vert g_{nj}^*-g_{0j}^*\Vert _\infty\le & {} c_1\Vert (X_j \phi _{nj}-X_j \phi _{0j})-(\overline{g}_{jw}(\phi _{nj},X_j)-\overline{g}_{jw}(\phi _{0j},X_j))\Vert _\infty \\\le & {} c_1\Vert \widetilde{g}_{nj}-g_{0j}\Vert _\infty +c_2\Big \Vert \sum _{i=1}^n \omega _i X_{(i)j}{\phi }_{nj}(X_{(i)j})-\delta \widetilde{g}_{nj}\Big \Vert _\infty \\&\quad +c_3\Big \Vert \sum _{i=1}^n \omega _i X_{(i)j}{\phi }_{0j}(X_{(i)j})-\delta g_{0j}\Vert _\infty +c_4\Vert \delta \widetilde{g}_{nj}-\delta g_{0j}\Big \Vert _\infty \\= & {} O_p(n^{-\nu p})+O_p(n^{-(1-\nu )/2})+O_p(n^{-1/2})+O_p(n^{-\nu p})\\= & {} O_p(n^{-\nu p}+n^{-(1-\nu )/2}), \end{aligned}$$

where $c_i$’s $i=1,\ldots , 4$ are finite constants. Thus, we have

$$\begin{aligned} \Vert g_n^*-g_0^*\Vert _\infty =\Vert g_n-g_0\Vert _\infty =O_p(d_n(n^{-\nu p}+n^{-(1-\nu )/2})). \end{aligned}$$

$\square$

Define $\widehat{g}_{nj}^*(X_j)=g_j^*(\widehat{\phi }_{nj},X_j)$ and $\widehat{g}_n^*(\varvec{X})=\sum _{j=1}^{d_n}\widehat{g}_{nj}^*(X_j)$, then we have the following lemma.

Lemma 2

Assume that Conditions (C1)–(C7) hold. If $0.25/p<\nu <0.5$, then $\Vert \widehat{g}_n^*-g_n^*\Vert ^2=o_p(d_n^2 q_n^{-1})$ and $\displaystyle \left\| \frac{1}{d_n}(\widehat{g}_n^*-g_n^*)\right\| _{\infty }=o_p(1)$.

Proof

Let $\eta _{nj}\in \Omega _n$ such that $\eta _{nj}(x)=\varvec{\theta }_{nj}^{*T}\varvec{\psi }_{q_n,m}(x)$ and $\Vert \eta _{nj}(x)\Vert ^2=O(q_n^{-1})$. Denote $h_n^*(\varvec{X})=\sum _{j=1}^{d_n} g_j^*(\eta _{nj},X_j)$, then we have $\displaystyle \Vert \frac{1}{d_n}h_n^*(\varvec{X})\Vert ^2=O_p(q_n^{-1})$. Define $H_n(\alpha )=Q_n(\varvec{\theta }_n+\alpha \varvec{\theta }_n^{*})$. To prove this lemma, it is sufficient to show that for any $\alpha _0>0$, $H'_n(\alpha _0)>0$ and $H'_n(-\alpha _0)<0$ with probability tending to one.

Note that

$$\begin{aligned} H_n(\alpha _0)= & {} \frac{1}{2n}\Vert Y^*-(g_n^*+\alpha _0 h_n^*)(\varvec{X})\Vert ^2 +\sum _{j=1}^{d_n}P_1(\Vert \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})\Vert ;\lambda _{1})\\&+\sum _{j=1}^{d_n}P_2(\Vert \varvec{\theta }_{nj} +\alpha _0 \varvec{\theta }_{nj}^{*}\Vert ;\lambda _{2}). \end{aligned}$$

Then

$$\begin{aligned} H'_n(\alpha _0)= & {} -{\mathbb {P}}_n\Big [h_n^*\big (Y^{*}-g_n^*-\alpha _0 h_n^*\big )\Big ] \\&+\sum _{j=1}^{d_n}P'_1(\Vert \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})\Vert ;\lambda _{1}) \frac{(\varvec{{C}_{\xi }}\varvec{\theta }_{nj}^{*})^T \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})}{\Vert \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})\Vert } \\&+\sum _{j=1}^{d_n}P'_2(\Vert \varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*}\Vert ;\lambda _{2})\frac{\varvec{\theta }_{nj}^{*T}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})}{\Vert \varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*}\Vert } \\\triangleq & {} H_1+H_2+H_3. \end{aligned}$$

We consider the first part

$$\begin{aligned} H_1= & {} -{\mathbb {P}}_n\big [h_n^*(Y^*-g_n^*)\big ]+\alpha _0{\mathbb {P}}_n(h_n^*\cdot h_n^*) \\= & {} -{\mathbb {P}}_n\big [h_n^*(Y^*-g_n^*)\big ]+\alpha _0{\mathbb {P}}\Vert h_n^*(\varvec{X})\Vert ^2+\alpha _0({\mathbb {P}}_n-{\mathbb {P}})(h_n^*\cdot h_n^*) \\= & {} -{\mathbb {P}}_n\big [h_n^*(Y^*-g_n^*)\big ]+c_0 \alpha _0 d_n^2 n^{-\nu }+O_p(n^{-1/2}d_n^2), \end{aligned}$$

where $c_0>0$ is a constant and the first term

$$\begin{aligned} {\mathbb {P}}_n\big [h_n^*(Y^*-g_n^*)\big ]= & {} ({\mathbb {P}}_n-{\mathbb {P}})\big [h_n^*(Y^*-g_n^*)\big ]+{\mathbb {P}}\big [h_n^*(Y^*-g_n^*)\big ] \\\triangleq & {} J_{1n}+J_{2n}. \end{aligned}$$

