Weighted quantile regression and testing for varying-coefficient models with randomly truncated data

Abstract

This paper develops a varying-coefficient approach to the estimation and testing of regression quantiles under randomly truncated data. In order to handle the truncated data, the random weights are introduced and the weighted quantile regression (WQR) estimators for nonparametric functions are proposed. To achieve nice efficiency properties, we further develop a weighted composite quantile regression (WCQR) estimation method for nonparametric functions in varying-coefficient models. The asymptotic properties both for the proposed WQR and WCQR estimators are established. In addition, we propose a novel bootstrap-based test procedure to test whether the nonparametric functions in varying-coefficient quantile models can be specified by some function forms. The performance of the proposed estimators and test procedure are investigated through simulation studies and a real data example.

Appendix

Lemma A.1

Let $(X_1,Y_1), \ldots , (X_n,Y_n)$ be independent and identically distributed (i.i.d.) random vectors. Assume that $E|Y|^r<\infty , sup_x\int |y|^rf(x,y){\hbox {d}}y<\infty$, where f denotes the joint density of (X, Y). Let K be a bounded positive function with a bounded support, satisfying a Lipschitz condition. Then

$$\begin{aligned} \sup _{x} \Bigg |\frac{1}{n}\sum ^n_{i=1}\big [K_h(X_i-x)Y_i-E(K_h(X_i-x)Y_i)\big ]\Bigg |=O_p\Bigg (\frac{\log ^{1/2}(1/h)}{\sqrt{nh}}\Bigg ), \end{aligned}$$

provided that $n^{2\varepsilon -1}h\rightarrow \infty$ for some $\varepsilon <1-r^{-1}$.

Lemma A.1 follows from the result by Mack and Silverman (1982).

Lemma A.2

(Lv et al. 2015) Suppose $A_n(s)$ is convex and can be represented as $\frac{1}{2}s^\mathrm{T}Vs+U_n^\mathrm{T}s+C_n+r_n(s)$, where V is symmetric and positive definite, $U_n$ is stochastically bounded, $C_n$ is arbitrary and $r_n(s)$ goes to zero in probability for each s. Then $\alpha _n$, the argmin of $A_n$, is only $o_p(1)$ away from $\beta _n=-V^{-1}U_n$, the argmin of $\frac{1}{2}s^\mathrm{T}Vs+U_n^\mathrm{T}s+C_n$. If also $U_n {\mathop {\rightarrow }\limits ^{\mathcal {D}}}U$, then $\alpha _n{\mathop {\rightarrow }\limits ^{\mathcal {D}}}-V^{-1}U$.

Let $\eta ^{*}_{i,k}=I(\varepsilon _i-c_{\tau _{k}}+r_i(u)\le 0)-\tau _k, ~\eta _{i,k}=I(\varepsilon _i-c_{\tau _{k}}\le 0)-\tau _k, ~\delta _n=\Big (\frac{\log (1/h)}{nh}\Big )^{1/2}$.

Proof of Theorem 2.1

The proof of Theorem 2.1 follows similar strategies in Theorem 2.2, and we omit the details here.

Proof of Theorem 2.2

Recall that $\{\hat{a}_{0,1}, \ldots , \hat{a}_{0,q}, \hat{\mathbf{a}}, \hat{b}_0, \hat{\mathbf{b}}\}$ minimizes

$$\begin{aligned} \sum _{k=1}^q\Bigg [\sum _{i=1}^n\frac{1}{G_n(Y_i)}\rho _{\tau _k}\big [Y_i-a_{0,k}-b_0(U_i-u)-X_i^\mathrm{T}\{\mathbf{a}+\mathbf{b}(U_i-u)\}\big ]K_h(U_i-u)\Bigg ]. \end{aligned}$$

