M-estimators for single-index model using B-spline

Abstract

The single-index model is an important tool in multivariate nonparametric regression. This paper deals with M-estimators for the single-index model. Unlike the existing M-estimator for the single-index model, the unknown link function is approximated by B-spline and M-estimators for the parameter and the nonparametric component are obtained in one step. The proposed M-estimator of unknown function is shown to attain the convergence rate as that of the optimal global rate of convergence of estimators for nonparametric regression according to Stone (Ann Stat 8:1348–1360, 1980; Ann Stat 10:1040–1053, 1982), and the M-estimator of parameter is $\sqrt{n}$-consistent and asymptotically normal. A small sample simulation study showed that the M-estimators proposed in this paper are robust. An application to real data illustrates the estimator’s usefulness.

Appendix

In this section, we prove the results of Theorems 1 and 2. Let

$$\begin{aligned} v_i(\theta _0)&= B^{\prime }(x_i^\tau \beta _0)^\tau \delta _0 x_i,\quad i=1,\ldots , n,\\ V(\theta _0)&= (v_1(\theta _0), \ldots , v_n(\theta _0))^\tau ,\\ Z_n&= (B(x_1^\tau \beta _0),\ldots ,B(x_n^\tau \beta _0))^\tau ,\quad P=Z_n(Z_n^\tau Z_n)^{-1}Z_n^\tau , \end{aligned}$$

$H_n^2=NZ_n^\tau Z_n,\quad z_i=H_n^{-1}B(x_i^\tau \beta _0), \lambda _n$ is the smallest eigenvalue of $N/nH_n^2$.

Lemma 6.1

If the Assumption 1–2 is satisfied, then

$$\begin{aligned} lim_{n\rightarrow \infty }\frac{1}{n} V(\theta _0)^\tau (I-P)V(\theta _0)=\Sigma , \end{aligned}$$

(6.1)

where $I$ is an identity matrix.

Proof

The proof of this Lemma 6.1 is referred to Gao and Liang (1997). $\square$

Lemma 6.2

Under Assumption 1, there exists a constant $c$ such that

$$\begin{aligned} \sup _{x\in X}|g(\beta _0^\tau x)-B(\beta _0^\tau x)^\tau \delta _0|\le ck_n^{-r}, \end{aligned}$$

(6.2)

where $k_n$ is the number of knots in the same order of $N$.

This Lemma justifies the approximation power of B-spline. Its proof follows readily from Corollary 6.21 in Schumaker (1981) .

Lemma 6.3

Assume $lim_{n\rightarrow \infty } n^{\gamma -1}k_n^2=0$ for some $\gamma \ge 0$, then with probability one

$$\begin{aligned} lim \, inf_n\lambda _n&> 0,\end{aligned}$$

(6.3)

$$\begin{aligned} max_{1\le i\le n}|z_i|^2&\le 2(l+3)N/(n\lambda _n). \end{aligned}$$

(6.4)

A complete proof can be found in Shi and Li (1995).

Proof of Theorem 1

By Lemma 6.2, there exists a constant $M$ such that

$$\begin{aligned} g(x_i^\tau \beta _0)=B(x_i^\tau \beta _0)^\tau \delta _0-R_{ni}, \sup _{1\le i\le n}|R_{ni}|\le M k_n^{-r}. \end{aligned}$$

(6.5)

Thus,

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n(\hat{g}(x_i^\tau \beta _0)- g(x_i^\tau \beta _0))^2\le \frac{2}{n}\sum _{i=1}^n (B(x_i^\tau \beta _0)^\tau (\hat{\delta }_0-\delta _0))^2+2M^2k_n^{-2r}.\quad \end{aligned}$$

(6.6)

Then, in order to prove (2.5) hold, it suffices to show that

$$\begin{aligned} \sum _{i=1}^n(B(x_i^\tau \beta _0)^\tau (\hat{\delta }_0-\delta _0))^2=O_p(k_n). \end{aligned}$$

(6.7)

Denote $V^*(\theta _0)=(I-P)V(\theta _0)= (v_1^*(\theta _0),\ldots ,v_n^*(\theta _0))^\tau , S_n=V^*(\theta _0)^\tau V^*(\theta _0)$.

