响应变量缺失下变系数部分非线性模型的统计推断

Statistical Inferences for Varying Coefficient Partially Nonlinear Model with Missing Responses

摘要: 本论文考虑响应变量缺失下变系数部分非线性模型的统计推断，我们采用完全数据方法下的轮廓非线性最小二乘来估计未知参数和非参函数，同时建立了估计量的渐近正态性。

Abstract: In this paper, we consider statistical inferences for varying coefficient partially nonlinear model with missing responses. The profile nonlinear least-squares estimation process based on the complete data method is employed to estimate the unknown parameter and the nonparametric function, and the asymptotic normality of the resulting estimators is proved.

1. 引言

近几年来，变系数部分非线性模型由于半参模型的灵活性和非参模型的易解释性受到了大量的关注，

$Y=X^{\text{T}}\theta \left(U\right)+g\left(Z,\beta \right)+\epsilon$，

其中Y是响应变量， $X,Z,U$ 是协向量，且 $X\in R^q$， $Z\in R^r$， $U\in R$， $\theta (\cdot )={\left({\theta }_1(\cdot )，\cdots ,{\theta }_q(\cdot )\right)}^{\text{T}}$ 是q维未知的系数函数向量， $g\left(\cdot ,\cdot \right)$ 为给定的非线性函数，参数 $\beta ={\left({\beta }_1,\cdots ,{\beta }_p\right)}^{\text{T}}$ 的维数没有必要一定和Z相等，随机误差 $\epsilon$ 满足 $E\left(\epsilon |X,U,Z\right)=0$， $Var\left(\epsilon |X,U,Z\right)={\sigma }^2$。

此模型最初由Li和Mei (2013) [1] 提出，且包含很多重要的子模型，像变系数模型、非线性模型、部分非线性模型和变系数部分线性模型等，因此对此模型的统计研究具有相当大的意义，他们提出用轮廓非线性最小二乘估计方法估计参数向量 $\beta$ 和系数函数向量 $\theta \left(U\right)$，Zhou，Zhao和Wang (2017) [2] 采用经验似然方法获得参数和非参部分的区间估计。

据我们所知，还没有文献研究过变系数部分非线性模型的缺失响应变量的推断，此论文用完全数据法处理缺失响应变量，并对VCPNLM模型推荐轮廓非线性最小二乘估计方法。

2. 基于完全数据下轮廓非线性最小二乘的估计

我们假定随机样本 $\left\{Y_i,X_i,U_i,Z_i,i=1,2,\cdots ,n\right\}$ 是从下面变系数部分非线性模型中产生

$Y_i=X_i^{\text{T}}\theta \left(U_i\right)+g\left(Z_i,\beta \right)+{\epsilon }_i$，

其中 $\left\{\left(X_i,U_i,Z_i\right),i=1,\cdots ,n\right\}$ 可以被完全观测到，响应变量 $Y_i$ 随机缺失，当 $Y_i$ 被观测到，指示变量 ${\delta }_i=1$，当 $Y_i$ 缺失，指示变量 ${\delta }_i=0$，选择概率函数为 $P\left({\delta }_i=1|Y_i,X_i,U_i,Z_i\right)=P\left({\delta }_i=1|X_i,U_i,Z_i\right)=\pi \left(X_i,U_i,Z_i\right)$，对于选择概率函数的估计，本论文我们选取logistic模型选择概率

$\pi \left(V_i,\omega \right)=\frac{\exp \left({\omega }_0+{\omega }_1^{\text{T}}{X}_i+{\omega }_2{U}_i+{\omega }_3^{\text{T}}{Z}_i\right)}{1+\exp \left({\omega }_0+{\omega }_1^{\text{T}}{X}_i+{\omega }_2{U}_i+{\omega }_3^{\text{T}}{Z}_i\right)}$，