In $J_{1n}$, $\Vert Y^*-g_n^*\Vert _\infty =\Vert Y^*-g_0^*+g_0^*-g_n^*\Vert _\infty \le O_p(1)+O_p(d_n (n^{-\nu p}+n^{-(1-\nu )/2}))$. Since $d_n^4/n\rightarrow 0,\ 0.25/p<\nu <0.5$, we have $\Vert \displaystyle \frac{1}{d_n^2}h_n^*(Y^*-g_n^*)\Vert _\infty \le M_0$ with a constant $M_0$. Let

$$\begin{aligned} \mu _0(\eta )=\left\{ \frac{1}{d_n^2}h_n^*\cdot (Y^*-g_n^*):\ \left\| \frac{1}{d_n}h_n^*\right\| \le \eta ,\left\| \frac{1}{d_n}(g_n^*-g_0^*)\right\| \le \eta \right\} . \end{aligned}$$

Then similar to Lemma A2 and Corollary A1 in Huang (1999), we have

$$\begin{aligned} \log N_\square (\varepsilon ,\mu _0(\eta ),L_2({\mathbb {P}}))\le c_0q_n\log (\eta /\varepsilon ), \end{aligned}$$

for any $\varepsilon <\eta$ with a constant $c_0$ and

$$\begin{aligned} J_\square (\eta ,\mu _0,L_2({\mathbb {P}}))\le c_0q_n^{1/2}\eta . \end{aligned}$$

Here we can take $\eta =q_n^{-1/2}$. Combining the results of Lemma 3.4.2 in van der Vaart and Wellner (1996) and Lemma A1 in Huang (1999), we get

$$\begin{aligned} J_{1n}=O_p(1)\cdot d_n^2\cdot n^{-1/2}\Big (q_n^{1/2}\eta +\frac{q_n}{\sqrt{n}}M_0\Big )=O_p\big (n^{-1/2}d_n^2\big ). \end{aligned}$$

We then consider $J_{2n}$ as

$$\begin{aligned} J_{2n}={\mathbb {P}}\big [h_n^*(g_n^*-g_0^*)\big ]=d_n^2\cdot {\mathbb {P}}\Big [\frac{h_n^*}{d_n}\cdot \frac{g_0^*-g_n^*}{d_n}\Big ], \end{aligned}$$

which gives that

$$\begin{aligned} |J_{2n}|\le O_p(1)\cdot d_n^2\cdot \big \Vert \frac{h_n^*}{d_n}\big \Vert \cdot \big \Vert \frac{g_0^*-g_n^*}{d_n}\big \Vert =O_p\big (d_n^2(n^{-(1/2+p)\nu }+n^{-1/2})\big ). \end{aligned}$$

Therefore,

$$\begin{aligned} H_1\ge c_0\alpha _0 d_n^2 n^{-\nu }+O_p(d_n^2 n^{-1/2})+O_p(d_n^2(n^{-(1/2+p)\nu }+n^{-1/2})). \end{aligned}$$

Next we focus on $H_2$ and $H_3$. Let $\varvec{B}_j(X_j)=(\varvec{\psi }_{q_n,m}(X_{1j}),\ldots ,\varvec{\psi }_{q_n,m}(X_{nj}))^T$. By Lemma 3 of Huang and Ma (2010), it follows that there are constants $0<c_3<c_4<\infty$ such that

$$\begin{aligned} c_3 q_n^{-1}\le \Lambda _{\min }\Big (\frac{\varvec{B}_j(X_j)^T \varvec{B}_j(X_j)}{n}\Big )\le \Lambda _{\max }\Big (\frac{\varvec{B}_j(X_j)^T \varvec{B}_j(X_j)}{n}\Big )\le c_4 q_n^{-1} \end{aligned}$$

with probability tending to one. Then we have $\Vert \varvec{\theta }_{nj}^{*}\Vert =O_p(1)$ and $\Vert \varvec{{C}_{\xi }}\varvec{\theta }_{nj}^{*}\Vert =O_p(1)$ by using of the fact that $\Vert \varvec{\theta }_{nj}^{*T} \varvec{\psi }_{q_n,m}(X_j)\Vert =O(q_n^{-1/2})$. Observing that

$$\begin{aligned} |H_2|= & {} \left| \sum _{j=1}^{d_n}P'_1(\Vert \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})\Vert ;\lambda _{1}) \frac{(\varvec{{C}_{\xi }}\varvec{\theta }_{nj}^{*})^T \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})}{\Vert \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})\Vert }\right| \\\le & {} \sum _{j=1}^{d_n}P'_1(\Vert \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})\Vert ;\lambda _{1})\frac{\big |(\varvec{{C}_{\xi }}\varvec{\theta }_{nj}^{*})^T \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})\big |}{\Vert \varvec{{C}_{\xi }}(\varvec{\theta }_{nj}+\alpha _0 \varvec{\theta }_{nj}^{*})\Vert }, \end{aligned}$$

by using of Condition (C9) and $\lambda _{1}=o(d_n n^{-\nu })$, we have

$$\begin{aligned} |H_2| \le P'_1(0+;\lambda _{1})\sum _{j=1}^{d_n}\Vert \varvec{{C}_{\xi }}\varvec{\theta }_{nj}^{*}\Vert \le O(\lambda _{1})O_p(d_n)=o_p(d_n^2 n^{-\nu }). \end{aligned}$$

The same arguments as above give that $|H_3|\le o_p(d_n^2 n^{-\nu })$ if $\lambda _{2}=o(d_n n^{-\nu })$.

