﻿ 单指标模型的加权复合分位数回归

# 单指标模型的加权复合分位数回归Single-Index Weighted Composite Quantile Regression

Abstract: Weighted composite quantile regression (WCQR), a robust estimation based on quantile regression, is becoming increasingly popular due to the nearly same efficiency as the semi-parametric maximum likelihood estimator. Recently, WCQR method has been promoted extensively to sin-gle-index model. However, the recent WCQR methods for single-index model are necessarily itera-tive, which seriously affects the computing speed. We propose a non-iterative estimation algorithm, and derive the asymptotic distribution of the proposed estimator. The simulation and empirical studies are conducted to illustrate the finite sample performance of the proposed methods.

1. 引言

$Y={g}_{0}\left({X}^{\text{T}}{\gamma }_{0}\right)+\epsilon$ (1.1)

2. 单指标模型的非迭代WCQR方法

2.1. NIWCQR方法

${\gamma }_{0}=\mathrm{arg}\underset{\gamma }{\mathrm{min}}\underset{k=1}{\overset{K}{\sum }}{w}_{k}E\left[{\rho }_{{\tau }_{k}}\left\{Y-{Q}_{{\tau }_{k}}\left(Y|{X}^{\text{T}}\gamma \right)\right\}\right]$ (2.1)

${Q}_{{\tau }_{k}}\left(Y|{X}^{\text{T}}{\gamma }_{0}\right)={Q}_{{\tau }_{k}}\left(Y|X\right)={H}_{{\tau }_{k}}\left({X}^{\text{T}}{\gamma }_{0}|{\gamma }_{0}\right)$$k=1,\cdots ,K$

${\gamma }_{0}=\mathrm{arg}\underset{\gamma }{\mathrm{min}}\underset{k=1}{\overset{K}{\sum }}{w}_{k}E\left[{\rho }_{{\tau }_{k}}\left\{Y-{H}_{{\tau }_{k}}\left({X}^{\text{T}}\gamma |\gamma \right)\right\}\right]$ (2.2)

${\left\{{Y}_{i},{X}_{i}\right\}}_{i=1}^{n}$ 是来自 $\left\{Y,X\right\}$ 独立同分布的样本，则(2.2)式右边的表达式可近似转化为 $\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}{w}_{k}{\rho }_{{\tau }_{k}}\left\{{Y}_{i}-{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\right\}$。对于每一个k我们可以得到 ${H}_{{\tau }_{k}}\left(·|\gamma \right)$ 的Nadaraya-Watson估计值 ${\stackrel{^}{H}}_{{\tau }_{k}}\left(t|\gamma \right)$ (见Christou & Akrites [10] (2016))

${\stackrel{^}{H}}_{{\tau }_{k}}\left(t|\gamma \right)=\frac{\underset{i=1}{\overset{n}{\sum }}{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|{X}_{i}\right){K}_{{h}_{k}}\left({X}_{i}^{\text{T}}\gamma -t\right)}{\underset{i=1}{\overset{n}{\sum }}{K}_{{h}_{k}}\left({X}_{i}^{\text{T}}\gamma -t\right)}$ (2.3)

${\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|X\right)={\stackrel{^}{F}}_{Y}^{-1}\left({\stackrel{^}{C}}_{{F}_{Y}|{F}_{1},\cdots ,{F}_{P}}^{-1}\left(\tau |{\stackrel{^}{F}}_{1}\left({x}_{1}\right),\cdots ,{\stackrel{^}{F}}_{p}\left({x}_{p}\right)\right)\right)$$k=1,\cdots ,K$

$\stackrel{¯}{\gamma }=\mathrm{arg}\underset{\gamma }{\mathrm{min}}\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}{w}_{k}{\rho }_{{\tau }_{k}}\left\{{Y}_{i}-{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\right\}$ (2.4)

$\left({\stackrel{^}{a}}_{1},\cdots ,{\stackrel{^}{a}}_{K},\stackrel{^}{b}\right)=\mathrm{arg}\underset{\left({a}_{1},\cdots ,{a}_{K},b\right)}{\mathrm{min}}\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}\left\{{Y}_{i}-{a}_{k}-b\left({X}_{i}^{\text{T}}\stackrel{^}{\gamma }-u\right)\right\}{K}_{h}\left({X}_{i}^{\text{T}}\stackrel{^}{\gamma }-u\right)$ (2.5)