其中 $V_i={\left(X_i^{\text{T}},U_i,Z_i^{\text{T}}\right)}^{\text{T}}$， $\omega ={\left({\omega }_0,{\omega }_1^{\text{T}},{\omega }_2,{\omega }_3^{\text{T}}\right)}^{\text{T}}$ 为未知参数向量，自然地，可以通过最大化下面的对数似然函数来估计 $\omega$，

$\ln L\left(\omega \right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left\{{\delta }_i\ln \left[\pi \left(V_i,\omega \right)\right]+\left(1-{\delta }_i\right)\ln \left[1-\pi \left(V_i,\omega \right)\right]\right\}}$，

此时 $\pi \left(V_i,\omega \right)$ 的拟似然估计量为 $\pi \left(V_i,\hat{\omega }\right)$，具体细节请参考陈盼盼，冯三营和薛留根(2015) [3]。

下面模型

${\delta }_i{Y}_i={\delta }_i{X}_i^{\text{T}}\theta \left(U_i\right)+{\delta }_{i}g\left(Z_i,\beta \right)+{\delta }_i{\epsilon }_i$ (2.1)

是完全数据下的模型，对于 $u_0$ 领域附近的u，对 ${\theta }_j\left(u\right)$ 应用泰勒展开可得， ${\theta }_j\left(u\right)\approx {\theta }_j\left(u_0\right)+{{\theta }^{\prime }}_j\left(u_0\right)\left(u-u_0\right)$，如果给定 $\beta$，这时可以用局部线性估计方法并最小化下面式子得到系数函数的 $\theta \left(u\right)$ 估计

${\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{\left\{Y_i-g\left(Z_i,\beta \right)-{\displaystyle \underset{j=1}{\overset{q}{\sum }}\left[{\theta }_j\left(u_0\right)+{{\theta }^{\prime }}_j\left(u_0\right)\right]}\left(U_i-u_0\right)X_{ij}\right\}}^2}{K}_{h_1}\left(U_i-u_0\right)$。 (2.2)

令 $Y={\left(Y_1,\cdots ,Y_n\right)}^n$， $g\left(Z,\beta \right)={\left(g\left(Z_1,\beta \right),\cdots ,g\left(Z_1,\beta \right)\right)}^{\text{T}}$， $M={\left(X_1^{\text{T}}\theta \left(U_1\right),\cdots ,X_1^{\text{T}}\theta \left(U_n\right)\right)}^{\text{T}}$， $\Psi \left(u_0\right)={\left(\theta {\left(u_0\right)}^{\text{T}},h_1{\theta }^{\prime }{\left(u_0\right)}^{\text{T}}\right)}^{\text{T}}$， $W_1\left(u_0\right)=diag\left(K_{1,h_{\cdot 1}}\left(U_1-u_0\right),\cdots ,K_{1,h_{\cdot 1}}\left(U_n-u_0\right)\right)$， $\Delta =diag\left({\delta }_1,\cdots ,{\delta }_n\right)$， $X_{h_1}\left(u_0\right)=\left(\begin{array}{cc}{X}_1^{\text{T}}& h_1^{-1}\left(U_1-u_0\right)X_1^{\text{T}}\\ ⋮& ⋮\\ X_n^{\text{T}}& h_1^{-1}\left(U_n-u_0\right)X_n^{\text{T}}\end{array}\right)$，

此时最小二乘问题(2.2)的解为

${\hat{\Psi }}_c\left(u_0;\beta \right)={\left[X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta X_{h_1}\left(u_0\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \left[Y-g\left(Z,\beta \right)\right]$，

在 $u_0$ 处系数函数向量 $\theta \left(u\right)$ 的估计量为

${\hat{\theta }}_c\left(u_0,\beta \right)=\left(I_{q\times q},0_{q\times q}\right){\left[X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta X_{h_1}\left(u_0\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \left[Y-g\left(Z,\beta \right)\right]$，