Consequently, $H'_n(\alpha _0)\ge c_0\alpha _0 d_n^2 n^{-\nu }+o_p(d_n^2 n^{-\nu })>0$ with probability tending to one. Similarly, we can prove that $H'_n(-\alpha _0)<0$ with probability tending to one. Therefore, the boundness of covariate $\varvec{X}$ in Condition (C2) ensures that

$$\begin{aligned} \Vert \widehat{g}_n^*-g_n^*\Vert ^2=o_p(d_n^2 q_n^{-1})=o_p(d_n^2 n^{-\nu }). \end{aligned}$$

Subsequently, Lemma 7 of Stone (1986) yields that $\displaystyle \Vert \frac{1}{d_n}(\widehat{g}_n^*-g_n^*)\Vert _{\infty }=o_p(1)$. $\square$

To verify the consistency of parameter estimation, we need the following lemma.

Lemma 3

Define $m_0(x,y^*;g^*)=(y^*-g^*(x))^2/d_n^2$. Denote $M_0={\mathbb {P}} m_0$ and $\displaystyle M_n={\mathbb {P}}_n m_0=\frac{1}{n}\Vert Y^*-g^*(\varvec{X})\Vert ^2/d_n^2$. Under the conditions of Lemma 1, for any function $g(\cdot )$ satisfying $E[\delta g(\varvec{X})]=0$, there exists a constant $c>0$ such that

$$\begin{aligned} {\mathbb {P}} m_0(\cdot ;g^*)-{\mathbb {P}} m_0(\cdot ;g_n^*)=c\left\| \frac{1}{d_n}(g^*-g_n^*)\right\| ^2+O_p(n^{-2\nu p}+n^{-(1-\nu )}). \end{aligned}$$

Proof

Let $h^*=g^*-g_0^*$ and

$$\begin{aligned} L(s)= & {} {\mathbb {P}} m_0(\cdot ;g_0^*+sh^*)-{\mathbb {P}} m_0(\cdot ;g_0^*) \\= & {} \frac{1}{d_n^2}\big [{\mathbb {P}}(Y^*-(g_0^*+sh^*))^2-{\mathbb {P}}(Y^*-g_0^*)^2\big ] \\= & {} \frac{1}{d_n^2}{\mathbb {P}}(-2sY^*h^*+2sg_0^* h^*+s^2 h^{*2}). \end{aligned}$$

Since $L'(0)=0$ and $L''(0)=2{\mathbb {P}}(h^{*2})/d_n^2$, there exists a constant $c>0$, such that $\displaystyle {\mathbb {P}} m_0(\cdot ;g^*)-{\mathbb {P}} m_0(\cdot ;g_0^*)=c\left\| \frac{1}{d_n}(g^*-g_0^*)\right\| ^2$. Similarly, we have

$$\begin{aligned} {\mathbb {P}} m_0(\cdot ;g_n^*)-{\mathbb {P}} m_0(\cdot ;g_0^*)=O_p(1)\left\| \frac{1}{d_n}(g_n^*-g_0^*)\right\| ^2. \end{aligned}$$

By Lemma 1, ${\mathbb {P}} m_0(\cdot ;g_n^*)-{\mathbb {P}} m_0(\cdot ;g_0^*)=O_p(n^{-2\nu p}+n^{-(1-\nu )})$. Combining the following equality

$$\begin{aligned} {\mathbb {P}} m_0(\cdot ;g^*)-{\mathbb {P}} m_0(\cdot ;g_n^*)=\Big ({\mathbb {P}} m_0(\cdot ;g^*)-{\mathbb {P}} m_0(\cdot ;g_0^*)\Big )+\Big ({\mathbb {P}} m_0(\cdot ;g_0^*)-{\mathbb {P}} m_0(\cdot ;g_n^*)\Big ) \end{aligned}$$

with the triangle inequality

$$\begin{aligned} \Vert g^*-g_n^*\Vert ^2-\Vert g_n^*-g_0^*\Vert ^2\le \Vert g^*-g_0^*\Vert ^2\le \Vert g^*-g_n^*\Vert ^2+\Vert g_n^*-g_0^*\Vert ^2, \end{aligned}$$

we have

$$\begin{aligned} {\mathbb {P}} m_0(\cdot ;g^*)-{\mathbb {P}} m_0(\cdot ;g_n^*)=c\left\| \frac{1}{d_n}(g^*-g_n^*)\right\| ^2+O_p(n^{-2\nu p}+n^{-(1-\nu )}), \end{aligned}$$

where $c>0$ is a finite constant. $\square$

Proof of Theorem 1

Let

$$\begin{aligned} V= & {} M_n(g^*)-M_n(g_n^*)-(M_0(g^*)-M_0(g_n^*)) \\= & {} {\mathbb {P}}_n m_0(\cdot ;g^*)-{\mathbb {P}}_n m_0(\cdot ;g_n^*)-({\mathbb {P}} m_0(\cdot ;g^*)-{\mathbb {P}} m_0(\cdot ;g_n^*)) \\= & {} ({\mathbb {P}}_n-{\mathbb {P}})(m_0(\cdot ;g^*)-m_0(\cdot ;g_n^*)), \end{aligned}$$

