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Composite quantile regression for single-index models with asymmetric errors

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Abstract

For the single-index model, a composite quantile regression technique is proposed in this paper to construct robust and efficient estimation. Theoretical analysis reveals that the proposed estimate of the single-index vector is highly efficient relative to its corresponding least squares estimate. For the single-index vector, the proposed method is always valid across a wide spectrum of error distributions; even in the worst case scenario, the asymptotic relative efficiency has a lower bound 86.4 %. Meanwhile, we employ weighted local composite quantile regression to obtain a consistent and robust estimate for the nonparametric component in the single-index model, which is adapted to both symmetric and asymmetric distributions. Numerical study and a real data analysis can further illustrate our theoretical findings.

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Acknowledgments

The research was supported by the National Natural Science Foundation of China (NSFC), Tianyuan fund for Mathematics (11426126), the Natural Science Foundation of Shandong Province, China (ZR2014AP007), and the Doctoral Scientific Research Foundation of Ludong University (LY2014001).

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Correspondence to Jing Sun.

Appendix: Proofs

Appendix: Proofs

For the asymptotic analysis, we need the following regularity conditions.

  1. (C1)

    The kernel \(K(\cdot )\) is a symmetric density function with bounded support and satisfies a Lipschitz condition.

  2. (C2)

    The density function of \(\varvec{X}^T \varvec{\beta }\) is positive and uniformly continuous for \(\varvec{\beta }\) in a neighborhood of \(\varvec{\beta }_0\). Further, the density of \(\varvec{X}^T \varvec{\beta }_0\) is continuous and bounded away from 0 and \(\infty \) on its support.

  3. (C3)

    The function \(g(\cdot )\) has a continuous and bounded second derivative.

  4. (C4)

    Assume that \(n\rightarrow \infty \), \(h\rightarrow 0\) and \(nh\rightarrow \infty \).

  5. (C5)

    The error \(\varepsilon \) has a positive density \(f(\cdot )\), which has finite Fisher information, that is, \(\int f(x)^{-1}[f'(x)]^2\mathrm {d}x<\infty \).

These conditions are quite mild and can be satisfied in many practical situations. The assumptions on the error in (C5) are the same as those for multiple linear rank regression Hettmansperger and McKean (2011). (C1)–(C4) are standard conditions, which are commonly used in the single-index regression model (Wu et al. 2010).

Throughout the appendix, we use the following notations for ease of exposition. Let \(\mathcal {T}\) be the \(\sigma \)-field generated by \(\{\varvec{X}_i,i=1,\ldots ,n\}\). Let \(\varvec{e}_k\) be a q-dimensional column vector with 1 at the kth position and 0 elsewhere.

Proof of Theorem 1

The proof follows along the same lines of the proof of both Theorem 1 of Fan and Zhu (2012) and Theorem 3.1 of Kai et al. (2011), although part of details differs much. To make it clear, we divide the proof into three steps.

Step (i)   For any given point \(\varvec{X}_0\), denote \(\varvec{a}_0=(a_{10},a_{20},\ldots ,a_{q0})^T\), \(\varvec{\theta }=(\varvec{a}_0^T,\varvec{b}_0^T)^T\) and \(\varvec{X}_{i0}=\varvec{X}_i-\varvec{X}_0\). The initial loss function is as follows:

$$\begin{aligned} L_n(\varvec{\theta })=\sum _{k=1}^q\sum _{i=1}^n\rho _{\tau _k} \{Y_i-a_{k0}-\varvec{X}_{i0}^T\varvec{b}_0\}K\big (\frac{\varvec{X}_{i0}^T\tilde{\varvec{\beta }}}{h}\big ). \end{aligned}$$