2.2. 渐近性

${R}_{2}\left(V\right)=\underset{k=1}{\overset{K}{\sum }}\underset{k\prime =1}{\overset{K}{\sum }}\frac{{v}_{k}{v}_{k\prime }{\tau }_{kk\prime }}{f\left({c}_{k}\right)f\left({c}_{k\prime }\right)}$${\tau }_{kk\prime }={\tau }_{k}\Lambda {\tau }_{k\prime }-{\tau }_{k}{\tau }_{k\prime }$${h}_{\mathrm{max}}={\mathrm{max}}_{1\le k\le K}\left\{{h}_{k}\right\}$${h}_{\mathrm{min}}={\mathrm{min}}_{1\le k\le K}\left\{{h}_{k}\right\}$

$\sqrt{n}\left(\stackrel{^}{\gamma }-{\gamma }_{0}\right)\stackrel{L}{\to }N\left(0,{S}^{-1}{R}_{1}\left(W\right)\right)$

${W}_{opt}=\mathrm{arg}\underset{W}{\mathrm{min}}{R}_{1}\left(W\right)={\left({f}^{\text{T}}{\Omega }^{-2}f\right)}^{-1/2}{\Omega }^{-1}f$ (2.6)

$\sqrt{n}\left(\stackrel{^}{\gamma }-{\gamma }_{0}\right)\stackrel{L}{\to }N\left(0,{S}^{-1}{\left({f}^{\text{T}}{\Omega }^{-1}f\right)}^{-1}\right)$ (2.7)

$\sqrt{nh}\left\{\stackrel{^}{g}\left(u|\stackrel{^}{\gamma }\right)-{g}_{0}\left(u\right)-\frac{1}{2}{{g}^{″}}_{0}\left(u\right){\mu }_{2}{h}^{2}\right\}\stackrel{L}{\to }N\left(0,\frac{{V}_{0}{R}_{2}\left(V\right)}{{f}_{U}\left(u\right)}\right)$

${V}_{opt}=\mathrm{arg}\underset{W}{\mathrm{min}}{R}_{2}\left(V\right)=\frac{\left({r}^{\text{T}}{A}^{-1}r\right){A}^{-1}1-\left({1}^{\text{T}}{A}^{-1}r\right){A}^{-1}r}{\left({r}^{\text{T}}{A}^{-1}r\right)\left({1}^{\text{T}}{A}^{-1}1\right)-{\left({1}^{\text{T}}{A}^{-1}r\right)}^{2}}$ (2.8)

3. 模拟研究

1) MAVE方法(参考Xia & Härdle [4] (2006))；

2) $\tau =0.5$ 的分位数回归(QR0.5) (参考Wu等 [6] 2010)；

3) WCQR方法其中 $K=9$ (WCQR9) (参考Jiang et al. [7] 2016a)；

4) 非迭代的最小二乘估计(NILSE) (参考Wang & Wu [17] (2013))；

5) 非迭代的的 $\tau =0.5$ 的分位数回归(NIQR0.5) (参考Christou & Akritas [10] 2016)；

6) 非迭代的复合分位数回归方法其中 $K=9$ (NICQR9)，其中 $W\equiv 1$

3.1. 模拟例子

$Y=\mathrm{sin}\left\{\pi \left({X}^{\text{T}}{\gamma }_{0}\right)\right\}+0.2\epsilon$ (3.1)

3.2. 实际例子：波士顿房价

RM：每家住户的平均房间数；

TAX：全部价值物业税(美元) \$10,000；

PTRATIO：街区学生与教师的比例；

LSTAT：地位较低的人口(%)。

${\stackrel{^}{y}}_{i}$${y}_{i}$ 的估计值。在表2中我们总结了上述模型的估计系数。值得注意的是，在四个协变量中PTRATIO对房价的影响最小，而LSTAT对房价的影响最大。表2还给出了所有估计方法的MAE和t (计算时间)。

Table 2. Estimated Single Index Factor for Boston House Prices and MSE

Figure 1. Estimated single index composite quantile regression for Boston housing data. The dots are the observations and the curve is the estimated link function

4. 定理证明

C1. 核函数 $K\left(.\right)$ 是一个对称的有界密度函数，并且有Lipschitz连续的二阶微分。

C2. $U={X}^{\text{T}}\gamma$ 的密度函数是正的并且在 ${\gamma }_{0}$ 的一个领域内对 $\gamma$ 一致连续。 ${X}^{\text{T}}{\gamma }_{0}$ 的密度函数连续有界。