By Lemma 3.4.2 of van der Vaart and Wellner (1996),

$$\begin{aligned} E\sup \limits _{\left\| \frac{1}{d_n}(g^*-g_n^*)\right\| \le \eta }|V|=n^{-1/2}\eta q_n^{1/2}. \end{aligned}$$

Then by Theorem 3.4.1 of van der Vaart and Wellner (1996), choosing the distance $d(\widehat{g}_n^*,g_n^*)=-[{\mathbb {P}} m_0(\cdot ;\widehat{g}_n^*)-{\mathbb {P}} m_0(\cdot ;g_n^*)]$ there, we have

$$\begin{aligned} -r_{1n}^2[{\mathbb {P}} m_0(\cdot ;\widehat{g}_n^*)-{\mathbb {P}} m_0(\cdot ;g_n^*)]=O_p(1), \end{aligned}$$

where $r_{1n}=O(n^{1/2}q_n^{-1/2})=O(n^{(1-\nu )/2})$. Therefore, ${\mathbb {P}} m_0(\cdot ;\widehat{g}_n^*)-{\mathbb {P}} m_0(\cdot ;g_n^*)=O_p(n^{-(1-\nu )})$. Thus Lemma 3 gives that $\Vert \frac{1}{d_n}(\widehat{g}_n^*-g_n^*)\Vert ^2=O_p(n^{-2\nu p}+n^{-(1-\nu )})$. Combining the result in Lemma 1 that $\Vert g_n^*-g_0^*\Vert _\infty ^2=O_p(d_n^2(n^{-2\nu p}+n^{-(1-\nu )}))$, we have

$$\begin{aligned} \Vert \widehat{g}_n^*-g_0^*\Vert ^2=O_p(d_n^2(n^{-2\nu p}+n^{-(1-\nu )})). \end{aligned}$$

By Conditions (C2)–(C4), it follows that

$$\begin{aligned} E\delta \big \Vert \varvec{\varvec{X}}_{M_1} (\widehat{\varvec{\phi }}_n(\varvec{\varvec{X}}_{M_1})-\tilde{\varvec{\phi }}_0(\varvec{\varvec{X}}_{M_1}))+\varvec{\varvec{X}}_{M_2} (\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)\big \Vert ^2=O(d_n^2(n^{-2\nu p}+n^{-(1-\nu )})). \end{aligned}$$

Denoting the projection of $\varvec{\varvec{X}}_{M_2}$ on $\varvec{\varvec{X}}_{M_1}$ as W, we have

$$\begin{aligned}&E\delta \big \Vert (\varvec{\varvec{X}}_{M_2}-W)(\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)+W(\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)+\varvec{\varvec{X}}_{M_1} (\widehat{\varvec{\phi }}_n-\tilde{\varvec{\phi }}_0)\big \Vert ^2 \\&\quad =E\delta \big \Vert (\varvec{\varvec{X}}_{M_2}-W)(\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)\big \Vert ^2+E\delta \big \Vert W(\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)+\varvec{\varvec{X}}_{M_1} (\widehat{\varvec{\phi }}_n-\tilde{\varvec{\phi }}_0)\big \Vert ^2 \\&\quad =O(d_n^2(n^{-2\nu p}+n^{-(1-\nu )})). \end{aligned}$$

By Condition (C6), we obtain

$$\begin{aligned} \Vert \widehat{\varvec{\beta }}_n-\varvec{\beta }_0\Vert ^2=O_p(d_n^2 (n^{-(1-\nu )}+n^{-2\nu p})). \end{aligned}$$

This in turn implies $E\delta \big \Vert \varvec{\varvec{X}}_{M_1} (\widehat{\varvec{\phi }}_n-\tilde{\varvec{\phi }}_0)\big \Vert ^2=O_p(d_n^2 (n^{-(1-\nu )}+n^{-2\nu p}))$. Therefore,

$$\begin{aligned} \Vert \widehat{\varvec{\phi }}_n-\tilde{\varvec{\phi }}_0\Vert ^2=O_p(d_n^2 (n^{-(1-\nu )}+n^{-2\nu p})). \end{aligned}$$

This completes the proof of Theorem 1. $\square$

Proof of Theorem 2

• (i) First, we prove the selection consistency of the variables. Let $\widetilde{\varvec{\theta }}_n=(\widetilde{\varvec{\theta }}_{n1}^T,\ldots ,\widetilde{\varvec{\theta }}_{nd_n}^T)^T$ with

  $$\begin{aligned} \widetilde{\varvec{\theta }}_{nj}= \left\{ \begin{array}{ll} \widehat{\varvec{\theta }}_{nj},&{}\text{ if } j\notin M_3\text{, }\\ 0,&{}\text{ if } j\in M_3\text{. } \end{array}\right. \end{aligned}$$

  Note that $\widehat{\varvec{\theta }}_n$ satisfies $\displaystyle \frac{\partial Q_n(\widehat{\varvec{\theta }}_n)}{\partial \varvec{\theta }}=\varvec{0}$. By the definition of $\widehat{\varvec{\theta }}_n$ and $\widetilde{\varvec{\theta }}_n$, we have