Define

$$\begin{aligned} \varvec{\theta }^{*}= & {} \sqrt{nh}\,(a_{10}-c_1-g(\varvec{X}_0^T \varvec{\beta }_0),\,a_{20}-c_2-g(\varvec{X}_0^T \varvec{\beta }_0),\,\ldots ,a_{q0}-c_q-g(\varvec{X}_0^T \varvec{\beta }_0),\\&(\varvec{b}_0-g'(\varvec{X}_0^T \varvec{\beta }_0)\varvec{\beta }_0)^T)^T,\\ \varvec{V}_{ik}^{*}= & {} \varvec{e}_k^T,\varvec{X}_{i0}^T)^T,\quad \Delta _{ik}^{*}=\frac{1}{\sqrt{nh}}(\varvec{V}_{ik}^{*})^T\varvec{\theta }^{*},\\ r_{i0}= & {} g(\varvec{X}_i^T \varvec{\beta }_0)-g(\varvec{X}_0^T \varvec{\beta }_0)-g'(\varvec{X}_0^T \varvec{\beta }_0)\varvec{X}_{i0}^T\varvec{\beta }_0,\\ \eta _{ik}^{0}= & {} I\{\varepsilon _i\le c_k-r_{i0}\}-\tau _k,\quad \eta _{ik}=I\{\varepsilon _i\le c_k\}-\tau _k. \end{aligned}$$

With the above notations, we can rewrite \(L_n(\varvec{\theta })\) as

$$\begin{aligned} \sum _{k=1}^q\sum _{i=1}^n[\rho _{\tau _k} \{\varepsilon _i-c_k+r_{i0}-\Delta _{ik}^{*}\}- \rho _{\tau _k} \{\varepsilon _i-c_k+r_{i0}\}]K\big (\frac{\varvec{X}_{i0}^T\tilde{\varvec{\beta }}}{h}\big )=L_n^{*}(\varvec{\theta }^{*})+R_n^{*}, \end{aligned}$$

where

$$\begin{aligned} L_n^{*}(\varvec{\theta }^{*})= & {} \sum _{k=1}^q\sum _{i=1}^n[\rho _{\tau _k} \{ \varepsilon _i-c_k+r_{i0}-\Delta _{ik}^{*}\}-\rho _{\tau _k} \{\varepsilon _i-c_k+r_{i0}\}] K\big (\frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big )\\ R_n^{*}= & {} \sum _{k=1}^q\sum _{i=1}^n[\rho _{\tau _k} \{ \varepsilon _i-c_k+r_{i0}-\Delta _{ik}^{*}\}-\rho _{\tau _k} \{\varepsilon _i-c_k+r_{i0}\}] D_i\big (\frac{\varvec{X}_{i0}}{h}\big )^T (\tilde{\varvec{\beta }}-\varvec{\beta }_0) \end{aligned}$$

with \(D_i's\) being uniformly bounded due to the Lipschitz continuity of the kernel. Note that for each fixed \(\varvec{\theta }^{*}\), \(R_n^{*}=O_p(\Vert \tilde{\varvec{\beta }}-\varvec{\beta }_0\Vert )=o_p(1)\). Thus we only need to study the main term \(L_n^{*}(\varvec{\theta }^{*})\). By applying the identity (Knight 1998),

$$\begin{aligned} \rho _\tau (x-y)-\rho _\tau (x)=y[I\{x\le 0\}-\tau ]+\int _0^y[I\{x\le z\}-I\{x\le 0\}]\,\mathrm {d}z, \end{aligned}$$

we have

$$\begin{aligned} L_n^{*}(\varvec{\theta }^{*})=(\varvec{W}_n^{*})^T \varvec{\theta }^{*}+\sum _{k=1}^q B_{nk}^{*}(\varvec{\theta }^{*}), \end{aligned}$$

where

$$\begin{aligned} \varvec{W}_n^{*}= & {} \frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{i=1}^n K\big (\frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big )\varvec{V}_{ik}^{*}\eta _{ik}^0,\\ B_{nk}^{*}(\varvec{\theta }^{*})= & {} \sum _{i=1}^n K\big (\frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big ) \int _0^{\Delta _{ik}^{*}} [I\{\varepsilon _i\le c_k-r_{i0}+t\}-I\{\varepsilon _i\le c_k-r_{i0}\}]\,\mathrm {d}t. \end{aligned}$$

Since \(B_{nk}^{*}(\varvec{\theta }^{*})\) is a summation of i.i.d. random variables of the kernel form, according to Lemma 7.1 of Kai et al. (2011), we have \(B_{nk}^{*}(\varvec{\theta }^{*})=E[B_{nk}^{*}(\varvec{\theta }^{*})]+O_p(\log ^{1/2}(1/h)/\sqrt{nh})\). The conditional expectation of \(\sum _{k=1}^qB_{nk}^{*}(\varvec{\theta }^{*})\) can be calculated as