C3. 函数 ${g}_{0}\left(·\right)$ 是连续有界的二次可微函数。

C4. 模型的误差 $\epsilon$ 的密度函数 $f\left(·\right)$ 是正的。

$\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}\left[{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)-{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\right]={ο}_{p}\left(1\right)$

$\begin{array}{l}\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}\left[{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)-{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\right]\\ =\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{\underset{j=1}{\overset{n}{\sum }}{K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}-\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\\ =\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\end{array}$

$\begin{array}{l}=\left\{\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left\{\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\right\}\equiv {T}_{1}+{T}_{2}\end{array}$

$\begin{array}{l}{T}_{1}=\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\end{array}$

${T}_{2}=\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)$

$\begin{array}{c}{T}_{1}=\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{Q}_{{\tau }_{K}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\\ =\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\left[{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|{X}_{j}\right)-{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right)\right]\frac{{K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}\left[\frac{1}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\right]\\ \equiv {T}_{11}+{T}_{12}\end{array}$

${T}_{11}=\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\left[{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|{X}_{j}\right)-{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right)\right]\frac{{K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}$

${T}_{12}=\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}\left[\frac{1}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\right]$

${\mathrm{sup}}_{X\in X}|{\stackrel{^}{Q}}_{{\tau }_{k}}\left(Y|X\right)-{Q}_{{\tau }_{k}}\left(Y|X\right)|={O}_{p}\left({n}^{-1/2}\right)$ 可得 ${T}_{11}={ο}_{P}\left(1\right)$，见Rémillard等 [19] (2017)。在C2和 $n{h}_{\mathrm{max}}^{4}={ο}_{p}\left(1\right)$${T}_{12}={ο}_{p}\left(1\right)$ 的条件下， ${T}_{1}={ο}_{P}\left(1\right)$

$\begin{array}{c}{T}_{2}=\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\frac{{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\\ =\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\left[{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right)-{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\right]\frac{{K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right){K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}\left[\frac{1}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}-\frac{1}{{\stackrel{^}{f}}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}\right]\\ =\frac{1}{{n}^{3/2}{h}_{k}}\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}\left[{Q}_{{\tau }_{k}}\left(Y|{X}_{j}\right)-{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)\right]\frac{{K}_{{h}_{k}}\left\{{\left({X}_{i}-{X}_{j}\right)}^{\text{T}}\gamma \right\}}{{f}_{\gamma }\left({X}_{i}^{\text{T}}\gamma \right)}+{ο}_{P}\left(1\right)\end{array}$

${L}_{n}\left({\gamma }^{*}\right)=\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}{w}_{k}\left\{{\rho }_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}-{\stackrel{˜}{H}}_{{\tau }_{k}}\left({X}_{i}|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)\right)-{\rho }_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right)\right\}$

${Y}_{i,{\tau }_{k}}^{*}={Y}_{i}-{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}{\gamma }_{0}|{\gamma }_{0}\right)+{O}_{p}\left({n}^{-1}\right)$

${L}_{n}\left({\gamma }^{*}\right)=E\left[{L}_{n}\left({\gamma }^{*}\right)|\chi \right]-\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}{w}_{k}\left\{\rho {\prime }_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right)-E\left[{{\rho }^{\prime }}_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right)|\chi \right]\right\}{\stackrel{˜}{H}}_{{\tau }_{k}}\left({X}_{i}|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)+{R}_{n}\left(\gamma *\right)$

$\begin{array}{l}E\left[{L}_{n}\left({\gamma }^{*}\right)|\chi \right]=\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}{w}_{k}E\left[{\rho }_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}-{\stackrel{˜}{H}}_{{\tau }_{k}}\left({X}_{i}|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)\right)-{\rho }_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right)|\chi \right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}{w}_{k}E\left[{{\rho }^{\prime }}_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right)|\chi \right]{\stackrel{˜}{H}}_{{\tau }_{k}}\left({X}_{i}|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{ }+\frac{1}{2}\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}{w}_{k}E\left[{{\rho }^{″}}_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right)|\chi \right]{\stackrel{˜}{H}}^{2}{}_{{\tau }_{k}}\left({X}_{i}|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)+{ο}_{P}\left(1\right)\end{array}$