  $$\begin{aligned}&Q_n(\widehat{\varvec{\theta }}_n)-Q_n(\widetilde{\varvec{\theta }}_n) \\= & {} \frac{\partial Q_n(\widehat{\varvec{\theta }}_{n})^T}{\partial \varvec{\theta }}(\widehat{\varvec{\theta }}_{n}-\widetilde{\varvec{\theta }}_n)-\frac{1}{2}(\widehat{\varvec{\theta }}_n-\widetilde{\varvec{\theta }}_n)^T\frac{\partial ^2 Q_n(\varvec{\theta }_n^*)}{\partial \varvec{\theta }\partial \varvec{\theta }^T}(\widehat{\varvec{\theta }}_n-\widetilde{\varvec{\theta }}_n)\\= & {} -\frac{1}{2}(\widehat{\varvec{\theta }}_n-\widetilde{\varvec{\theta }}_n)^T\frac{\partial ^2 Q_n(\varvec{\theta }_n^*)}{\partial \varvec{\theta }\partial \varvec{\theta }^T}(\widehat{\varvec{\theta }}_n-\widetilde{\varvec{\theta }}_n)\\= & {} -\frac{1}{2}\widehat{\varvec{\theta }}_{nM_3}^T \frac{\partial ^2 \ell _n(\varvec{\theta }_n^*)}{\partial \varvec{\theta }_{M_3}\partial \varvec{\theta }_{M_3}^T}\widehat{\varvec{\theta }}_{nM_3}-\frac{1}{2}\sum \limits _{j\in M_3}(\varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj})^T\big (P''_1(\Vert \varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj}\Vert ;\lambda _{1})+o_p(1)\big )(\varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj}) \\&-\frac{1}{2}\sum \limits _{j\in M_3}\widehat{\varvec{\theta }}_{nj}^T\big (P''_2(\Vert \widehat{\varvec{\theta }}_{nj}\Vert ;\lambda _{2})+o_p(1)\big )\widehat{\varvec{\theta }}_{nj}, \end{aligned}$$

  where $\varvec{\theta }_n^*$ is between $\widehat{\varvec{\theta }}_n$ and $\widetilde{\varvec{\theta }}_n$.

  Since $\widehat{\varvec{\theta }}_n$ is the minimizer of $Q(\varvec{\theta })$, we have $Q(\widehat{\varvec{\theta }}_n)\le Q(\widetilde{\varvec{\theta }}_n)$, which implies that

  $$\begin{aligned} \frac{1}{2}\widehat{\varvec{\theta }}_{nM_3}^T \frac{\partial ^2 \ell _n(\varvec{\theta }_n^*)}{\partial \varvec{\theta }_{M_3}\partial \varvec{\theta }_{M_3}^T}\widehat{\varvec{\theta }}_{nM_3}\ge & {} -\frac{1}{2}\sum \limits _{j\in M_3}(\varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj})^T\big (P''_1(\Vert \varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj}\Vert ;\lambda _{1})+o_p(1)\big )(\varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj})\nonumber \\&-\frac{1}{2}\sum \limits _{j\in M_3}\widehat{\varvec{\theta }}_{nj}^T\big (P''_2(\Vert \widehat{\varvec{\theta }}_{nj}\Vert ;\lambda _{2})+o_p(1)\big )\widehat{\varvec{\theta }}_{nj}. \end{aligned}$$

  (15)

  Note that the left hand of Eq (15)

  $$\begin{aligned} I_1\le c \widehat{\varvec{\theta }}_{nM_3}^T E(X_{M_3}^TX_{M_3})\widehat{\varvec{\theta }}_{nM_3}\le c\rho _n^*\Vert \widehat{\varvec{\theta }}_{nM_3}\Vert ^2 \end{aligned}$$

  for some constant c by the continuity of the B-spline functions and the definition of $\rho _n^*$. And using Condition (C9), there exist constants a, b and c such that the right hand of Eq (15)

  $$\begin{aligned} I_2\ge c (\lambda _{1}^a+\lambda _{2}^a) \Vert \widehat{\varvec{\theta }}_{nM_3}\Vert ^{2-b}. \end{aligned}$$

  Thus, by the results of Theorem 1, we obtain that

  $$\begin{aligned} O_p(1)(d_n^2(n^{-(1-\nu )}+n^{-2\nu p}))^{b/2}\ge \Vert \widehat{\varvec{\theta }}_{nM_3}\Vert ^{b}\ge O_p(1)\frac{\lambda _{1}^a+\lambda _{2}^a}{\rho _n^*}. \end{aligned}$$

  This shows that under the condition that $\displaystyle \frac{\lambda _{1}^a}{\rho _n^*(d_n^2(n^{-(1-\nu )}+n^{-2\nu p}))^{b/2}}$ and $\displaystyle \frac{\lambda _{2}^a}{\rho _n^*(d_n^2(n^{-(1-\nu )}+n^{-2\nu p}))^{b/2}}$ goes to infinity,

  $$\begin{aligned} P(\Vert \widehat{\varvec{\theta }}_{nM_3}\Vert >0)\le P\Big (\frac{\lambda _{1}^a+\lambda _{2}^a}{\rho _n^*(d_n^2(n^{-(1-\nu )}+n^{-2\nu p}))^{b/2}}\le O_p(1)\Big )\rightarrow 0. \end{aligned}$$

  Next, we prove the structure selection consistency. Assume that ${\mathop {\varvec{\theta }}\limits _{\sim }}_{n}=({\mathop {\varvec{\theta }}\limits _{\sim }}_{n1}^{T},\ldots ,{\mathop {\varvec{\theta }}\limits _{\sim }}_{nd_{n}}^T)^T$ with

  $$\begin{aligned} {\mathop {\varvec{\theta }}\limits _{\sim }}_{nM_{2}}= \left\{ \begin{array}{ll} \widehat{\varvec{\theta }}_{nj},&{}\text{ if } j\notin M_2\text{, }\\ {\mathop {\varvec{\theta }}\limits _{\sim }}_{nj}, s.t. \ \varvec{{C}_{\xi }}{\mathop {\varvec{\theta }}\limits _{\sim }}_{nj}=0,&{}\text{ if } j\in M_2\text{. } \end{array}\right. \end{aligned}$$