Let

$$\begin{aligned} \begin{aligned} \theta (\beta , \delta )&= \left( \begin{array}{c} \theta _1\\ \theta _2 \end{array}\right) =\left( \begin{array}{c} S_n^{\frac{1}{2}}(\beta -\beta _0)\\ H_n(\delta -\delta _0)N^{-\frac{1}{2}}+H_n^{-1}N^{\frac{1}{2}}Z_n^\tau V(\theta _0)(\beta -\beta _0) \end{array}\right) ,\\ \hat{\theta }&=\left( \begin{array}{c} \hat{\theta }_1\\ \hat{\theta }_2 \end{array}\right) =\theta (\hat{\beta }, \hat{\delta }). \end{aligned} \end{aligned}$$

(6.8)

In order to attain (6.7), we first show that $||\hat{\theta }||=O_p(k_n^{1/2})$. Let $\tilde{v_i}(\theta _0)=S_n^{-\frac{1}{2}}v_i^*(\theta _0), \tilde{B}(x_i^\tau \beta _0)=H_n^{-1}B(x_i^\tau \beta _0)N^{1/2}$.

Then, we have that

$$\begin{aligned} \sum _{i=1}^n\rho (y_i-B(x_i^\tau \beta )^\tau \delta )= \sum _{i=1}^n\rho (e_i-R_{ni}-\tilde{v_i}(\theta _0)^\tau \theta _1 -\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2+R^*(\theta )),\nonumber \\ \end{aligned}$$

(6.9)

which is minimized at $\hat{\theta }$, where $R^*(\theta )=B^{\prime }(\beta _0^\tau x_i)^\tau \delta _0 x_i^\tau S_n^{-1/2}\theta _1-B^{\prime }(x_i^\tau \beta ^*)^\tau \delta x_i^\tau$ $S_n^{-1/2}\theta _1, \beta ^*$ is between $\beta$ and $\beta _0$.

According to the properties of B-spline, there exist constant vectors $s_j j=1, \ldots , N$, for any $h_j(x_i^\tau \beta _0)$,

$$\begin{aligned} h_j(x_i^\tau \beta _0)=B(x_i^\tau \beta _0)^\tau s_j+\tilde{R}_{nij}, \end{aligned}$$

(6.10)

where $\tilde{R}_{nij}$ is the remainder of $h_j(x_i^\tau \beta _0)$ approximating by B-spline. According to Lemma 6.2, $g^{\prime }(x_i^\tau \beta _0)x_i=B^{\prime }(x_i^\tau \beta _0)^\tau \delta _0 x_i+\tilde{R}_{ni}, \sup _{1\le ni}|\tilde{R}_{ni}|\le k_n^{-(r-1)}$ by Assumption 3, we know that there exist matrices $G$ and $W_n$ such that

$$\begin{aligned} V^*(\theta _0)=(I-P)(Z_nG+W_n), \end{aligned}$$

where $W_n=\tilde{R}_n+U_n, \tilde{R}_n=(\tilde{R}_{nij})_{n\times N}, U_n=(u_{ij})_{n\times N}, G=(s_1,\ldots ,s_N)$. Hence, any column of $V^*(\theta _0)$ is of the order of $O_p(n^{1/2})$,

Therefore, according to (6.8), Lemmas 6.1, 6.2 and 6.3

$$\begin{aligned}&|\delta -\delta _0|=|N^{1/2}H_n^{-1}\theta _2-NH_n^{-2}Z_n^\tau V(\theta _0)S_n^{-1/2}\theta _1|\nonumber \\&\quad \le |N^{1/2}H_n^{-1}\theta _2|+|NH_n^{-2}(z_1,\ldots , z_n)V(\theta _0)S_n^{-1/2}\theta _1|\nonumber \\&\quad \le L\lambda _n^{-1/2}(N^{1/2}n^{-1/2}+Nn^{-1/2}). \end{aligned}$$

(6.11)