$$\begin{aligned}&\sum _{k=1}^q E[B_{nk}^{*}(\varvec{\theta }^{*})|\mathcal {T}\,]=\sum _{k=1}^q\sum _{i=1}^n K\big ( \frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big ) \int _0^{\Delta _{ik}^{*}}[F(c_k-r_{i0}+t)-F(c_k-r_{i0})]\mathrm {d}t\\&\quad =\sum _{k=1}^q\sum _{i=1}^n K\big (\frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big )f(c_k-r_{i0})\frac{1}{2}(\Delta _{ik}^{*})^2 +O_p(\log ^{1/2}(1/h)/\sqrt{nh})\\&\quad =\frac{1}{2}(\varvec{\theta }^{*})^T \varvec{S}_n^{*}\varvec{\theta }^{*}+O_p(\log ^{1/2}(1/h)/\sqrt{nh}), \end{aligned}$$

where \(\varvec{S}_n^{*}=\frac{1}{nh}\sum _{k=1}^q\sum _{i=1}^n K(\varvec{X}_{i0}^T\varvec{\beta }_0/h) f(c_k-r_{i0})\varvec{V}_{ik}^{*}(\varvec{V}_{ik}^{*})^T\). Then we have

$$\begin{aligned} L^{*}_n(\varvec{\theta }^{*}) =(\varvec{W}_n^{*})^T \varvec{\theta }^{*}+\dfrac{1}{2}(\varvec{\theta }^{*})^T E[\varvec{S}_n^{*}]\varvec{\theta }^{*}+O_p(\log ^{1/2}(1/h)/\sqrt{nh}). \end{aligned}$$
(9)

Note that

$$\begin{aligned}&E[\varvec{S}_n^{*}]=E\Big [\frac{1}{h}\sum _{k=1}^q K\big (\frac{\varvec{X}_{10}^T\varvec{\beta }_0}{h}\big )f(c_k-r_{10})\varvec{V}_{1k}^{*}(\varvec{V}_{1k}^{*})^T\Big ]\\&\quad =E\bigg \{\frac{1}{h}K\big (\frac{\varvec{X}_{10}^T\varvec{\beta }_0}{h}\big ) \sum _{k=1}^q f(c_k-r_{10}) \left[ \begin{array}{cc} \varvec{e}_k\varvec{e}_k^T &{} \varvec{e}_k\varvec{X}_{10}^T \\ \varvec{X}_{10}\varvec{e}_k^T &{} \varvec{X}_{10}\varvec{X}_{10}^T \\ \end{array}\right] \bigg \}. \end{aligned}$$

By conditioning on \(\varvec{X}_{10}^T\varvec{\beta }_0\), we can calculate

$$\begin{aligned}&E\Big [\frac{1}{h}\sum _{k=1}^q K\big (\frac{\varvec{X}_{10}^T\varvec{\beta }_0}{h}\big )f(c_k-r_{10})\varvec{e}_k\varvec{e}_k^T\Big ]\\&\quad =f_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\sum _{k=1}^q f(c_k)\varvec{e}_k\varvec{e}_k^T(1+O(h^2)),\\&\quad E\Big [\frac{1}{h}\sum _{k=1}^q K\big (\frac{\varvec{X}_{10}^T\varvec{\beta }_0}{h}\big )f(c_k-r_{10})\varvec{e}_k\varvec{X}_{10}^T\Big ]\\&\quad =f_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\sum _{k=1}^q f(c_k)\varvec{e}_k(\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)-\varvec{X}_0)^T(1+O(h^2)),\\&E\Big [\frac{1}{h}\sum _{k=1}^q K\big (\frac{\varvec{X}_{10}^T\varvec{\beta }_0}{h}\big )f(c_k-r_{10})\varvec{X}_{10}\varvec{X}_{10}^T\Big ]\\&\quad =f_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\kappa (\varvec{\omega }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\varvec{X}_0^T-\varvec{X}_0(\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0))^T+\varvec{X}_0\varvec{X}_0^T)\\&\quad \times (1+O(h^2)). \end{aligned}$$