  Then we have

  $$\begin{aligned}&\frac{1}{2}\Big (\widehat{\varvec{\theta }}_{nM_2}-{\mathop {\varvec{\theta }}\limits _{\sim }}_{nM_2}\Big )^T \frac{\partial ^2 \ell _n(\varvec{\theta }_n^0)}{\partial \varvec{\theta }_{M_2}\partial \varvec{\theta }_{M_2}^T}\Big (\widehat{\varvec{\theta }}_{nM_2}-{\mathop {\varvec{\theta }}\limits _{\sim }}_{nM_2}\Big )\nonumber \\\ge & {} -\frac{1}{2}\sum \limits _{j\in M_2}(\varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj})^T\big (P''_1(\Vert \varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj}\Vert ;\lambda _{1})+o_p(1)\big )(\varvec{{C}_{\xi }}\widehat{\varvec{\theta }}_{nj})\nonumber \\&-\frac{1}{2}\sum \limits _{j\in M_2}\Big (\widehat{\varvec{\theta }}_{nj}-{\mathop {\varvec{\theta }}\limits _{\sim }}_{nj}\Big )^T \big (P''_2(\Vert \widehat{\varvec{\theta }}_{nj}\Vert ;\lambda _{2})+o_p(1)\big ) \Big (\widehat{\varvec{\theta }}_{nj}-{\mathop {\varvec{\theta }}\limits _{\sim }}_{nj}\Big ), \end{aligned}$$

  (16)

  where $\varvec{\theta }_n^0$ is between $\widehat{\varvec{\theta }}_n$ and ${\mathop {\varvec{\theta }}\limits _{\sim }}_n$. The left hand of equation (16)

  $$\begin{aligned} II_1\le & {} O_p(1)\Big (\varvec{C}_\xi (\widehat{\varvec{\theta }}_{nM_2}-{\mathop {\varvec{\theta }}\limits _{\sim }}_{nM_2})\Big )^T \cdot \frac{\partial ^2 \ell _n(\varvec{\theta }_n^0)}{\partial \varvec{\theta }_{M_2}\partial \varvec{\theta }_{M_2}^T}\cdot \Big (\varvec{C}_\xi (\widehat{\varvec{\theta }}_{nM_2}-{\mathop {\varvec{\theta }}\limits _{\sim }}_{nM_2})\Big )\\= & {} O_p(1)(\varvec{C}_\xi \widehat{\varvec{\theta }}_{nM_2})^T \cdot \frac{\partial ^2 \ell _n(\varvec{\theta }_n^0)}{\partial \varvec{\theta }_{M_2}\partial \varvec{\theta }_{M_2}^T}\cdot (\varvec{C}_\xi \widehat{\varvec{\theta }}_{nM_2})\\\le & {} O_p(1)\rho _n^*\Vert \widehat{\varvec{\theta }}_{nM_2}\Vert ^2. \end{aligned}$$

  Similarly, we can obtain that the right hand of equation (16)

  $$\begin{aligned} II_2\ge c (\lambda _{1}^a+\lambda _{2}^a) \Vert \widehat{\varvec{\theta }}_{nM_2}\Vert ^{2-b}. \end{aligned}$$

  Therefore,

  $$\begin{aligned} P(\Vert \widehat{\varvec{\theta }}_{nM_2}\Vert >0)\le P\Big (\frac{\lambda _{1}^a+\lambda _{2}^a}{\rho _n^*(d_n^2(n^{-(1-\nu )}+n^{-2\nu p}))^{b/2}}\le O_p(1)\Big )\rightarrow 0. \end{aligned}$$

  The selection consistency of variable and structure is concluded.

• (ii) Let the column and row vectors of covariate matrix $\varvec{X}^*$ are $X_1^*,\ldots , X_{d_n}^*$ and $X_{(1)}^*,\ldots , X_{(n)}^*$, respectively. Define

  $$\begin{aligned} \overline{X}_w=\frac{\sum \nolimits _{i=1}^n \omega _i X_{(i)}}{\sum \nolimits _{i=1}^n \omega _i}, \quad X_{(i)}^*=(n\omega _i)^{1/2}(X_{(i)}-\overline{X}_w), \end{aligned}$$
  $$\begin{aligned} U(\varvec{W};\varvec{\beta },\widehat{\varvec{\phi }}_n)\triangleq (-\varvec{X}_{M_2}^{*})\Big (Y^*-\sum \limits _{j\in M_1}\hat{g}_{nj}^*(X_{j})-\varvec{X}_{M_2}^* \varvec{\beta }\Big ), \end{aligned}$$
  $$\begin{aligned} \widehat{U}_n(\varvec{\beta })\triangleq \frac{1}{n}\sum \limits _{i=1}^{n}U(\varvec{W}_i;\varvec{\beta },\widehat{\varvec{\phi }}_n), \end{aligned}$$

  with $\varvec{W}\triangleq (\omega ,\varvec{X},Y)$. Then $\widehat{\varvec{\beta }}_n$ satisfies the estimating equation $\widehat{U}_n(\widehat{\varvec{\beta }})=0$ by the definition of $\widehat{\varvec{\beta }}_n$ and $\widehat{\varvec{\phi }}_n$.