令 $\hat{M}=\left(\begin{array}{c}{X}_1^{\text{T}}{\hat{\theta }}_c\left(U_1;\beta \right)\\ ⋮\\ X_n^{\text{T}}{\hat{\theta }}_c\left(U_n;\beta \right)\end{array}\right)\widehat{=}{S}_{h_1}\left[Y-g\left(\beta \right)\right]$，

其中 $S_{h_1}=\left(\begin{array}{c}\left(\begin{array}{cc}{X}_1^{\text{T}}& 0_{1\times q}\end{array}\right){\left[X_{h_1}^{\text{T}}\left(U_1\right)W_1\left(U_1\right)\Delta X_{h_1}\left(U_1\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(U_1\right)W_1\left(U_1\right)\Delta \\ ⋮\\ \left(\begin{array}{cc}{X}_n^{\text{T}}& 0_{1\times q}\end{array}\right){\left[X_{h_1}^{\text{T}}\left(U_n\right)W_1\left(U_n\right)\Delta X_{h_1}\left(U_n\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(U_n\right)W_1\left(U_n\right)\Delta \end{array}\right)$，

最小化下面关于 $\beta$ 的轮廓非线性最小二乘函数

$Q_c\left(\beta \right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{\left\{Y_i-X_i^{\text{T}}{\hat{\theta }}_c\left(U_i,\beta \right)-g\left(Z_i,\beta \right)\right\}}^2}$，

可得完全数据方法下轮廓非线性最小二乘估计量 ${\hat{\beta }}_c$，此时把 ${\hat{\beta }}_c$ 代入 ${\hat{\Psi }}_c\left(u_0;\beta \right)$ 和 ${\hat{\theta }}_c\left(u_0,\beta \right)$ 可得 $\Psi \left(u_0;\beta \right)$ 的估计量 ${\hat{\Psi }}_c\left(u_0;\hat{\beta }\right)$ 和 $\theta \left(u_0,\beta \right)$ 的估计量 ${\hat{\theta }}_c\left(u_0,\hat{\beta }\right)$。

引入以下标记：

${\mu }_{1,j}={\displaystyle \int u^j}{K}_{1,h_1}\left(u\right)\text{d}u$， ${\nu }_{1,j}={\displaystyle \int u^j}{K}_{1,h_1}^2\left(u\right)\text{d}u$， $\Gamma \left(u\right)=E\left(\pi \left(V\right)X{X}^{\text{T}}|U=u\right)$， $\Phi \left(u\right)=E\left(\pi \left(V\right)X{g}^{\prime }{\left(Z,{\beta }_0\right)}^{\text{T}}|U=u\right)$，

定理 2.1 设条件C1-C8成立， ${\beta }_0$ 为参数真值，则 $\sqrt{n}\left({\hat{\beta }}_c-{\beta }_0\right)\stackrel{D}{\to }N\left(0,{\sigma }^2{\Sigma }^{-1}\right)$，

其中， $\Sigma =E\left[\pi \left(V\right)g^{\prime }\left(Z,{\beta }_0\right)g^{\prime }{\left(Z,{\beta }_0\right)}^{\text{T}}\right]-E\left[\Phi {\left(U\right)}^{\text{T}}{\Gamma }^{-1}\left(U\right)\Phi \left(U\right)\right]$。

定理 2.2 设条件C1-C8成立，对于任意的 $u_0\in \Omega$，有

$\sqrt{n{h}_1}\left[{\hat{\Psi }}_c\left(u_0;{\hat{\beta }}_c\right)-{\Psi }_0\left(u_0\right)-\frac{1}{2}{h}_1{}^2{\mu }_{1，2}\left(\begin{array}{c}{{\theta }^{″}}_0\left(u_0\right)\\ 0\end{array}\right)\right]\stackrel{D}{\to }N\left(0,{\Sigma }_1\right)$

其中， ${\Sigma }_1={\sigma }^{2}f{\left(u_0\right)}^{-1}{\Gamma }^{-1}\left(u_0\right)\otimes \left(\begin{array}{cc}{\nu }_{1,0}& 0\\ 0& {\nu }_{1,2}{\mu }_{1,2}^{-2}\end{array}\right)$。