Thus we obtain \(E[\varvec{S}_n^{*}]=f_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\varvec{S}^{*}(1+O(h^2))\), where \(\varvec{S}^{*}= \left[ \begin{array}{cc} \varvec{S}^{*}_{11}&{}\quad \varvec{S}^{*}_{12}\\ (\varvec{S}^{*}_{12})^T&{}\quad \varvec{S}^{*}_{22} \end{array}\right] \) with

$$\begin{aligned} \varvec{S}^{*}_{11}= & {} \sum _{k=1}^q f(c_k)\varvec{e}_k\varvec{e}_k^T,\quad \varvec{S}^{*}_{12}=\sum _{k=1}^q f(c_k)\varvec{e}_k(\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)-\varvec{X}_0)^T, \\ \varvec{S}^{*}_{22}= & {} \kappa (\varvec{\omega }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\varvec{X}_0^T-\varvec{X}_0(\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0))^T+\varvec{X}_0\varvec{X}_0^T). \end{aligned}$$

According to the convexity lemma (Pollard 1991), the minimizer of (9), defined as \(\hat{\varvec{\theta }}^{*}\), can be expressed as \(\hat{\varvec{\theta }}^{*}=-f^{-1}_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)(\varvec{S}^{*})^{-1}\varvec{W}_n^{*}+o_p(1),\) which holds uniformly for \(\varvec{X}_0\). Define

$$\begin{aligned} \varvec{\Pi }_{n2}^{*}=\frac{1}{\sqrt{nh}}\sum _{i=1}^n K\big (\frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big )\big (\sum _{k=1}^q \eta _{ik}^0\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)),\\ \varvec{\Pi }_{n2}=\frac{1}{\sqrt{nh}}\sum _{i=1}^n K\big (\frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big )\big (\sum _{k=1}^q \eta _{ik}\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)). \end{aligned}$$

Note that

$$\begin{aligned} (\varvec{S}^{*})^{-1}= \left[ \begin{array}{cc} \star &{} \star \\ -\kappa ^{-1}\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_0^T \varvec{\beta }_0)\big (\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)-\varvec{X}_0\big ) \sum \limits _{k=1}^q \varvec{e}_k^T &{} \quad \kappa ^{-1}\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_0^T \varvec{\beta }_0) \end{array}\right] , \end{aligned}$$

then by some simple calculations, it’s easy to show

$$\begin{aligned} \sqrt{nh}\,(\hat{\varvec{b}}_0-g'(\varvec{X}_0^T \varvec{\beta }_0)\varvec{\beta }_0) =-\frac{\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_0^T \varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\kappa }\varvec{\Pi }_{n2}^{*}+o_p(1). \end{aligned}$$
(10)

Note that \(Cov(\eta _{ik},\eta _{ik'})=\tau _{kk'}\) and \(Cov(\eta _{ik},\eta _{jk'})=0 \text{ if } i\ne j\). It’s easy to show

$$\begin{aligned} E[\varvec{\Pi }_{n2}]=\varvec{0}\quad \text{ and }\quad Var[\varvec{\Pi }_{n2}]\rightarrow \nu _0\varrho f_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\varvec{W}_{ \varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0). \end{aligned}$$

According to the Cramer-Wald theorem, the central limit theorem for \(\varvec{\Pi }_{n2}\) holds, which implies that

$$\begin{aligned} \varvec{\Pi }_{n2}\mathop {\longrightarrow }\limits ^{D}N\{\varvec{0},\,\nu _0\varrho f_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\varvec{W}_{ \varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\}. \end{aligned}$$

Moreover, we have

$$\begin{aligned}&Var[\varvec{\Pi }_{n2}-\varvec{\Pi }_{n2}^{*}|\mathcal {T}\,]\\&\quad =\frac{1}{nh}\sum _{i=1}^n K^2\big (\frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0))(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0))^T\times \\&Var \big [\sum _{k=1}^q(\eta _{ik}-\eta _{ik}^{0})|\mathcal {T}\big ]\\&\quad \le \frac{q}{nh}\sum _{i=1}^n K^2\big (\frac{\varvec{X}_{i0}^T\varvec{\beta }_0}{h}\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0))(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0))^T\times \\&\quad \mathop {\mathrm{max}}\limits _{1\le k\le q}\{F(c_k+|r_{i0}|)-F(c_k)\}=o_p(1). \end{aligned}$$