Thus, when $|\theta |\le L, \delta =\delta _0+O_p(Nn^{-1/2})$. Because the derivative of $g(\cdot )$ is continuous and bounded, by Lemma 6.2, have

$$\begin{aligned}&|B^{\prime }(\beta _0^\tau x_i)^\tau \delta _0-B^{\prime }(\beta ^{*\tau } x_i)^\tau \delta |=|B^{\prime }(\beta _0^\tau x_i)^\tau \delta _0-B^{\prime }(\beta ^{*\tau } x_i)^\tau \delta _0\nonumber \\&\qquad +O_p(Nn^{-1/2})| \le |B^{\prime }(\beta _0^\tau x_i)^\tau \delta _0-B^{\prime }(\beta ^{*\tau } x_i)^\tau \delta _0|+O_p(Nn^{-1/2})\nonumber \\&\quad =|g^{\prime }(\beta _0^\tau x_i)+\tilde{R}_{ni}-g^{\prime }(\beta ^{*\tau } x_i)-\tilde{R}_{ni}^*|+O_p(Nn^{-1/2})=o_p(1). \end{aligned}$$

(6.12)

As the support of $X$ is convex, so when $|\theta |\le L$

$$\begin{aligned} |R_i^*(\theta )|&= |B^{\prime }(\beta _0^\tau x_i)^\tau \delta _0x_i^{\tau }S_n^{-1/2} \theta _1-B^{\prime } (\beta _0^{*\tau }x_i)^\tau \delta x_i^\tau S_n^{-1/2}\theta _1|\nonumber \\&= |B^{\prime }(\beta _0^\tau x_i)^\tau \delta _0-B^{\prime }(\beta _0^{*\tau }x_i)^\tau \delta ||x_i^\tau S_n^{-1/2}\theta _1|=O_p(n^{-1/2}). \end{aligned}$$

(6.13)

Therefore

$$\begin{aligned} \sup _i\sup _{||\theta ||\le L} max\left\{ k_n^{1/2}|\tilde{v_i} (\theta _0)^\tau \theta _1|,k_n^{1/2}|\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2|,|R_{ni}|, |R^*(\theta )|\right\} =o_p(1).\nonumber \\ \end{aligned}$$

(6.14)

According to Assumption 4–5, by similar argument to those for the below Lemma 6.4 of this paper, we have for any $L>0$, that

$$\begin{aligned}&\sup _{||\theta ||\le L}k_n^{-1}\left| \sum _{i=1}^n[\rho (e_i-k_n^{1/2} \tilde{v_i}(\theta _0)^\tau \theta _1-k_n^{1/2}\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni}+R^*(\theta ))\right. \nonumber \\&\quad -\rho (e_i-R_{ni})-E\{\rho (e_i-k_n^{1/2}\tilde{v_i} (\theta _0)^\tau \theta _1-k_n^{1/2}\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni}+R^*(\theta ))\nonumber \\&\quad \left. -\rho (e_i-R_{ni})\}+(k_n^{1/2}\tilde{v_i} (\theta )^\tau \theta _1+k_n^{1/2}\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2+R^*(\theta ))\psi (e_i)]\right| =o_p(1).\nonumber \\ \end{aligned}$$

(6.15)

Note that $|k_n^{-1/2}\sum _{i=1}^n\tilde{v_i}(\theta _0)R_{ni}b| =||k_n^{-1/2}S_n^{-\frac{1}{2}}(W_n^\tau (I-P)+G^\tau Z_n^\tau (I-P))r_n||=o_p(1),$ where $r_n$ is a vector formed by $R_{ni}b$ and $R^*(\theta )=O_p(n^{-1/2})=o_p(R_{ni})$ for any $L>0$.

By Assumption 4–5 and the fact $\sum _{i=1}^n\tilde{v_i} (\theta _0)\tilde{B}(x_i^\tau \beta _0)^\tau =0$, we have

$$\begin{aligned}&k_{n}^{-1}\sum _{i=1}^nE[\rho (e_i-(k_n^{1/2} \tilde{v_i} (\theta _0)^\tau \theta _1+k_n^{1/2}\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2+R^*(\theta ))-R_{ni})\nonumber \\&\qquad -\rho (e_i-R_{ni})]= k_n^{-1}\sum _{i=1}^nE\int \limits _{-R_{ni}}^{-(k_n^{1/2} \tilde{v_i}(\theta _0)^\tau \theta _1+k_n^{1/2}\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2+R_{ni}+R^*(\theta ))}\psi (e_i+s)ds\nonumber \\&\quad =k_n^{-1}\sum _{i=1}^n[k_n\{(\tilde{v_i}(\theta _0)^\tau \theta _1)^2+(\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2)^2\} b/2+(k_n^{1/2}\tilde{v_i}(\theta _0)^\tau \theta _1+k_n^{1/2 }\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2\nonumber \\&\qquad +R^*(\theta ))R_{ni}b]\{1+o(1)\}\!=\!\frac{1}{2}\left( b\theta _1^\tau \theta _1+\theta _2^\tau \sum _{i=1}^n\tilde{B}(x_i^\tau \beta _0) \tilde{B}(x_i^\tau \beta _0)^\tau \theta _2)+0(||\theta ||^2\right) . \nonumber \\ \end{aligned}$$