Denote

$$\begin{aligned} \hat{\xi }&=\sqrt{nh}\left( \begin{array}{ccc} \hat{a}_{0,1}-\alpha _0(u)-c_{\tau _{1}}\\ \vdots \\ \hat{a}_{0,q}-\alpha _0(u)-c_{\tau _{q}}\\ \hat{\mathbf{a}}-\alpha (u)\\ h(\hat{b}_0-\alpha '_0(u))\\ h(\hat{\mathbf{b}}-\alpha '(u))\\ \end{array}\right) , \xi =\sqrt{nh}\left( \begin{array}{ccc} a_{0,1}-\alpha _0(u)-c_{\tau _{1}}\\ \vdots \\ a_{0,q}-\alpha _0(u)-c_{\tau _{q}}\\ \mathbf{a}-\alpha (u)\\ h(b_0-\alpha '_0(u))\\ h(\mathbf{b}-\alpha '(u)) \end{array}\right) ,\\ N_{i,k}&= \left( \begin{array}{c} \mathbf{e_k}\\ X_i\\ \frac{U_i-u}{h}\\ X_i \frac{U_i-u}{h} \end{array}\right) , \end{aligned}$$

$\mathbf{e_k}$ is a q-vector with 1 at the kth position and 0 elsewhere. We write $Y_i-a_{0,k}-b_0(U_i-u)-X_i^\mathrm{T}\{\mathbf{a}+\mathbf{b}(U_i-u)\}=\varepsilon _i-c_{\tau _{k}}+r_i(u)-N^\mathrm{T}_{i,k}\xi /\sqrt{nh}:=\varepsilon _i-c_{\tau _{k}}+r_i(u)-\Delta _{i,k}$, where $r_i(u)=\alpha _0(U_i)-\alpha _0(u)-\alpha '_0(u)(U_i-u)+X_i^\mathrm{T}\{\alpha (U_i)-\alpha (u)-\alpha '(u)(U_i-u)\}, K_i(u)=K(\frac{U_i-u}{h})$, then $\hat{\xi }$ will be the minimizer of

$$\begin{aligned} Q_n(\xi )=\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G_n(Y_i)}\bigg [\rho _{\tau _k}(\varepsilon _i-c_{\tau _{k}}+r_i(u)-N^\mathrm{T}_{i,k}\xi /\sqrt{nh})-\rho _{\tau _k}(\varepsilon _i-c_{\tau _{k}}+r_i(u))\bigg ]. \end{aligned}$$

Following the identity by Knight (1998),

$$\begin{aligned} \rho _\tau (u-v)-\rho _\tau (u)=-v\psi _\tau (u)+\int _0^v\big [I(u\le s)-I(u\le 0)\big ]{\hbox {d}}s, \end{aligned}$$

where $\psi _\tau (u)=\tau -I(u\le 0)$. Then we obtain

$$\begin{aligned} Q_n(\xi )=&\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G_n(Y_i)}\Bigg [ \frac{N^\mathrm{T}_{i,k}\xi }{\sqrt{nh}}\{I(\varepsilon _i-c_{\tau _{k}}+r_i(u))-\tau _k\}\\&+\int _{0}^{\Delta _{i,k}}\big [I(\varepsilon _i-c_{\tau _{k}}+r_i(u)\le z)-I(\varepsilon _i-c_{\tau _{k}}\le 0)\big ]{\hbox {d}}z\Bigg ]\\ =&\,\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G_n(Y_i)}N_{i,k}^\mathrm{T}\xi \{I(\varepsilon _i-c_{\tau _{k}}+r_i(u)\le 0)-\tau _k\}\\&+\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G_n(Y_i)}\int _{0}^{\Delta _{i,k}}\big [I(\varepsilon _i-c_{\tau _{k}}+r_i(u)\le z)-I(\varepsilon _i-c_{\tau _{k}}+r_i(u)\le 0)\big ]{\hbox {d}}z\\ :=\,&W_{n,k}^\mathrm{T}(u)\xi +\sum _{k=1}^qB_{n,k}(\xi ), \end{aligned}$$

where $W_{n,k}(u)=\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G_n(Y_i)}\eta ^{*}_{i,k}N_{i,k}, \eta ^{*}_{i,k}=I(\varepsilon _i-c_{\tau _{k}}+r_i(u)\le 0)-\tau _k$.

Firstly, we prove that $E\{\sum _{k=1}^qB_{n,k}(\xi )\}=\frac{1}{2}\xi ^\mathrm{T}\frac{f_U(u)}{\theta }S(u)\xi$. Let $\widetilde{B}_{n,k}(\xi )=\sum _{i=1}^n\frac{K_i(u)}{G(Y_i)}\int _{0}^{\Delta _{i,k}}\big \{I(\varepsilon _i\le c_{\tau _{k}}-r_i(u)+z)-I(\varepsilon _i\le c_{\tau _{k}}-r_i(u))\big \}{\hbox {d}}z$, $\Delta (u,x,\mu )$ and $r_i(u,x,\mu )$ be equal to $N_{i,k}^\mathrm{T}\xi /\sqrt{nh}$ and $r_i(u)$, where $X_i, U_i$ are replaced by $x,\mu$.