Combining this with (10), we immediately obtain

$$\begin{aligned} \sqrt{nh}\,(\hat{\varvec{b}}_0-g'(\varvec{X}_0^T \varvec{\beta }_0)\varvec{\beta }_0) =-\frac{\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_0^T \varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_0^T \varvec{\beta }_0)\kappa }\varvec{\Pi }_{n2}+o_p(1) \end{aligned}$$
(11)

and

$$\begin{aligned} \sqrt{nh}\,(\hat{\varvec{b}}_0-g'(\varvec{X}_0^T\varvec{\beta }_0)\varvec{\beta }_0) \mathop {\longrightarrow }\limits ^{D}N\big \{\varvec{0},\,\frac{\varrho }{\kappa ^2}\nu _0f^{ -1}_{\varvec{\beta }_0} (\varvec{X}_0^T\varvec{\beta }_0)\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_0^T\varvec{\beta }_0)\big \}. \end{aligned}$$

Step (ii)   At the given point \(\varvec{X}_j,j=1,\ldots ,n\), we can approximate \(\hat{\varvec{b}}_j\) using (11), that is,

$$\begin{aligned} \hat{\varvec{b}}_j= & {} g'(\varvec{X}_j^T \varvec{\beta }_0)\varvec{\beta }_0-\frac{\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_j^T \varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_j^T \varvec{\beta }_0)\kappa }\frac{1}{nh} \sum _{i=1}^n K\big (\frac{\varvec{X}_{ij}^T\varvec{\beta }_0}{h}\big )\big (\sum _{k=1}^q \eta _{ik}\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_j^T \varvec{\beta }_0))\\&\times (1+o_p(1)). \end{aligned}$$

Then we have

$$\begin{aligned} \frac{1}{n}\sum _{j=1}^{n}\hat{\varvec{b}}_j\hat{\varvec{b}}_j^T=\frac{1}{n}\sum _{j=1}^{n}(g'(\varvec{X}_j^T \varvec{\beta }_0))^2\varvec{\beta }_0\varvec{\beta }_0^T-(\varvec{\beta }_0\varvec{M}_n^T+\varvec{M}_n\varvec{\beta }_0^T)(1+o_p(1)), \end{aligned}$$
(12)

where

$$\begin{aligned} \varvec{M}_n=\frac{1}{n^2h}\sum _{j=1}^n\frac{g'(\varvec{X}_j^T \varvec{\beta }_0)\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_j^T \varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_j^T \varvec{\beta }_0)\kappa }\sum _{i=1}^n K\big (\frac{\varvec{X}_{ij}^T\varvec{\beta }_0}{h}\big )\big (\sum _{k=1}^q \eta _{ik}\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_j^T \varvec{\beta }_0)). \end{aligned}$$

We shall prove later

$$\begin{aligned} \varvec{M}_n=\kappa ^{-1}\frac{1}{n}\sum _{i=1}^ng'(\varvec{X}_i^T \varvec{\beta }_0)\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_i^T \varvec{\beta }_0)\big (\sum _{k=1}^q \eta _{ik}\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_i^T \varvec{\beta }_0))+o_p(n^{-1/2}). \end{aligned}$$
(13)

From (13), we have \(\sqrt{n}\varvec{M}_n\mathop {\longrightarrow }\limits ^{D}N\{\varvec{0},\frac{\varrho }{\kappa ^2} E[(g'(\varvec{X}^T \varvec{\beta }_0))^2\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}^T \varvec{\beta }_0)]\}\), which uses the following equality

$$\begin{aligned} E[(g'(\varvec{X}^T \varvec{\beta }_0))^2\varvec{W}_{\varvec{\beta }_0}^{+} (\varvec{X}^T \varvec{\beta }_0)\varvec{W}_{0}(\varvec{X})\varvec{W}_{\varvec{\beta }_0}^{+} (\varvec{X}^T \varvec{\beta }_0)]=E[(g'(\varvec{X}^T \varvec{\beta }_0))^2\varvec{W}_{\varvec{\beta }_0}^{+} (\varvec{X}^T \varvec{\beta }_0)]. \end{aligned}$$

Combining (12) with \(\varvec{M}_n=O_p(n^{-1/2})\), we obtain

$$\begin{aligned} \frac{1}{n}\sum _{j=1}^{n}\hat{\varvec{b}}_j\hat{\varvec{b}}_j^T= & {} E[(g'(\varvec{X}^T \varvec{\beta }_0))^2]\varvec{\beta }_0\varvec{\beta }_0^T-(\varvec{\beta }_0\varvec{M}_n^T+\varvec{M}_n\varvec{\beta }_0^T)+o_p(n^{-1/2})\\= & {} E[(g'(\varvec{X}^T \varvec{\beta }_0))^2]\big (\varvec{\beta }_0-\{ E[(g'(\varvec{X}^T \varvec{\beta }_0))^2]\}^{-1}\varvec{M}_n+o_p(n^{-1/2})\big )\\= & {} \big (\varvec{\beta }_0-\{ E[(g'(\varvec{X}^T \varvec{\beta }_0))^2]\}^{-1}\varvec{M}_n+o_p(n^{-1/2})\big )^T. \end{aligned}$$