(6.16)

By the Tchebychev inequality and Assumption 4, we obtain

$$\begin{aligned} P\left( \Bigg |\sum _i^n\tilde{v_i}(\theta _0)^\tau \theta _1\psi (e_i)\Bigg |\ge Lk_n^{-1/2}\right)&< \frac{E(\sum _i^n\tilde{v_i}(\theta _0)^\tau \theta _1\psi (e_i))^2}{L^2k_n} \\&= \frac{||\theta _1||^2E\psi (e_1)^2}{L^2k_n}\rightarrow 0. \end{aligned}$$

Hence, $\sup _{||\theta ||=L} k_n^{-1/2}|\sum _{i=1}^n \tilde{v_i}(\theta _0)^\tau \theta _1\psi (e_i)|=O_p(L)$.

Similarly, $\sup _{||\theta ||=L} k_n^{-1/2}|\sum _{i=1}^n \tilde{B}(x_i^\tau \beta _0)^\tau \theta _2\psi (e_i)|=O_p(L)$.

Thus, it follows from (6.13) that, for sufficiently large $L$,

$$\begin{aligned}&\inf _{||\theta ||=L}k_n^{-1}\sum _{i=1}^n[E\{\rho (e_i-k_n^{1/2} \tilde{v_i}(\theta _0)^\tau \theta _1-k_n^{1/2}\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni}+R^*(\theta ))\nonumber \\&\quad -\rho (e_i-R_{ni})\}-(k_n^{1/2}\tilde{v_i}(\theta )^\tau \theta _1+k_n^{1/2}\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2+R^*(\theta ))\psi (e_i)]>0,\nonumber \\ \end{aligned}$$

(6.17)

with probability tending to 1 as $n\rightarrow \infty$. Combining (6.12) and (6.14) yields

$$\begin{aligned} \inf _{||\theta ||\!=\!L}\sum _{i=1}^n\rho (e_i\!-\!k_n^{1/2} \tilde{v_i} (\theta _0)^\tau \theta _1-k_n^{1/2} \tilde{B} (x_i^\tau \beta _0)^\tau \theta _2\!-\! R_{ni}+ R^*(\theta ))\!>\! \sum _{i=1}^n\rho (e_i-R_{ni}), \end{aligned}$$

which implies,by the convexity of $\rho$, that

$$\begin{aligned} \inf _{||\theta ||\ge Lk_n^{1/2}}\sum _{i=1}^n\rho (e_i- \tilde{v_i}(\theta _0)^\tau \theta _1-\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni}+R^*(\theta ))>\sum _{i=1}^n\rho (e_i-R_{ni}). \end{aligned}$$

We then conclude that $||\hat{\theta }||=O_p(k_n^{1/2})$. By (6.8), we have

$$\begin{aligned} |S_n^{1/2}(\hat{\beta }_0-\beta _0)|^2=(\hat{\beta }_0-\beta _0)^{\tau }S_n (\hat{\beta }_0-\beta _0)=O_p(k_n). \end{aligned}$$

Thus

$$\begin{aligned}&|\hat{\beta }_0-\beta _0|=O_p(n^{-r/(2r+1)}),\\&\sum _{i=1}^n(B(x_i^\tau \beta _0)^\tau (\hat{\delta }_0-\delta _0))^2 = O_p(k_n). \end{aligned}$$

Hence,

$$\begin{aligned}&\sum _{i=1}^n(B(x_i^\tau \hat{\beta }_0)^\tau (\hat{\delta }_0-\delta _0))^2 \le 2\sum _{i=1}^n\{[B(\hat{\beta }_0^\tau x_i)-B(\beta _0^\tau x_i)](\hat{\delta }_0-\delta _0)\}^2\\&\qquad +2\sum _{i=1}^n(B(\beta _0^\tau x_i)(\hat{\delta }_0-\delta _0))^2\\&\quad =2\sum _{i=1}^n[B^{\prime }(\tilde{\beta })^\tau (\hat{\delta }_0-\delta _0) x_i^\tau (\hat{\beta }_0-\beta _0)]+O_p(k_n)=O_p(k_n). \end{aligned}$$