特殊地， $\sqrt{n{h}_1}\left[{\hat{\theta }}_c\left(u_0,{\hat{\beta }}_c\right)-{\theta }_0\left(u_0\right)-\frac{1}{2}{h}_1^2{\mu }_{1，2}{{\theta }^{″}}_0\left(u_0\right)\right]\stackrel{D}{\to }N\left(0,{\sigma }^2{\nu }_{1，0}f{\left(u_0\right)}^{-1}\Gamma {\left(u_0\right)}^{-1}\right)$。

3. 本论文需要的正则条件及证明

本论文需要的正则假设条件如下，这些正则假设也可以从Li和 Mei (2013) [1]，Xiao和Chen (2018) [4] 以及其他一些文献中找到：

C1：随机变量U具有有界支撑 $\Omega$，其密度函数 $f\left(u\right)$ 在 $\Omega$ 上具有有界的二阶导数且在支撑 $\Omega$ 上有界远离0。

C2：所有的系数函数 $\left\{{\theta }_j\left(u\right),j=1,\cdots ,q\right\}$ 具有连续二阶导数。

C3：对于任何Z， $g\left(Z,\beta \right)$ 是一个关于 $\beta$ 的连续函数且二阶导数连续。

C4：存在一个正数 $s>2.5$ 使得 $E\left({\Vert X\Vert }^{2s}\right)<\infty$， $E\left({\Vert g^{\prime }\left(Z,\beta \right)\Vert }^{2s}\right)<\infty$，且存在某个 $0<\delta <2-s^{-1}$ 使得 $n^{2\delta -1}{h}_1\to \infty$，其中 $\Vert \cdot \Vert$ 为欧式范数。

C5： $\beta$ 的某一邻域内满足 $E\left[g^{\prime }{\left(\beta \right)}^{\otimes 2}\right]<\infty$， $E\left[E{\left(g^{\prime }\left(\beta \right)|U\right)}^{\otimes 2}\right]<\infty$， $E\left[{\Vert g^{\prime }\left(\beta \right)\Vert }^4\right]<\infty$， $E{\Vert g^{\prime }\left(Z,\beta \right)\Vert }^4<\infty$， $E{\Vert Vech{g}^{″}\left(Z,\beta \right)\Vert }^4<\infty$。

C6：核权函数 $K_1\left(\text{·}\right)$ 和 $K_2\left(\text{·}\right)$ 具有紧支撑的对称密度函数，并满足Lipschitz条件，且函数 ${\mu }^3{K}_1\left(\mu \right),{\mu }^3{K^{\prime }}_1(\; \mu \; )$

有界， ${\displaystyle \int {\mu }^4}{K}_1\left(\mu \right)\text{d}\mu <\infty$，当 $n\to \infty$，窗宽 $h_i$ 满足 $n{h}_i^8\to 0$， $n{h}_i^2/{\left(\log n\right)}^2\to \infty$， $h_i\to 0$， $n{h}_i\to \infty$， $i=1,2$。

C7： $\Gamma \left(U\right)$ 对任何 $u\in \Omega$ 是非奇异的， $\Gamma \left(U\right),{\Gamma }^{*}\left(U\right),\Gamma {\left(U\right)}^{-1},{\Gamma }^{*}{\left(U\right)}^{-1},\Phi \left(U\right),{\Phi }^{*}\left(U\right)$ 均满足Lipschitz。

C8：缺失概率 $\pi \left(V,\omega \right)$ 关于 $\omega$ 二阶可导且存在常数 $C_0$，使得 $\inf \pi \left(V,\omega \right)\ge C_0\ge 0$。