This immediately implies \(\hat{\varvec{\beta }}_{\scriptscriptstyle CQR}=\varvec{\beta }_0-\{ E[(g'(\varvec{X}^T \varvec{\beta }_0))^2]\}^{-1}\varvec{M}_n+o_p(n^{-1/2})\) and \(\sqrt{n}(\hat{\varvec{\beta }}_{\scriptscriptstyle CQR}-\varvec{\beta }_0)\) has an asymptotically normal distribution \(N(\varvec{0},\varvec{\Sigma }_{\scriptscriptstyle CQR})\) with

$$\begin{aligned} \varvec{\Sigma }_{\scriptscriptstyle CQR}=\frac{\varrho }{\kappa ^2}\frac{E[(g'(\varvec{X}^T \varvec{\beta }_0))^2\varvec{W}_{\varvec{\beta }_0}^{+}(\varvec{X}^T \varvec{\beta }_0)]}{\{ E[(g'(\varvec{X}^T \varvec{\beta }_0))^2]\}^2}. \end{aligned}$$

Step (iii)   The remaining part of the proof is to show (13). To obtain the dominant term of \(\varvec{M}_n\), it’s sufficient to derive that of \(\varvec{\alpha }^T \varvec{M}_n\) for any p-dimensional non-random column \(\varvec{\alpha }\). We rewrite

$$\begin{aligned}&\varvec{\alpha }^T \varvec{M}_n\\&\quad =\frac{1}{n^2h}\sum _{j=1}^n\frac{g'(\varvec{X}_j^T \varvec{\beta }_0)\varvec{\alpha }^T\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_j^T\varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_j^T \varvec{\beta }_0)\kappa }\sum _{i=1}^n K\big (\frac{\varvec{X}_{ij}^T\varvec{\beta }_0}{h}\big )\big (\sum _{k=1}^q \eta _{ik}\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_j^T \varvec{\beta }_0))\\&\quad = \sum _{1\le i,j\le n} \xi _n(\varvec{W}_i,\varvec{W}_j), \end{aligned}$$

where \(\xi _n(\varvec{W}_i,\varvec{W}_j)=\zeta _n(\varvec{W}_i,\varvec{W}_j)+\zeta _n(\varvec{W}_j,\varvec{W}_i)\) is a U-statistics with kernel

$$\begin{aligned} \zeta _n(\varvec{W}_1,\varvec{W}_2)= & {} \frac{1}{n^2h}\frac{g'(\varvec{X}_1^T \varvec{\beta }_0)\varvec{\alpha }^T\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_1^T \varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0)\kappa }(\varvec{X}_2-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0))K\big (\frac{\varvec{X}_{21}^T\varvec{\beta }_0}{h}\big )\\&\times \big (\sum _{k=1}^q \eta _{2k}\big ) \end{aligned}$$

and \(\varvec{W}_i=(\varepsilon _i,\varvec{X}_i)\). It’s easy to see that \(E[\xi _n(\varvec{W}_i,\varvec{W}_j)]=E[\zeta _n(\varvec{W}_i,\varvec{W}_j)]=0\). Let \(\phi _n(\varvec{t})=E[\xi _n(\varvec{W}_1,\varvec{t})]=E[\zeta _n(\varvec{W}_1,\varvec{t})]\) with \(\varvec{t}=(t_{\varepsilon },\,\varvec{t}_{\varvec{X}})\) and \(Q_n=(n-1)\sum _{i=1}^n \phi _n(\varvec{W}_i)\). Using the Hoeffding composition of \(\varvec{\alpha }^T \varvec{M}_n\) (Serfling 1980), we have

$$\begin{aligned} E[(\varvec{\alpha }^T \varvec{M}_n-Q_n)^2]= & {} \frac{n(n-1)}{2}(E[\xi _n^2(\varvec{W}_1,\varvec{W}_2)]- 2E[\phi _n^2(\varvec{W}_1)])\\\le & {} \frac{n(n-1)}{2}E[\xi _n^2(\varvec{W}_1,\varvec{W}_2)]. \end{aligned}$$