As the derivative of $g(\cdot )$ is bounded and the support of $X$ is compact, so

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^n(\hat{g}(\hat{\beta }_0^\tau x_i)-g(\beta _0^\tau x_i))^2=\frac{1}{n}\sum _{i=1}^n [\hat{g}(\hat{\beta }_0^\tau x_i)-g(\hat{\beta }_0^\tau x_i)+g(\hat{\beta }_0^\tau x_i)\nonumber \\&\qquad -g(\beta _0^\tau x_i))^2 \le \frac{2}{n}\sum _{i=1}^n[\hat{g}(\hat{\beta }_0^\tau x_i)-g(\hat{\beta }_0^\tau x_i)]^2+\frac{2}{n}\sum _{i=1}^n[ g(\hat{\beta }_0^\tau x_i)-g(\beta _0^\tau x_i)]^2\nonumber \\&\quad =\frac{2}{n}\sum _{i=1}^n[\hat{g}(\hat{\beta }_0^\tau x_i)-g(\hat{\beta }_0^\tau x_i)]^2+\frac{2}{n}\sum _{i=1}^n g^{\prime }(\beta _0^{*\tau }x_i)^2|x_i||\hat{\beta }_0-\beta _0|^2=O_p (k_n^{-2r}).\nonumber \\ \end{aligned}$$

(6.18)

The proof of Theorem 1 is completed.

To obtain asymptotic normality of $\hat{\beta }_0$, we shall make use of several asymptotic linearization results which are similar to Lemmas 6.3 and 6.4 of He and Shi (1996). The proofs is similar to the argument in He and Shi (1996).

Lemma 6.4

Under the Assumption of Theorem 2, we have for any $L>0$ and $M_0>0$

$$\begin{aligned}&\sup _{|\theta _1|\le M_0, |\theta _2|\le L k_n^{1/2}}\left| \sum _{i=1}^n[\rho (e_i-\tilde{v_i} (\theta _0)^\tau \theta _1-\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni})\right. \nonumber \\&\quad -\rho (e_i-\tilde{B}(x_i^\tau \beta _0) ^\tau \theta _2-R_{ni})+\tilde{v_i}(\theta _0)\theta _1\psi (e_i) -E_e(\rho (e_i-\tilde{v_i}(\theta _0)^\tau \theta _1\nonumber \\&\quad \left. -\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni})-\rho (e_i-\tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni}))]\right| =o_p(1),\qquad \end{aligned}$$

(6.19)

where $E_e$ is the expectation with respect to $e$.

Proof

Because $f_e(\cdot )$ is continuous at 0, there exist two positive numbers $\omega _1,\omega _2$ such that when $|t|\le \omega _2$, $|f_e(t)|<\omega _1$ holds. By (6.13), (6.14) and Lemma 6.2, have

$$\begin{aligned} d_n&= k_n^{1/2}|\tilde{v}_i(\theta _0)^\tau \theta _1+\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta )|\le k_n^{1/2}|\tilde{v}_i(\theta _0)^\tau \theta _1|\\&+k_n^{1/2}[|\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2|+|R_{ni}|+|R_i^*(\theta )|]=O_p(n^{-1/2}k_n)=O_p(1). \end{aligned}$$

In order to prove the (6.19), it suffices to show that given $\varepsilon >0$,

$$\begin{aligned}&P\biggr \{\sup _{||\theta _1||\le 1,||\theta _2||\le L}\left| \sum _{i=1}^n [\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2} \tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}\right. \nonumber \\&\quad +R_i^*(\theta ))-\rho (e_i+R_i^*(\theta )-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_ni)+M\tilde{v}_i(\theta _0)\theta _1\psi (e_i)\nonumber \\&\quad -E_e(\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\nonumber \\&\quad \left. -\rho (e_i+R_i^*(\theta )-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_ni))]\right| \ge \varepsilon , d_n\le \omega _2\biggr \}\rightarrow 0.\qquad \quad \qquad \end{aligned}$$

(6.20)