Since $\widetilde{B}_{n,k}(\xi )$ is a summation of i.i.d. random variables of the kernel form, according to Lemma A.1, we have $\widetilde{B}_{n,k}(\xi )=E[\widetilde{B}_{n,k}(\xi )]+O_p(\delta _n)$. The expectation of $\widetilde{B}_{n,k}(\xi )$ is

$$\begin{aligned} E\{\widetilde{B}_{n,k}(\xi )\}=&\sum _{k=1}^q\sum _{i=1}^nE\Bigg [\frac{K_i(u)}{G(Y_i)}\int _{0}^{\Delta _{i,k}}\big \{I(\varepsilon _i\le c_{\tau _{k}}-r_i(u)+z)-I(\varepsilon _i\le c_{\tau _{k}}-r_i(u))\big \}{\hbox {d}}z\Bigg ]\\ =\,&\sum _{k=1}^q\sum _{i=1}^n\int \int \int \frac{1}{G(y)}K\left( \frac{\mu -u}{h}\right) \int _{0}^{\Delta (u,x,\mu )}\\ {}&\big [I(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{\tau _{k}}-r_i(u,x,\mu )+z)\\&-I(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{\tau _{k}}-r_i(u,x,\mu ))\big ]{\hbox {d}}zf^{*}(x,\mu ,y){\hbox {d}}x{\hbox {d}}\mu {\hbox {d}}y\\ =\,&\frac{1}{\theta }\sum _{k=1}^q\sum _{i=1}^n\mathbb {E}\bigg \{K_i(u)\int _{0}^{\Delta _{i,k}}\big \{I(\varepsilon _i\le c_{\tau _{k}}-r_i(u)+z)-I(\varepsilon _i\le c_{\tau _{k}}-r_i(u))\big \}{\hbox {d}}z\bigg \}\\ =\,&\frac{1}{\theta }\sum _{k=1}^q\sum _{i=1}^n\mathbb {E}\bigg \{K_i(u)\mathbb {E}\big \{\int _{0}^{\Delta _{i,k}}\big [\big \{I(\varepsilon _i\le c_{\tau _{k}}-r_i(u)+z)\\ {}&-I(\varepsilon _i\le c_{\tau _{k}}-r_i(u))\big \}{\hbox {d}}z\big \}|U\big ]\big \}\bigg \}\\ =\,&\frac{1}{\theta }\sum _{k=1}^q\sum _{i=1}^n\mathbb {E}\bigg \{K_i(u)\int _{0}^{\Delta _{i,k}}\big [F_{\varepsilon _i}(c_{\tau _{k}}-r_i(u)+z)-F_{\varepsilon _i}(c_{\tau _{k}}-r_i(u))\big ]{\hbox {d}}z\bigg \}\\ =\,&\frac{1}{\theta }\sum _{k=1}^q\sum _{i=1}^n\mathbb {E}\bigg \{K_i(u)\int _{0}^{\Delta _{i,k}}\big [f_{\varepsilon _i}(c_{\tau _{k}}-r_i(u))z+o(1)\big ]{\hbox {d}}s\bigg \}\\ =\,&\frac{1}{2\theta }\xi ^\mathrm{T}\mathbb {E}\bigg \{\frac{1}{nh}\sum _{k=1}^q\sum _{i=1}^nK_i(u)f_{\varepsilon _i}(c_{\tau _{k}}-r_i(u))N_{i,k}N_{i,k}^\mathrm{T}\bigg \}\xi +O_p(\delta _n)\\ :=\,&\frac{1}{2\theta }\xi ^\mathrm{T}S_n(u)\xi +O_p(\delta _n). \end{aligned}$$