By conditioning on \((\varvec{X}_1^T \varvec{\beta }_0,\varvec{X}_2^T \varvec{\beta }_0)\) and some calculations, we have

$$\begin{aligned}&E[\xi _n^2(\varvec{W}_1,\varvec{W}_2)]\le 4E[\zeta _n^2(\varvec{W}_1,\varvec{W}_2)]\\&\quad =4E\Big [\Big (\frac{1}{n^2h}\frac{g'(\varvec{X}_1^T \varvec{\beta }_0)\varvec{\alpha }^T\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_1^T \varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0)\kappa }(\varvec{X}_2-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0))K\big (\frac{\varvec{X}_{21}^T\varvec{\beta }_0}{h}\big )\big (\sum _{k=1}^q \eta _{2k}\big )\Big )^2\Big ]\\&\quad =\frac{\varrho }{\kappa ^2}\frac{4}{n^4h^2}\varvec{\alpha }^T E\Big [\frac{(g'(\varvec{X}_1^T \varvec{\beta }_0))^2}{f^2_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0)}\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_1^T \varvec{\beta }_0)(\varvec{X}_2-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0))(\varvec{X}_2-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0))^T\\&\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_1^T \varvec{\beta }_0)K^2\big (\frac{\varvec{X}_{21}^T\varvec{\beta }_0}{h}\big )\Big ]\varvec{\alpha }\\&\quad =O(n^{-4}h^{-1}). \end{aligned}$$

Thus \(E[(\varvec{\alpha }^T \varvec{M}_n-Q_n)^2]=O(n^{-2}h^{-1})\). Because \(nh\rightarrow \infty \), we have \(\varvec{\alpha }^T \varvec{M}_n=Q_n+o_p(n^{-1/2})\). To further study \(Q_n\), we calculate \(\phi _n(\varvec{t})\) now. Note that

$$\begin{aligned}&\phi _n(\varvec{t})=E[\zeta _n(\varvec{W}_1,\varvec{t})]\\&\quad =E\Big [\dfrac{1}{n^2h}\frac{g'(\varvec{X}_1^T \varvec{\beta }_0)\varvec{\alpha }^T\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_1^T \varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0)\kappa }(\varvec{t}_{\varvec{X}}-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0))K\big (\frac{\varvec{t}_{\varvec{X}}^T\varvec{\beta }_0-\varvec{X}_1^T \varvec{\beta }_0}{h}\big )\\&\qquad \times \big (\sum _{k=1}^q[I\{t_{\varepsilon }\le c_k\}-\tau _k]\big )\Big ]\\&\quad =\frac{\sum _{k=1}^q[I\{t_{\varepsilon }\le c_k\}-\tau _k]}{\kappa }\frac{1}{n^2h}\varvec{\alpha }^T E\Big [\frac{g'(\varvec{X}_1^T \varvec{\beta }_0)}{f_{\varvec{\beta }_0} (\varvec{X}_1^T \varvec{\beta }_0)}\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_1^T \varvec{\beta }_0)(\varvec{t}_{\varvec{X}}-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_1^T\varvec{\beta }_0))\\&\qquad \times K\big (\frac{\varvec{t}_{\varvec{X}}^T\varvec{\beta }_0-\varvec{X}_1^T \varvec{\beta }_0}{h}\big )\Big ]\\&\quad =\frac{\sum _{k=1}^q[I\{t_{\varepsilon }\le c_k\}-\tau _k]}{\kappa }\frac{1}{n^2h}\varvec{\alpha }^T hg'(\varvec{t}_{\varvec{X}}^T\varvec{\beta }_0)\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{t}_{\varvec{X}}^T\varvec{\beta }_0)(\varvec{t}_{\varvec{X}}-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{t}_{\varvec{X}}^T\varvec{\beta }_0))\\&\qquad \times (1+o(1))\\&\quad =\frac{\sum _{k=1}^q[I\{t_{\varepsilon }\le c_k\}-\tau _k]}{\kappa }\varvec{\alpha }^T\frac{1}{n^2}g'(\varvec{t}_{\varvec{X}}^T\varvec{\beta }_0)\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{t}_{\varvec{X}}^T\varvec{\beta }_0)(\varvec{t}_{\varvec{X}}-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{t}_{\varvec{X}}^T\varvec{\beta }_0))(1+o(1)). \end{aligned}$$