Denote $\Gamma =\{\theta _1:|\theta _1|\le 1, \theta _1\in R^{p}\}$ as a union of $K_n$ disjoint parts $\Gamma _1,\ldots ,\Gamma _{k_n}$ such that the diameter of each part does not exceed $q_0=k_n\varepsilon /(8b_2M\omega _2n)$. Then $K_n\le (2\sqrt{p}/q_0+1)^p$. Choose $\theta _{1j}\in \Gamma _j$, for each of $j=1,\ldots , K_n$, by Assumptions 5–6 and Lemma 6.3, we have

$$\begin{aligned}&\min _{1\le j\le K_n}\Bigg |\sum _{i=1}^n[\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\nonumber \\&\qquad \!-\!\rho (e_i\!-\!Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2\!-\!R_{ni}\!+\!sR_i^*(\theta ))\!+\!M\tilde{v}_i(\theta _0)^\tau \theta _1\psi (e_i)\!-\!\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _{1j}\nonumber \\&\qquad -Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))+ \rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\nonumber \\&\qquad -M\tilde{v}_i(\theta _0)^\tau \theta _{1j}\psi (e_i)- E(\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2} \tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta )) \nonumber \\&\qquad -\rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta )))+E(\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _{1j}\nonumber \\&\qquad -Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))- \rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2\nonumber \\&\qquad -R_{ni}+R_i^*(\theta )))]\Bigg |I(d_n\le \omega _2) \le n(b_2\omega _2M\max _{1\le i\le n}|\tilde{v}_i(\theta _0)|\min _{1\le j\le K_n}|\theta _1-\theta _{1j}|\nonumber \\&\qquad +bM\omega _2\max _{1\le i\le n}|\tilde{v}_i(\theta _0)|\min _{1\le j\le K_n}|\theta _1-\theta _{1j}|\max _{1\le i\le n}(M|\tilde{v}_i(\theta _0)|+Lk_n^{1/2}|\tilde{B}(\beta _0^\tau x_i)||\theta _2| \nonumber \\&\qquad +|R_{ni}|+|R_i^*(\theta )|) \le 2b_2nM\omega _2\max _{1\le i\le n}|\tilde{v}_i(\theta _0)|\min _{1\le j\le K_n}|\theta _1-\theta _{1j}|\nonumber \\&\quad \le 2b_2nM\omega _2\max _{1\le i\le n}|\tilde{v}_i(\theta _0)|k_n\varepsilon /8b_2M\omega _2n<\varepsilon /4, \end{aligned}$$

(6.21)

where $I(\cdot )$ is a indicator function. According to Assumption 6 and the mean theorem of differential, have

$$\begin{aligned}&\sup _{\theta _1\in \Gamma }|\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\nonumber \\&\qquad -\rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))+\tilde{v}_i(\theta _0)^\tau \theta _1\psi (e_i)|I(d_n\le \omega _2) \nonumber \\&\quad =\sup _{\theta _1\in \Gamma }|\psi (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1^*-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\tilde{v}_i(\theta _0)^\tau \theta _1M\nonumber \\&\qquad -\tilde{v}_i(\theta _0)^\tau \theta _1\psi (e_i)|I(d_n\le \omega _2)\le b_2\max _{1\le i\le n}|\tilde{v}_i(\theta _0)|M\omega _2\le b_2\omega _2Mk_n^{1/2}n^{-1/2}. \nonumber \\ \end{aligned}$$

(6.22)

It follows the Assumption 6 again that,

$$\begin{aligned}&|\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\nonumber \\&\qquad -\rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))+M\tilde{v}_i(\theta _0)^\tau \theta _1\psi (e_i)|\nonumber \\&\quad \le b_2|\tilde{v}_i(\theta _0)^\tau \theta _1|MI(|e_i|\le |\tilde{v}_i(\theta _0)|M+Lk_n^{1/2}|\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2|+|R_{ni}|+|R_i^*(\theta )|).\nonumber \\ \end{aligned}$$

(6.23)

Therefore, given $X^{*}=(X_1,\ldots ,X_n)$, by (6.23), we have

$$\begin{aligned}&\sum _{i=1}^n Var[\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\nonumber \\&\qquad \!-\!\rho (e_i\!-\!Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}\!+\!R_i^*(\theta ))\!+\!M\tilde{v}_i(\theta _0)^\tau \theta _1\psi (e_i)|X^*] \nonumber \\&\quad \le b_2^2M^2\omega _2^2k_nn^{-1}. \end{aligned}$$