Further, we can prove that $\mathbb {E}\{S_n(u)\}=f_U(u)S(u)+O(h^2)$, where $S(u)=\text{ diag }\{S_1(u),c\mu _2S_2(u)\}$, $S_2(u)=\mathbb {E}\{(1,X^\mathrm{T})^\mathrm{T}(1,X^\mathrm{T})|U=u\}$, $c_{\tau _{k}}=F_{\varepsilon }^{-1}(\tau _k)$,

$$\begin{aligned} S_1(u)=\mathbb {E}\bigg \{\bigg (\begin{array}{c} C~~\mathbf{c}X^\mathrm{T}\\ X^\mathrm{T}{} \mathbf{c}~~ cXX^\mathrm{T} \end{array}\bigg )|U=u\bigg \}, \end{aligned}$$

C is a $q\times q$ diagonal matrix with $C_{jj}=f_{\varepsilon }(c_{\tau _{j}})$, $\mathbf{c}=(f_{\varepsilon }(c_{\tau _{1}}), \ldots , f_{\varepsilon }(c_{\tau _{q}}))^\mathrm{T}$, $c=\sum ^q_{k=1}f_{\varepsilon }(c_{\tau _{k}})$. Similarly, we can obtain $\text{ Var }\{{\widetilde{B}_{n,k}(\xi )}\}=o(1)$. Then $\widetilde{B}_{n,k}(\xi )=\frac{1}{2}\xi ^\mathrm{T}\frac{f_U(u)}{\theta }S(u)\xi +O_p(\delta _n)$. According to Lemma 5.2 in Liang and Baek (2016), we have

$$\begin{aligned} \text{ sup }_y|G_n(y)-G(y)|=O_p(n^{-1/2}). \end{aligned}$$

(A.1)

By some calculations, we can obtain

$$\begin{aligned} | B_{n,k}(\xi )- \widetilde{B}_{n,k}(\xi )|=O_p(h^{\frac{1}{2}})=o_p(1). \end{aligned}$$

(A.2)

Thus,

$$\begin{aligned} Q_{n}(\xi )&=W_{n,k}^\mathrm{T}(u)\xi +E[\widetilde{B}_{n,k}(\xi )]+O_p(\delta _n)\\&=W_{n,k}^\mathrm{T}(u)\xi +\frac{1}{2}\xi ^\mathrm{T}\frac{f_U(u)}{\theta }S(u)\xi +O_p(\delta _n+h^2). \end{aligned}$$

According to Lemma A.2, the minimizer of $Q_{n}(\xi )$ can be expressed as

$$\begin{aligned} \hat{\xi }=-\theta f^{-1}_U(u)S^{-1}(u)W_{n,k}(u)+o_p(1). \end{aligned}$$

(A.3)

Therefore,

$$\begin{aligned} \sqrt{nh}\left( \begin{array}{ccc} \hat{a}_{0,1}-\alpha _0(u)-c_{\tau _{1}}\\ \vdots \\ \hat{a}_{0,q}-\alpha _0(u)-c_{\tau _{q}}\\ \hat{a}-\alpha (u) \end{array}\right) =-\theta f^{-1}_U(u)S^{-1}_1(u)W^*_{n,k}(u)+o_p(1), \end{aligned}$$

(A.4)

where $W^*_{n,k}(u)=\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G_n(Y_i)}\eta ^{*}_{i,k}(e_k^\mathrm{T},X_i^\mathrm{T})^\mathrm{T}$.

Denote $\widetilde{W}^*_{n,k}(u)=\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G(Y_i)}\eta _{i,k}(e_k^\mathrm{T},X_i^\mathrm{T})^\mathrm{T}:=(w_{11}, \ldots , w_{1q},w_{21})^\mathrm{T}$, where $w_{1k}=(nh)^{-1/2}\sum _{i=1}^n\frac{K_i(u)}{G(Y_i)}\eta _{i,k}, k=1,\ldots ,q$, and $w_{21}=(nh)^{-1/2}\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G(Y_i)}\eta _{i,k}X_i$. Note that $\text{ Cov }(\eta _{i,k}, \eta _{i,k^{'}})=\tau _{kk^{'}}=\tau _k\wedge \tau _{k^{'}}-\tau _k\tau _{k^{'}}$ and $\text{ Cov }(\eta _{i,k}, \eta _{j,k^{'}})=0$ if $i\ne j$. Then

$$\begin{aligned} E(w_{1k})&=\frac{1}{\sqrt{nh}}\sum _{i=1}^nE\bigg \{{\frac{K_i(u)}{G(Y_i)}\eta _{i,k}}\bigg \}\\&=\frac{1}{\theta \sqrt{nh}}\sum _{i=1}^n\mathbb {E}\bigg [K_i(u)\mathbb {E}\big \{(I(\varepsilon _i\le c_{\tau _{k}})-\tau _k)|U\big \}\bigg ]\\&=\frac{1}{\theta \sqrt{nh}}\sum _{i=1}^n\mathbb {E}\bigg [K_i(u)\{F_{\varepsilon _i}(c_{\tau _{k}})-\tau _k\}\bigg ]=0. \end{aligned}$$