在证明定理2.1和定理2.2之前先给出下面引理1和引理2，

引理1 假定条件C1~C8成立，则当 $n\to \infty$，有

$\frac{1}{n}{X}_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta X_{h_1}\left(u\right)=f\left(u\right)\Gamma \left(u\right)\otimes \left(\begin{array}{cc}1& 0\\ 0& {\mu }_{1，2}\end{array}\right)\left[1+O_p\left(c_n\right)\right]$ (3.1)

$\frac{1}{n}{X}_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta g^{\prime }\left({\beta }_0\right)=f\left(u\right)\Phi \left(u\right)\otimes {\left(1,0\right)}^{\text{T}}\left[1+O_p\left(c_n\right)\right]$ (3.2)

$\frac{1}{n}{X}_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta M_0=\Gamma \left(u\right){\theta }_0\left(u\right)f\left(u\right)\otimes {\left(1,0\right)}^{\text{T}}\left[1+O_p\left(c_n\right)\right]$ (3.3)

$\frac{1}{n}{X}_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta \epsilon =O_p\left({\left\{\frac{\log \left(1/h_1\right)}{n{h}_1}\right\}}^{\frac{1}{2}}\right)$ (3.4)

其中 $c_n=h_1^2+{\left\{\frac{\log \left(1/h_1\right)}{n{h}_1}\right\}}^{\frac{1}{2}}$。

证明：容易得到以下结果

$\begin{array}{l}\frac{1}{n}{X}_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta X_{h_1}\left(u\right)\\ =\left(\begin{array}{cc}\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{X}_i{X}_i^{\text{T}}{K}_{1,h_1}\left(U_i-u\right)}& \frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{X}_i{X}_i^{\text{T}}\frac{U_i-u}{h_1}{K}_{1,h_1}\left(U_i-u\right)}\\ \frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{X}_i{X}_i^{\text{T}}\frac{U_i-u}{h_1}{K}_{1,h_1}\left(U_i-u\right)}& \frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{X}_i{X}_i^{\text{T}}{\left(\frac{U_i-u}{h_1}\right)}^2{K}_{1,h_1}\left(U_i-u\right)}\end{array}\right)\end{array}$

根据Mack和Silverman (1982) [5] 命题1及迭代期望，有

$\begin{array}{c}\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{X}_i{X}_i^{\text{T}}{K}_{1,h_1}\left(U_i-u\right)}=\Gamma \left(u\right)f\left(u\right)+O\left(h_1^2\right)+O_p\left({\left\{\frac{\log \left(1/h_1\right)}{n{h}_1}\right\}}^{\frac{1}{2}}\right)\\ =\Gamma \left(u\right)f\left(u\right)\left[1+O_p\left(c_n\right)\right]\end{array}$ (3.5)

根据和(3.5)相似的证明方式，有

$\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{X}_i{X}_i^{\text{T}}\frac{U_i-u}{h_1}{K}_{1,h_1}\left(U_i-u\right)}=O\left(h_1^2\right)+O_p\left({\left\{\frac{\log \left(1/h_1\right)}{n{h}_1}\right\}}^{\frac{1}{2}}\right)$

$\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{X}_i{X}_i^{\text{T}}{\left(\frac{U_i-u}{h_1}\right)}^2{K}_{1,h_1}\left(U_i-u\right)}=\Gamma \left(u\right)f\left(u\right){\mu }_{1，2}\left[1+O_p\left(c_n\right)\right]$

因此，(3.1)式得证，利用相似的方式可证(3.2)，(3.3)和(3.4)式。

引理2 假定条件C1~C8成立，我们有

$\frac{1}{n}{g}^{\prime }{\left({\beta }_0\right)}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)g^{\prime }\left({\beta }_0\right)=\Sigma \left\{1+o_p\left(1\right)\right\}$ (3.6)

其中 $\Sigma =E\left[\pi \left(V\right)g^{\prime }\left(Z,{\beta }_0\right)g^{\prime }{\left(Z,{\beta }_0\right)}^{\text{T}}\right]-E\left[\Phi {\left(U\right)}^{\text{T}}{\Gamma }^{-1}\left(U\right)\Phi \left(U\right)\right]$