(6.24)

By (6.21), (6.22), (6.23), (6.24) and Bernstein inequality, given any $\varepsilon >0$, we have

$$\begin{aligned}&P\left\{ \sup _{|\theta _1|\le 1}\left| \sum _{i=1}^n [\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\right. \right. \nonumber \\&\qquad -\rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))+M\tilde{v}_i(\theta _0)^\tau \theta _1\psi (e_i)\nonumber \\&\qquad -E_{e_i}(\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\nonumber \\&\qquad -\left. \rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta )))]\Bigg |\le \varepsilon , d_n\le \omega _2|X^*\right\} \nonumber \\&\quad \le \sum _{j=1}^{K_n}P\left\{ \Bigg |\sum _{i=1}^n[\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\right. \nonumber \\&\qquad -\rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))+M\tilde{v}_i(\theta _0)^\tau \theta _1\psi (e_i)\nonumber \\&\qquad -E_{e_i}(\rho (e_i-M\tilde{v}_i(\theta _0)^\tau \theta _1-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))\nonumber \\&\qquad \left. -\rho (e_i-Lk_n^{1/2}\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta )))]\Bigg |\le \varepsilon /2, d_n\le \omega _2|X^*\right\} \nonumber \\&\quad \le 2K_n exp\left\{ -n(\varepsilon /2n)^2/2\left( \frac{b_2^2M^2 \omega _1\omega _2k_n^{1/2}n^{-1/2}}{n}+\frac{3M^3b_1b_2\omega _2k_n n^{-1/2}}{n}\right) \right\} \nonumber \\&\quad =2exp\left\{ -\gamma \frac{n^{1/2}}{k_n^{1/2}}\left( 1-\frac{k_n^{1/2}}{n^{1/2}\gamma }pln\left( 2\sqrt{p}/q_{0}+1\right) \right) \right\} , \end{aligned}$$

(6.25)

where $\gamma =\varepsilon ^2/8b_2^2M^2\omega _1\omega _2$. As $k_n=n^{1/(2r+1)}$, so (6.19) holds. The Lemma 6.4 is shown. $\square$

Lemma 6.5

If Assumptions of Theorem 2 hold, then

$$\begin{aligned}&\sum _{i=1}^nE_e(\rho (e_i-\tilde{v_i}(\theta _0)^\tau \theta _1- \tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni})\\&\quad -\rho (e_i- \tilde{B}(x_i^\tau \beta _0)^\tau \theta _2-R_{ni})) =\frac{b\theta _1^\tau \theta _1}{2}+r_n(\theta _1,\theta _2), \end{aligned}$$

where $\sup _{|\theta _1|\le M_0,|\theta _2\le Lk_n^{1/2}}|r(\theta _1,\theta _2)|=o_p(1)$.

The proof of Lemma 6.5 is similar to those of (6.16), so the proof is omitted.

Proof of Theorem 2

Let $\hat{\theta }^\tau =(\hat{\theta }_1^\tau , \hat{\theta }_2^\tau )$ as $\hat{\theta }$ in the proof of Theorem 1, and $\tilde{\theta _1}=b^{-1}\sum _{i=1}^n\tilde{v_i}(\theta _0)\psi (e_i)$. Following Lemmas 6.4, 6.5 and triangle inequality, for any $L>0$ and $\varepsilon >0$, we obtain

$$\begin{aligned}&\sup _{|\theta _1-\tilde{\theta }_1|=\varepsilon ,\theta _1\in R^p}I(|\tilde{\theta }_1|\le L,|\theta _2|\le Lk_n^{1/2}) \left| \sum _{i=1}^n(\rho (e_i-\tilde{v}_i(\theta _0)^\tau \theta _1\!-\!\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2\right. \nonumber \\&\qquad \left. \!-\!R_{ni} \!+\!R_i^*(\theta ))\!-\!\rho (e_i\!-\!\tilde{v}_i(\theta _0)^\tau \tilde{\theta }_1-\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni}+R_i^*(\theta )))\!-\!\frac{b\varepsilon ^2}{2} \right| \nonumber \\&\quad \!\le \! 2\sup _{|\theta _1|\!\le \! L\!+\!\varepsilon ,|\theta _2|\le Lk_n^{1/2}}\left| \sum _{i=1}^n(\rho (e_i-\tilde{v}_i(\theta _0)^\tau \theta _1-\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni} +R_i^*(\theta ))\right. \nonumber \\&\quad \quad -\rho (e_i-\tilde{v}_i(\theta _0)^\tau \tilde{\theta }_1-\tilde{B}(\beta _0^\tau x_i)^\tau \theta _2-R_{ni}+R_i^*(\theta ))+\tilde{v}_i(\theta _0)^\tau \theta _1\psi (e_i))\nonumber \\&\quad \quad -\frac{b\theta _1^\tau \theta }{2}\Bigg | =o_p(1). \end{aligned}$$