Therefore it’s easy to derive

$$\begin{aligned}&\varvec{\alpha }^T \varvec{M}_n=Q_n+o_p(n^{-1/2})=(n-1)\sum _{i=1}^n \phi _n(\varvec{W}_i)+o_p(n^{-1/2})\\&\quad = \varvec{\alpha }^T \kappa ^{-1}\frac{1}{n}\sum _{i=1}^ng'(\varvec{X}_i^T \varvec{\beta }_0)\varvec{W}_{ \varvec{\beta }_0}^{+}(\varvec{X}_i^T \varvec{\beta }_0)\big (\sum _{k=1}^q\eta _{ik}\big )(\varvec{X}_i-\varvec{\mu }_{\varvec{\beta }_0} (\varvec{X}_i^T \varvec{\beta }_0))+o_p(n^{-1/2}), \end{aligned}$$

which holds for any non-random vector \(\varvec{\alpha }\). Then \(\mathrm {(A5)}\) is obtained and the proof is completed. \(\square \)

Proof of Proposition 1

It follows from the proof of Theorem 3.1 of Zou and Yuan (2008) that

$$\begin{aligned} \lim _{q\rightarrow \infty }\mathrm {ARE}(q,f)=12\sigma ^2\tau ^2. \end{aligned}$$

Moreover, according to the result of Hodges and Lehmann (1956), we immediately complete the proof. \(\square \)

Proof of Theorem 2

Note that

$$\begin{aligned} \sqrt{nh}\{\hat{g}(t_0)-g(t_0)\}=\sqrt{nh}\{\hat{g}(t_0)-\hat{g}^0(t_0)\} +\sqrt{nh}\{\hat{g}^0(t_0)-g(t_0)\}, \end{aligned}$$
(14)

where \(\hat{g}^0(t_0)\) is the weighted local linear CQR estimator of \(g(\cdot )\) when the index coefficient \(\varvec{\beta }_0\) is known. By Theorem 3.1 in Sun et al. (2013), we have

$$\begin{aligned} \sqrt{nh}\{\hat{g}^0(t_0)-g(t_0)-\frac{1}{2}g''(t_0)\mu _2h^2\} \mathop {\longrightarrow }\limits ^{D}N\big \{0,\,\frac{\nu _0\,\sigma ^2}{f_{T_0}(t_0)}J_1(q)\big \}. \end{aligned}$$

By Theorem 1 in Jiang et al. (2012), the first part of on the right side of (14), \(\sqrt{nh}\{\hat{g}(t_0)-\hat{g}^0(t_0)\}\) can be shown as \(o_p(1)\). This completes the proof. \(\square \)

Proof of Theorem 3

Note that

$$\begin{aligned}&\sqrt{nh}\{\hat{g}(\varvec{X}^T\hat{\varvec{\beta }}_{\scriptscriptstyle CQR})-g(\varvec{X}^T\varvec{\beta }_0)\}\\&\quad =\sqrt{nh}\{\hat{g}(\varvec{X}^T\hat{\varvec{\beta }}_{\scriptscriptstyle CQR})-\hat{g}^0(\varvec{X}^T\varvec{\beta }_0)\}+\sqrt{nh}\{\hat{g}^0(\varvec{X}^T\varvec{\beta }_0)-\hat{g}^0(\varvec{X}^T\varvec{\beta }_0)\}. \end{aligned}$$

By the Taylor theorem, \(\sqrt{nh}\{\hat{g}(\varvec{X}^T\hat{\varvec{\beta }}_{\scriptscriptstyle CQR})-\hat{g}^0(\varvec{X}^T\varvec{\beta }_0)\}=\sqrt{nh}\,O_p(\Vert \hat{\varvec{\beta }}_{\scriptscriptstyle CQR}-\beta _0\Vert )=o_p(1)\). Thus, Theorem 3 is the result of Theorem 2. \(\square \)

Proof of Theorem 4

It can be immediately obtained from Theorem 3.2 in Sun et al. (2013). Details are omitted. \(\square \)

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Sun, J. Composite quantile regression for single-index models with asymmetric errors. Comput Stat 31, 329–351 (2016). https://doi.org/10.1007/s00180-016-0645-7

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