Similarly, we can obtain $E(w_{21})=0$. On the other hand,

$$\begin{aligned} \text{ Cov }&(w_{1k}, w_{1k^{'}})=E(w_{1k}w_{1k^{'}})=\frac{1}{nh}\sum _{i=1}^nE\bigg \{{\frac{K^2_i(u)}{G^2(Y_i)}\eta _{i,k}\eta _{i,k^{'}}}\bigg \}\\ =&\frac{1}{h}\int \int \int \frac{K^2(\frac{\mu -u}{h})}{G^2(y)}\{I(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{\tau _{k}})-\tau _k\}\\&\{I(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{k^{'}})-\tau _{k^{'}}\}f^{*}(\mu ,x,y){\hbox {d}}\mu {\hbox {d}}x {\hbox {d}}y\\ =&\frac{1}{\theta h}\int \int \int \frac{K^2(\frac{\mu -u}{h})}{G(y)}\{I(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{\tau _{k}})-\tau _k\}\\&\{I(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{\tau _{k^{'}}})-\tau _{k^{'}}\}f(\mu ,x,y){\hbox {d}}\mu {\hbox {d}}x {\hbox {d}}y\\ =&\frac{1}{\theta h}\int \int \int \frac{K^2(\frac{\mu -u}{h})f(\mu ,x,y)}{G(y)}\Big \{I(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{\tau _{k}}\wedge c_{\tau _{k^{'}}})-\\&\tau _kI(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{\tau _{k^{'}}})-\tau _{k^{'}}I(y\le \alpha _0(\mu )+x^\mathrm{T}\alpha (\mu )+c_{k})+\tau _k\tau _{k^{'}}\Big \}{\hbox {d}}\mu {\hbox {d}}x {\hbox {d}}y\\ \rightarrow&\frac{1}{\theta }\int \int \int \frac{K^2(t)f(u,x,y)}{G(y)}\Big \{I(y\le \alpha _0(u)+x^\mathrm{T}\alpha (u)+c_{\tau _{k}}\wedge c_{\tau _{k^{'}}})-\\&\tau _kI(y\le \alpha _0(u)+x^\mathrm{T}\alpha (u)+c_{\tau _{k^{'}}})-\tau _{k^{'}}I(y\le \alpha _0(u)+x^\mathrm{T}\alpha (u)+c_{k})+\tau _k\tau _{k^{'}}\Big \}{\hbox {d}}x{\hbox {d}}t{\hbox {d}}y+o_p(1)\\ =&\frac{\nu _0 f_U(u)}{\theta }\lambda ^0_{kk^{'}}(u):=\frac{\nu _0 f_U(u)}{\theta }A_{11}(u). \end{aligned}$$

Similarly, we can obtain that $\text{ Cov }(w_{1k}, w_{21})=\frac{\nu _0 f_U(u)}{\theta }\sum _{k^{'}=1}^q\lambda ^1_{kk^{'}}(u):=\frac{\nu _0 f_U(u)}{\theta }A_{12}(u)$ and $\text{ Var }(w_{21})=\frac{\nu _0 f_U(u)}{\theta }\sum _{k=1}^q\sum _{k^{'}=1}^q\lambda ^2_{kk^{'}}(u):=\frac{\nu _0 f_U(u)}{\theta }A_{22}(u)$.