  Let $\displaystyle U_n(\varvec{\beta })\triangleq \frac{1}{n}\sum \nolimits _{i=1}^{n}U(\varvec{W}_i;\varvec{\beta },\tilde{\varvec{\phi }}_0)$ and $\widetilde{\varvec{\beta }}_n$ be the root of $U_n(\varvec{\beta })=0$. We then show that $\widehat{\varvec{\beta }}_n$ has the same distribution with $\widetilde{\varvec{\beta }}_n$. The Fréchet derivative of $U(\varvec{W};\varvec{\beta }_0,\varvec{\phi })$ at $\tilde{\varvec{\phi }}_0$ in the direction $\varvec{h}$ is given as

  $$\begin{aligned} D(\varvec{W},\varvec{h})= & {} \lim \limits _{\alpha \rightarrow 0}\frac{U(\varvec{W};\varvec{\beta }_0,\tilde{\varvec{\phi }}_0+\alpha \varvec{h})-U(\varvec{W};\varvec{\beta }_0,\tilde{\varvec{\phi }}_0)}{\alpha } \\= & {} \varvec{X}_{M_2}^{*T}\varvec{X}_{M_1}^{*} \varvec{h}, \end{aligned}$$

  with $\varvec{h}\in \{h_1+\ldots +h_{|M_1|},h_j\in \tilde{{\mathcal {H}}},\ j\in M_1\}$.

  The relation $\Vert (\widehat{\varvec{\phi }}_n-\tilde{\varvec{\phi }}_0)/d_n\Vert =O_p(n^{-(1-\nu )/2}+n^{-\nu p})=o_p(n^{-1/4})$ ensures that the linear assumption 5.1 in Newey (1994) is satisfied. Then by Lemma 3.4.2 of van der Vaart and Wellner (1996), we have

  $$\begin{aligned} \sqrt{n}({\mathbb {P}}_n-{\mathbb {P}})\{D(\varvec{W};\widehat{\varvec{\phi }}_n-\tilde{\varvec{\phi }}_0)\}\xrightarrow {P}0. \end{aligned}$$

  It follows that the stochastic equicontinuity assumption 5.2 holds. For $\varvec{\phi }$ close enough to $\tilde{\varvec{\phi }}_0$, a straightforward calculation yields that $ED(\varvec{W};\varvec{\phi }-\tilde{\varvec{\phi }}_0)=0$ by using Condition (C8). Then the mean square continuity assumption 5.3 holds with $\alpha (\varvec{W})=0$. By Lemma 5.1 of Newey (1994), $\widehat{\varvec{\beta }}_n$ and $\widetilde{\varvec{\beta }}_n$ have the same distribution.

  Next, we seek for the asymptotic distribution of $\widetilde{\varvec{\beta }}_{nM_2}$. Let $\iota _n=n^{-1/2},\ V_{1n}(\varvec{a})=Q_n(\varvec{\beta }_0+\iota _n(\varvec{a}^T,\varvec{0}^T)^T,\tilde{\varvec{\phi }}_0)-Q_n(\varvec{\beta }_0,\tilde{\varvec{\phi }}_0)$, where $\varvec{a}=(a_1,\ldots ,a_{|M_2|})^T$ is a $|M_2|$-dimensional constant vector and $\varvec{0}$ is a $|M_3|$-dimensional zero vector. By part (i) of Theorem 2, $\widetilde{\varvec{\beta }}_n-\varvec{\beta }_0=\iota _n(\widehat{\varvec{a}}_n^T,\varvec{0}^T)^T$ with probability converging to one, where $\widehat{\varvec{a}}_n=\text {argmin}\{V_{1n}(\varvec{a}): \varvec{a}\in {\mathbb {R}}^{|M_2|}\}$. Letting $\widetilde{\varvec{\theta }}_n$ be the estimator corresponding to $\widetilde{\varvec{\beta }}_n$, then similar to Theorem 2 (i), we also have $\varvec{{C}_{\xi }}\widetilde{\varvec{\theta }}_{nj}=0\ (j\in M_2)$ with probability converging to one.

  Note that

  $$\begin{aligned} V_{1n}(\varvec{a})= & {} Q_n(\varvec{\beta }_0+\iota _n(\varvec{a}^T,\varvec{0}^T)^T,\tilde{\varvec{\phi }}_0)-Q_n(\varvec{\beta }_0,\tilde{\varvec{\phi }}_0) \\= & {} \Big (\iota _n \varvec{a}^T U_n(\varvec{\beta }_0)+ \frac{\iota _n^2}{2} \varvec{a}^T U_n'(\varvec{\beta }_0)\varvec{a}\Big ) \\&+\left( \sum _{j\in M_2}P_1(\Vert \varvec{{C}_{\xi }}\widetilde{\varvec{\theta }}_{nj}\Vert ;\lambda _{1})-\sum _{j\in M_2}P_1(\Vert \varvec{{C}_{\xi }}\varvec{\theta }_{0j}\Vert ;\lambda _{1})\right) \\&+\left( \sum _{j\in M_2}P_2(\Vert \widetilde{\varvec{\theta }}_{nj}\Vert ;\lambda _{2n}) -\sum _{j\in M_2}P_2(\Vert \varvec{\theta }_{0j}\Vert ;\lambda _{2n})\right) \\\triangleq & {} A_{1n}(\varvec{a})+A_{2n}(\varvec{a})+A_{3n}(\varvec{a}). \end{aligned}$$