$\frac{1}{n}{g}^{\prime }{\left({\beta }_0\right)}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)\left(Y-g\left({\beta }_0\right)\right)={\xi }_n+O_p\left(c_n^2\right)$ (3.7)

其中 ${\xi }_n=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i\left\{g^{\prime }\left(Z_i,{\beta }_0\right)-E\left[g^{\prime }\left(Z_i,{\beta }_0\right)X_i^{\text{T}}|U=U_i\right]\Gamma {\left(U_i\right)}^{-1}{X}_i\right\}{\epsilon }_i}$

证明：结合(3.1)和(3.2)式，我们可得

$\left(X^{\text{T}},0\right){\left[X_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta X_{h_1}\left(u\right)\right]}^{-1}{X}_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta g^{\prime }\left({\beta }_0\right)=X^{\text{T}}\Gamma {\left(u\right)}^{-1}\Phi \left(u\right)\left[1+O_p\left(c_n\right)\right]$

根据弱大数定律和上面的式子，有

$\frac{1}{n}{g}^{\prime }{\left({\beta }_0\right)}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)g^{\prime }\left({\beta }_0\right)=\Sigma \left(1+o_p\left(1\right)\right)+O_p\left(c_n^2\right)E\left[\Phi {\left(U\right)}^{\text{T}}{\Gamma }^{-1}\left(U\right)\Phi \left(U\right)\right]\left[1+o_p\left(1\right)\right]$

在条件C6下， $c_n^2=o\left(1\right)$，因此(3.6)式成立。下一步我们证明(3.7)式

令 $M_0={\left(X_1^{\text{T}}{\theta }_0\left(U_1\right),\cdots ,X_n^{\text{T}}{\theta }_0\left(U_n\right)\right)}^{\text{T}}$，根据(3.1)式和(3.3)式，有

$\left(X^{\text{T}},0\right){\left[X_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta X_{h_1}\left(u\right)\right]}^{-1}{X}_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta M_0=X^{\text{T}}{\theta }_0\left(u\right)\left[1+O_p\left(c_n\right)\right]$

因此， $\frac{1}{n}{g}^{\prime }{\left({\beta }_0\right)}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)M_0=O_p\left(c_n^2\right)$

再根据(3.1)式，(3.4)式和 $E\left(\epsilon |V\right)=0$，有

$\left(X^{\text{T}},0\right){\left[X_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta X_{h_1}\left(u\right)\right]}^{-1}{X}_{h_1}{\left(u\right)}^{\text{T}}{W}_1\left(u\right)\Delta \epsilon =O_p(\; c\; n\; )$

此时

$\frac{1}{n}{g}^{\prime }{\left({\beta }_0\right)}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)\epsilon ={\xi }_n+O_p\left(c_n^2\right)$

因此

$\frac{1}{n}{g}^{\prime }{\left({\beta }_0\right)}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)\left(Y-g\left({\beta }_0\right)\right)={\xi }_n+O_p\left(c_n^2\right)$，(3.7)式子得证。

定理2.1的证明：根据Li和Mei (2013) [1] 定理1的证明，可类似证明 ${\hat{\beta }}_c$ 的一致性，即对于充分小的a，有

$\underset{n\to \infty }{\lim }P\left[\underset{\Vert t\Vert =a}{\sup }{Q}_c\left({\beta }_0+t\right)>Q_c\left({\beta }_0\right)\right]=1$ (3.8)

以及 $t^{\text{T}}{Q^{″}}_c\left({\beta }^{*}\right)t=2n\left\{t^{\text{T}}\Sigma t+O_p\left({\Vert t\Vert }^3\right)\right\}$，此处细节省略，下面证明 ${\hat{\beta }}_c$ 的渐近正态性，