(6.26)

Notice that when $L\rightarrow \infty , P\{|\tilde{\theta }_1|\le L\}\rightarrow 1, P\{|\hat{\theta }_2|\le Lk_n^{1/2}\}\rightarrow 1$ hold. Then

$$\begin{aligned}&\lim _{n\rightarrow \infty }P\left\{ \sup _{|\theta -\tilde{\theta }|=\varepsilon , \theta \in R^p}\left| \sum _{i=1}^n(\rho (e_i-\tilde{v}_i(\theta _0)^\tau \theta _1-\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni} +R_i^*(\theta ))\right. \right. \\&\left. \left. \quad -\rho (e_i-\tilde{v}_i(\theta _0)^\tau \tilde{\theta }_1-\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni}+R_i^*(\theta )))- \frac{b\varepsilon ^2}{2} \right| \le \varepsilon \right\} =0. \end{aligned}$$

Therefore, when $|\theta _1-\tilde{\theta }_1|=\varepsilon ,\theta _1\in R^p$, we obtain

$$\begin{aligned}&\sum _{i=1}^n\rho (e_i-\tilde{v}_i(\theta _0)^\tau \theta _1-\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni} +R_i^*(\theta ))\\&\quad =\sum _{i=1}^n\rho (e_i-\tilde{v}_i(\theta _0)^\tau \tilde{\theta }_1 -\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni}+R_i^*(\theta ))) +\frac{b\varepsilon ^2}{2}+o_p(1). \end{aligned}$$

It follows from the above formula that

$$\begin{aligned}&\lim _{n\rightarrow \infty }P\left\{ \inf _{|\theta -\tilde{\theta }|=\varepsilon , \theta \in R^p}\sum _{i=1}^n\rho (e_i-\tilde{v}_i(\theta _0)^\tau \theta _1-\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni}\right. \nonumber \\&\left. \quad +R_i^*(\theta )) >\sum _{i=1}^n\rho (e_i-\tilde{v}_i(\theta _0)^\tau \tilde{\theta }_1-\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni}+R_i^*(\theta ))\right\} =1.\qquad \qquad \quad \end{aligned}$$

(6.27)

According to Corollary 25 of Eggleston (1958), we have

$$\begin{aligned}&\lim _{n\rightarrow \infty }P\left\{ \inf _{|\theta -\tilde{\theta }|\ge \varepsilon ,\theta \in R^p}\sum _{i=1}^n\rho (e_i-\tilde{v}_i(\theta _0)^\tau \theta _1-\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni}\right. \nonumber \\&\left. \quad +R_i^*(\theta )) >\sum _{i=1}^n\rho (e_i-\tilde{v}_i(\theta _0)^\tau \tilde{\theta }_1-\tilde{B}(\beta _0^\tau x_i)^\tau \hat{\theta }_2-R_{ni}+R_i^*(\theta ))\right\} =1.\nonumber \\ \end{aligned}$$

(6.28)

By the definition of $\hat{\theta }$, we obtain

$$\begin{aligned} \hat{\theta }_1=\tilde{\theta }_1+o_p(1)= \frac{1}{b}\sum _{i=1}^n\tilde{v_i}(\theta _0)\psi (e_i)+o_p(1). \end{aligned}$$

(6.29)

According to Lemma 6.1, the central limit theorem and Slutsky’s theorem, we have

$$\begin{aligned} \sqrt{n}(\hat{\beta }_0-\beta _0)\rightarrow N(0,\sigma ^2\Sigma ^{-1}). \end{aligned}$$