$\begin{array}{c}{L}_{n}\left({\gamma }^{*}\right)=-\underset{k=1}{\overset{K}{\sum }}{w}_{k}\underset{i=1}{\overset{n}{\sum }}{{\rho }^{\prime }}_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right){\stackrel{˜}{H}}_{{\tau }_{k}}\left({X}_{i}|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)+\frac{1}{2}\underset{k=1}{\overset{K}{\sum }}{w}_{k}{f}_{\epsilon }\left({c}_{k}\right)\underset{i=1}{\overset{n}{\sum }}{\stackrel{˜}{H}}^{2}{}_{{\tau }_{k}}\left({X}_{i}|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)+{ο}_{p}\left(1\right)\\ =-\underset{k=1}{\overset{K}{\sum }}{w}_{k}\underset{i=1}{\overset{n}{\sum }}{{\rho }^{\prime }}_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right)\left[{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\left({\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)-{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}{\gamma }_{0}|{\gamma }_{0}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}\underset{k=1}{\overset{K}{\sum }}{w}_{k}{f}_{\epsilon }\left({c}_{k}\right)\underset{i=1}{\overset{n}{\sum }}\left[{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\left({\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)-{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}{\gamma }_{0}|{\gamma }_{0}\right)\right]+{ο}_{p}\left(1\right)\end{array}$

$\begin{array}{l}\underset{i=1}{\overset{n}{\sum }}\left[{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\left({\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)-{\stackrel{^}{H}}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}{\gamma }_{0}|{\gamma }_{0}\right)\right]\\ =\underset{i=1}{\overset{n}{\sum }}\left[{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\left({\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)-{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}{\gamma }_{0}|{\gamma }_{0}\right)\right]+{ο}_{P}\left({n}^{-1/2}\right)\end{array}$

$\begin{array}{l}{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\left({\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)|{\gamma }^{*}/\sqrt{n}+{\gamma }_{0}\right)-{H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}{\gamma }_{0}|{\gamma }_{0}\right)\\ =\frac{{\gamma }^{*}}{\sqrt{n}}{\frac{\partial {H}_{{\tau }_{k}}\left({X}_{i}^{\text{T}}\gamma |\gamma \right)}{\partial \gamma }|}_{{\gamma }_{0}}+{O}_{p}\left({n}^{-1}\right)\\ =\frac{1}{\sqrt{n}}{{g}^{\prime }}_{0}\left({X}_{i}^{\text{T}}{\gamma }_{0}\right){\left({X}_{i}-E\left[X|{X}^{\text{T}}{\gamma }_{0}\right]\right)}^{\text{T}}{\gamma }^{*}+{O}_{p}\left({n}^{-1}\right)\end{array}$

${L}_{n}\left({\gamma }^{*}\right)=-{W}_{n}{\gamma }^{*}+\frac{1}{2}{\left\{{\gamma }^{*}\right\}}^{\text{T}}\left\{\underset{k=1}{\overset{K}{\sum }}{w}_{k}{f}_{\epsilon }\left({c}_{k}\right)\right\}{S}_{n}{\gamma }^{*}+{ο}_{p}\left(1\right)$

${W}_{n}=\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum }}\underset{k=1}{\overset{K}{\sum }}{w}_{k}{{\rho }^{\prime }}_{{\tau }_{k}}\left({Y}_{i,{\tau }_{k}}^{*}\right)g\prime \left({X}_{i}^{\text{T}}{\gamma }_{0}\right){\left({X}_{i}-E\left[X|{X}^{\text{T}}{\gamma }_{0}\right]\right)}^{\text{T}}$

${S}_{n}=\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{\left\{g\prime \left({X}_{i}^{\text{T}}{\gamma }_{0}\right)\right\}}^{2}\left({X}_{i}-E\left[X|{X}^{\text{T}}{\gamma }_{0}\right]\right){\left({X}_{i}-E\left[X|{X}^{\text{T}}{\gamma }_{0}\right]\right)}^{\text{T}}$

${L}_{n}\left({\gamma }^{*}\right)=-{W}_{n}{\gamma }^{*}+\frac{1}{2}{\left\{{\gamma }^{*}\right\}}^{\text{T}}\left\{\underset{k=1}{\overset{K}{\sum }}{w}_{k}{f}_{\epsilon }\left({c}_{k}\right)\right\}S{\gamma }^{*}+{ο}_{p}\left(1\right)$

${\stackrel{^}{\gamma }}^{*}=-{\left\{\underset{k=1}{\overset{K}{\sum }}{w}_{k}f\left({c}_{k}\right)\right\}}^{-1}{S}^{-1}{W}_{n}+{ο}_{p}\left(1\right)$

5. 结论

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