  Since $\widetilde{\varvec{\beta }}_{n M_2}-\varvec{\beta }_{0}=\iota _n\varvec{a}=\varvec{\psi }_{q_n,m}(\varvec{X}_{M_2})^T(\widetilde{\varvec{\theta }}_{n M_2}-\varvec{\theta }_{0 M_2})$, we have

  $$\begin{aligned} \widetilde{\varvec{\theta }}_{nj}-\varvec{\theta }_{0j}=(\varvec{\psi }_{q_n,m}(X_j)\varvec{\psi }_{q_n,m}(X_j)^T)^{-1}\varvec{\psi }_{q_n,m}(X_j)\iota _na_j,\ j=s_1+1,\ldots ,s_2. \end{aligned}$$

  It follows that

  $$\begin{aligned} A_{3n}(\varvec{a})= & {} \sum _{j\in M_2}P_2(\Vert \widetilde{\varvec{\theta }}_{nj}\Vert ;\lambda _{2n})-\sum _{j\in M_2}P_2(\Vert \varvec{\theta }_{0j}\Vert ;\lambda _{2n}) \\= & {} \sum _{j\in M_2}\left[ P_2'(\Vert \varvec{\theta }_{0j}\Vert ;\lambda _{2n})\frac{\varvec{\theta }_{0j}^T}{\Vert \varvec{\theta }_{0j}\Vert }+o_p(1)\right] \\&\big [(\varvec{\psi }_{q_n,m}(X_j)\varvec{\psi }_{q_n,m}(X_j)^T)^{-1}\varvec{\psi }_{q_n,m}(X_j)\iota _na_j\big ]. \end{aligned}$$

  By Condition (C7), we have

  $$\begin{aligned} |A_{3n}(\varvec{a})|\le & {} d_n P_2'(0+;\lambda _{2n}) \sqrt{\Vert ({\varvec{\psi }}_{q_{n,m}}(X_j){\varvec{\psi }}_{q_{n,m}}(X_j)^T)^{-1}\Vert }\iota _n a_j\\= & {} O_p(d_n^2 n^{-\nu }) O_p(n^{-(1-\nu )/2})=o_p(1). \end{aligned}$$

  Similarly, we can get that $A_{2n}(\varvec{a})\xrightarrow {p}0$.

  Hence, $\widehat{\varvec{a}}_n=\text {argmin}\{V_{1n}(\varvec{a}): \varvec{a}\in {\mathbb {R}}^{|M_2|}\}=\text {argmin}\{A_{1n}(\varvec{a}): \varvec{a}\in {\mathbb {R}}^{|M_2|}\}$ and so we only care about the minimum of $nA_{1n}(\varvec{a})$. Similar to Huang et al. (2010), we have

  $$\begin{aligned} nA_{1n}(\varvec{a})= & {} \varvec{a}^T \big (\sqrt{n}U_n(\varvec{\beta }_0)\big )+ \frac{1}{2}\varvec{a}^T U_n'(\varvec{\beta }_0)\varvec{a}\\\triangleq & {} \varvec{a}^T T_1+ \varvec{a}^T T_2 \varvec{a}. \end{aligned}$$

  It can be seen that $T_2 \xrightarrow {p} \Sigma _2$ and $\varvec{u}\Sigma _3^{-1/2} T_1$ is distributed asymptotically by N(0, 1) for any $\varvec{u}\in {\mathbb {R}}^{|M_2|}$ with $\Vert \varvec{u}\Vert =1$ , where $\Sigma _3=Var(\delta \gamma _0(Y)(Y-g_0(\varvec{X}))\varvec{X}_{M_2}+(1-\delta )\gamma _1(Y)-\gamma _2(Y))$ with the following notations that

  $$\begin{aligned}&\tilde{H}^{11}(\varvec{x},y)=P(\varvec{X}\le \varvec{x}, Y\le y, \delta =1), \quad \tilde{H}^0=P(Y\le y,\delta =0),\\&\gamma _0(y)=\exp \Bigg (\int _0^{y-}\frac{\tilde{H}^0(dw)}{1-H(w)}\Bigg ),\\&\gamma _{1,j}(y)=\frac{1}{1-H(y)}\int I(z>y)e(z,\varvec{x})x_j\gamma _0(z)\tilde{H}^{11}(d\varvec{x},dz),\\&\gamma _{2,j}(y)=\iint \frac{I(v<y,v<z)e(z,\varvec{x})x_j\gamma _0(z)}{[1-H(v)]^2}\tilde{H}^0(dv)\tilde{H}^{11}(d\varvec{x},dz),\\&\gamma _l(y)=(\gamma _{l,j}; j\in M_2), ~~ l=1,2. \end{aligned}$$

  Let $\hat{\varvec{a}}=\text {argmin}\{V_1(\varvec{a})=\varvec{a}^T T_1+\frac{1}{2}\varvec{a}^T \Sigma _2 \varvec{a}: \varvec{a}\in {\mathbb {R}}^{|M_2|}\}$. According to the continuous mapping theorem of Kim and Pollard (1990), $\sqrt{n} \varvec{u}\Sigma ^{-1/2}(\widehat{\varvec{\beta }}_{nM_2}-\varvec{\beta }_{0})$ has the same asymptotical distribution as $\varvec{u}\Sigma ^{-1/2}\hat{\varvec{a}}\xrightarrow {d} N(0,1)$ for any $\varvec{u}\in {\mathbb {R}}^{|M_2|}$ with $\Vert \varvec{u}\Vert =1$, where $\Sigma =\Sigma _2^{-1}\Sigma _3\Sigma _2^{-1}$. This completes the proof of Theorem 2.

$\square$