又 $Q_c\left(\beta \right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{\left\{Y_i-X_i^{\text{T}}{\hat{\theta }}_c\left(U_i,\beta \right)-g\left(Z_i,\beta \right)\right\}}^2}={\left[Y-g\left(\beta \right)\right]}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)\left[Y-g\left(\beta \right)\right]$

和

${Q^{\prime }}_c\left({\beta }_0\right)=-2g^{\prime }{\left({\beta }_0\right)}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)\left(Y-g\left({\beta }_0\right)\right)$，

由(3.7)式，我们可以得到

$\frac{1}{n}{Q^{\prime }}_c\left({\beta }_0\right)=-\frac{2}{n}{g}^{\prime }{\left({\beta }_0\right)}^{\text{T}}{\left(I_n-S_{h_1}\right)}^{\text{T}}\Delta \left(I_n-S_{h_1}\right)\left(Y-g\left({\beta }_0\right)\right)=-2{\xi }_n+O_p\left(c_n^2\right)$

根据泰勒展开

$0={Q^{\prime }}_c\left({\hat{\beta }}_c\right)={Q^{\prime }}_c\left({\beta }_0\right)+{Q^{″}}_c{\left({\beta }^{*}\right)}^{\text{T}}\left({\hat{\beta }}_c-{\beta }_0\right)$，

结合 $\frac{1}{2n}{Q^{″}}_c\left({\beta }^{*}\right)=\Sigma \left\{1+o_p\left(1\right)\right\}$，有 $\sqrt{n}\Sigma \left\{1+o_p\left(1\right)\right\}\left({\hat{\beta }}_c-{\beta }_0\right)=\sqrt{n}{\xi }_n+o_p\left(1\right)$，

注意到， $\sqrt{n}{\xi }_n=\frac{1}{\sqrt{n}}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i\left\{g^{\prime }\left(Z_i,{\beta }_0\right)-E\left[g^{\prime }\left(Z_i,{\beta }_0\right)X_i^{\text{T}}|U=u_i\right]\Gamma {\left(U_i\right)}^{-1}{X}_i\right\}{\epsilon }_i}$，

根据Slutsky定理和中心极限定理，可得 $\sqrt{n}\left({\hat{\beta }}_c-{\beta }_0\right)\stackrel{D}{\to }N\left(0,{\sigma }^2{\Sigma }^{-1}\right)$。

定理2.2的证明：根据 ${\hat{\Psi }}_c\left(u_0;{\hat{\beta }}_c\right)$ 的定义，我们可以得到

$\begin{array}{c}{\hat{\Psi }}_c\left(u_0;{\hat{\beta }}_c\right)={\left[X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta X\left(u_0\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta M_0\\ +{\left[X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta X_{h_1}\left(u_0\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \epsilon \\ -{\left[X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta X_{h_1}\left(u_0\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \left[g\left(Z,{\hat{\beta }}_c\right)-g\left(Z,{\beta }_0\right)\right]\\ \stackrel{def}{=}{I}_1+I_2-I_3\end{array}$

易知 $I_3={\left[X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta X_{h_1}\left(u_0\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \left[g^{\prime }\left({\beta }_0\right)\left({\hat{\beta }}_c-{\beta }_0\right)+O_p{\Vert {\hat{\beta }}_c-{\beta }_0\Vert }^2\right]=O_p\left(n^{-\frac{1}{2}}\right)$

下一步计算再计算 $I_1$，因为

$\begin{array}{c}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta M_0=X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta X_{h_1}\left(u_0\right){\Psi }_0\left(u_0\right)+\frac{1}{2}{h}_1^2{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta A\left(u_0\right)\\ +X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta I_{n}o\left(h_1^2\right)\end{array}$

其中， $A\left(u_0\right)=\left(\begin{array}{c}{\left(\frac{U_1-u_0}{h_1}\right)}^2{X}_1^{\text{T}}{{\theta }^{″}}_0\left(u_0\right)\\ ⋮\\ \left(\frac{U_1-u_0}{h_1}\right)X_n^{\text{T}}{{\theta }^{″}}_0\left(u_0\right)\end{array}\right)$。