By the Cramér–Wald theorem and the central limit theorem, we have

$$\begin{aligned} \widetilde{W}^*_{n,k}(u){\mathop {\rightarrow }\limits ^{\mathcal {D}}}N\Big (0,\frac{\nu _0 f_U(u)}{\theta }A(u)\Big ). \end{aligned}$$

Define $\overline{W}^*_{n,k}(u)=\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G(Y_i)}\eta ^*_{i,k}(e_k^\mathrm{T},X_i^\mathrm{T})^\mathrm{T}:=(\bar{w}_{11}, \ldots , \bar{w}_{1q},\bar{w}_{21})^\mathrm{T},$ where $\bar{w}_{1k}=\frac{1}{\sqrt{nh}}\sum _{i=1}^n\frac{K_i(u)}{G(Y_i)}\eta ^*_{i,k}, k=1,\ldots ,q,$ and $\bar{w}_{21}=\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G(Y_i)}\eta ^*_{i,k}X_i$.

By some calculations, we have

$$\begin{aligned}&\text{ Var }(\bar{w}_{1k}-\tilde{w}_{1k})\le \frac{C_0}{\theta G(a_F)nh}\sum _{i=1}^n\mathbb {E}\{K^2_i(u)(\eta ^*_{i,k}-\eta _{i,k})^2\}=o(1)~\text{ and }~\\&\quad \text{ Var }(\bar{w}_{21}-\tilde{w}_{21})=o(1). \end{aligned}$$

Thus $\text{ Var }\{\overline{W}^*_{n,k}(u)-\widetilde{W}^*_{n,k}(u)\}=o(1)$. By Slutsky’s theorem, we have

$$\begin{aligned} \overline{W}^*_{n,k}(u)-E\{\overline{W}^*_{n,k}(u)\}{\mathop {\rightarrow }\limits ^{\mathcal {D}}}N\Big (0,\frac{\nu _0 f_U(u)}{\theta }A(u)\Big ). \end{aligned}$$

(A.5)

Note that

$$\begin{aligned} W^*_{n,k}(u)=W^*_{n,k}(u)-\overline{W}^*_{n,k}(u)+\big [\overline{W}^*_{n,k}(u)-E\{\overline{W}^*_{n,k}(u)\}\big ]+E\{\overline{W}^*_{n,k}(u)\}. \end{aligned}$$

(A.6)

Similar to the proof of (A.2), we have $W^*_{n,k}(u)-\overline{W}^*_{n,k}(u)=o_p(1)$. Thus,

$$\begin{aligned} W^*_{n,k}(u)-E\{\overline{W}^*_{n,k}(u)\}=\overline{W}^*_{n,k}(u)-E\{\overline{W}^*_{n,k}(u)\}+o_p(1). \end{aligned}$$

(A.7)

Next we calculate the mean of $\overline{W}^*_{n,k}(u)$. In fact

$$\begin{aligned} \frac{1}{\sqrt{nh}}E(\overline{W}^*_{n,k}(u))&=\frac{1}{nh}E\bigg \{\sum _{k=1}^q\sum _{i=1}^n\frac{K_i(u)}{G(Y_i)}\eta ^*_{i,k}( e^\mathrm{T}_k,X_i^\mathrm{T})^\mathrm{T}\bigg \}\nonumber \\&=\frac{1}{nh\theta }\sum _{k=1}^q\sum _{i=1}^n\mathbb {E}\big [K_i(u)\mathbb {E}\{(I(\varepsilon _i-c_{\tau _{k}}+ r_i(u)\le 0)-\tau _k)|U,X\}(e^\mathrm{T}_k,X_i^\mathrm{T})^\mathrm{T}\big ]\nonumber \\&=\frac{1}{nh\theta }\sum _{k=1}^q\sum _{i=1}^n\mathbb {E}\big [K_i(u)\{F_{\varepsilon }(c_{\tau _{k}}- r_i(u))-F_{\varepsilon }(c_{\tau _{k}})\}(e^\mathrm{T}_k,X_i^\mathrm{T})^\mathrm{T}\big ]\nonumber \\&=-\frac{1}{nh\theta }\sum _{k=1}^q\sum _{i=1}^n\mathbb {E}\big \{K_i(u)f_{\varepsilon }(c_{\tau _{k}})r_i(u)\big (1+o(1)\big )(e^\mathrm{T}_k,X_i^\mathrm{T})^\mathrm{T}\big \}\nonumber \\&=-\frac{\mu _2h^2}{2\theta }f_U(u)S_1(u)\bigg (\begin{array}{ccc} \alpha _0''(u)\\ \alpha ''(u) \end{array}\bigg )+o_p(h^2). \end{aligned}$$

(A.8)

Combining (A.4)–(A.8), the proof of Theorem 2.2 is completed.