又 $X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta A\left(u_0\right)=nf\left(u_0\right)\Gamma \left(u_0\right)\otimes {\left({\mu }_{1,2},0\right)}^{\text{T}}\left[1+O_p\left(c_n\right)\right]{\theta }^{″}\left(u_0\right)$，

$X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta I_{n}o\left(h_1^2\right)=nf\left(u_0\right)E\left(\pi \left(V\right)X|U=u_0\right)\otimes {\left(1,0\right)}^{\text{T}}\left[1+O_p\left(c_n\right)\right]o\left(h_1^2\right)$，

此时 $I_1={\left[X_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta X_{h_1}\left(u_0\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta M_0={\Psi }_0\left(u_0\right)+\frac{1}{2}{h}_1^2{\mu }_{1，2}\left(\begin{array}{c}{{\theta }^{″}}_0\left(u_0\right)\\ 0\end{array}\right)+o_p\left(h_1^2\right)$

因此， $\sqrt{n{h}_1}\left[{\hat{\Psi }}_c\left(u_0;{\hat{\beta }}_c\right)-{\Psi }_0\left(u_0\right)-\frac{1}{2}{h}_1^2{\mu }_{1，2}\left(\begin{array}{c}{{\theta }^{″}}_0\left(u_0\right)\\ 0\end{array}\right)\right]=\sqrt{n{h}_1}{I}_2+O_p\left(\sqrt{n{h}_1^5}+\sqrt{h_1}\right)$，

容易推导 $I_2=\frac{1}{n}f{\left(u_0\right)}^{-1}\Gamma {\left(u_0\right)}^{-1}\otimes \left(\begin{array}{cc}1& 0\\ 0& {\mu }_{1,2}^{-1}\end{array}\right){\left[1+O_p\left(c_n\right)\right]}^{-1}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \epsilon$，

又因为

$\frac{1}{n}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \epsilon =\left(\begin{array}{c}\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{\epsilon }_i{X}_i{K}_{1,h_1}\left(U_i-u_0\right)}\\ \frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\delta }_i{\epsilon }_i{X}_i\frac{U_i-u_0}{h_1}{K}_{1,h_1}\left(U_i-u_0\right)}\end{array}\right)$

和 $\sqrt{n{h}_1}\frac{1}{n}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \epsilon \stackrel{D}{\to }N\left(0,{\Sigma }^{*}\right)$，其中 ${\Sigma }^{*}={\sigma }^2\Gamma \left(u_0\right)f\left(u_0\right)\otimes \left(\begin{array}{cc}{\nu }_{1，0}& 0\\ 0& {\nu }_{1,2}\end{array}\right)$，

最后我们可以得到

$\begin{array}{l}\sqrt{n{h}_1}{I}_2=f{\left(u_0\right)}^{-1}{\Gamma }^{-1}\left(u_0\right)\otimes \left(\begin{array}{cc}1& 0\\ 0& {\mu }_{1，2}^{-1}\end{array}\right){\left[1+O_p\left(c_n\right)\right]}^{-1}\sqrt{n{h}_1}\frac{1}{n}{X}_{h_1}^{\text{T}}\left(u_0\right)W_1\left(u_0\right)\Delta \epsilon \\ \stackrel{D}{\to }N\left(0,{\sigma }^{2}f{\left(u_0\right)}^{-1}{\Gamma }^{-1}\left(u_0\right)\otimes \left(\begin{array}{cc}{\nu }_{1，0}& 0\\ 0& {\nu }_{1，2}{\mu }_{1，2}^{-2}\end{array}\right)\right)\end{array}$

现在定理2.2的证明完成。

致谢

感谢本论文所引用文献的作者为本论文的创作提供了宝贵的思想，也同时感谢身边的人给我提供的帮助。