Weighted composite quantile regression for single index model with missing covariates at random

Abstract

This paper considers weighted composite quantile estimation of the single-index model with missing covariates at random. Under some regularity conditions, we establish the large sample properties of the estimated index parameters and link function. The large sample properties of the parametric part show that the estimator with estimated selection probability have a smaller limiting variance than the one with the true selection probability. However, the large sample properties of the estimated link function indicate that whether weights were estimated or not has no effect on the asymptotic variance. Studies of simulation and the real data analysis are presented to illustrate the behavior of the proposed estimators.

Appendix

Lemma 1

Let $(X_{1}, Y_{1})$, $(X_{2}, Y_{2}),\ldots , (X_{n}, Y_{n})$ be independent and identically distributed random vectors, where $Y_{i}$ are scalar random variables. Furthermore, assume that $E|Y|^{s}<\infty$, and $\sup _{x}\int |Y|^{s}f(x,y)dy<\infty$, where $f(\cdot ,\cdot )$ denotes the joint density of (X, Y). Let K be a bounded positive function with a bounded support, satisfying the Lipschitz condition. Given that $n^{2\varepsilon -1}h\rightarrow \infty$ for some $\varepsilon <1-s^{-1}$, then

$$\begin{aligned} \sup _{x}|\frac{1}{n}\sum _{i=1}^{n}[K_{h}(X_{i}-x)Y_{i}-E(K_{h}(X_{i}-x)Y_{i})]|=O_{p}(\sqrt{\log (1/h)/(nh)}). \end{aligned}$$

Proof

This follows immediately from the result obtained by Mack and Silverman (1982). $\square$

Lemma 2

(Quadratic Approximation Lemma) Suppose $A_{n}(s)$ is convex and can be represented as $\frac{1}{2}s^{T}Vs+U^{T}_{n}s+C_{n}+r_{n}(s)$ , 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 minimizer of $A_{n}(s)$, is only $o_{p}(1)$ away from $\beta _{n}=-V^{-1}U_{n}$, the minimizer of $\frac{1}{2}s^{T}Vs+U^{T}_{n}s+C_{n}$. If also $U_{n}\xrightarrow {d} U$, then $\alpha _{n}\xrightarrow {d} -V^{-1}U$.

The proof of Lemma 2 is available from Hjort and Pollard (2011). The proof of the Quadratic Approximation Lemma indicates that the positive definiteness of the matrix V can be reduced into semi-positive definiteness, even to the existence of a generalized inverse, see Wu et al. (2010).

Lemma 3

Suppose that Conditions (C1)–(C7) hold. Let ${\hat{\beta }}^{0}$ be the initial estimator of $\beta _{0}$. As $h\rightarrow 0$ and $\delta _{n}\rightarrow 0$, we have

$$\begin{aligned}&\sqrt{nh}\left( \begin{array}{c} {\hat{a}}_{WCQR1}(u,{\hat{\beta }}^{0})-g(u)-c_{1}+g^{'}(u)E(X|u)^{T}{\tilde{\beta }}-\frac{h^{2}}{2}g^{''}(u)\mu _{2}+O(h^{2}\delta _{\beta }+\delta ^{2}_{\beta }+h^{3})\\ ... \\ {\hat{a}}_{WCQRq}(u,{\hat{\beta }}^{0})-g(u)-c_{q}+g^{'}(u)E(X|u)^{T}{\tilde{\beta }}-\frac{h^{2}}{2}g^{''}(u)\mu _{2}+O(h^{2}\delta _{\beta }+\delta ^{2}_{\beta }+h^{3})\\ h({\hat{g}}_{WCQR}'(u,{\hat{\beta }}^{0})-g'(u)+O(\delta _{\beta }+h^{2}))\\ \end{array} \right) \\&\quad =-\frac{S^{-1}}{f_{U_{0}}(u)}W_{n1}+o_{p}(1), \end{aligned}$$

where ${\tilde{\beta }}={\hat{\beta }}^{0}-\beta _{0}$, $\delta _{n}=\sqrt{\log (1/h)/(nh)}$ and $\delta _{\beta }=|{\hat{\beta }}^{0}-\beta _{0}|$.

Proof

Recall that $({\hat{a}}_{WCQR1}(u,{\hat{\beta }}^{0}),...,{\hat{a}}_{WCQRq}(u,{\hat{\beta }}^{0}),{\hat{g}}_{WCQR}'(u,{\hat{\beta }}^{0}))$ are obtained by minimizing the following objective function

$$\begin{aligned}&\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{\pi (Y_{i})}\rho _{\tau _{k}}[Y_{i}-a_{k}-b(X_{i}^{T}{\hat{\beta }}^{0}-u)]K_{i}(u), \end{aligned}$$

(A.1)

where $K_{i}(u)=K_{h}(X_{i}^{T}{\hat{\beta }}^{0}-u).$ Let ${\hat{\theta }}=\sqrt{nh}\{({\hat{a}}_{WCQR1}(u,{\hat{\beta }}^{0})-g(u)-c_{1}),...,({\hat{a}}_{WCQRq}(u,{\hat{\beta }}^{0})-g(u)-c_{q}),h({\hat{g}}_{WCQR}'(u,{\hat{\beta }}^{0})-g'(u))\}$, $r_{i}(u)=g(X_{i}^{T}\beta _{0})-g(u)-g^{'}(u)(X_{i}^{T}{\hat{\beta }}^{0}-u)$, $r^{0}_{i}(u)=g(X_{i}^{T}{\hat{\beta }}^{0})-g(u)-g^{'}(u)(X_{i}^{T}{\hat{\beta }}^{0}-u)$, $\eta _{ik}=I(\epsilon _{i}-c_{k}<0)-\tau _{k}$, $\eta _{ik}(u)=I(\epsilon _{i}-c_{k}+r_{i}(u)<0)-\tau _{k}$, $X_{ik}=(e_{k},(X_{i}^{T}{\hat{\beta }}^{0}-u)/h)^{T}$, $X^{*}_{k}=(e_{k},(X^{T}{\hat{\beta }}^{0}-u)/h)^{T}$, $e_{k}$ is a q-vector with 1 on the kth position and 0 elsewhere. Then, ${\hat{\theta }}$ is also the minimizer of

$$\begin{aligned} L_{n}(\theta )=\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{\pi (Y_{i})}\left\{ \rho _{\tau _{k}}[\epsilon _{i}-c_{k}{+}r_{i}(u){-}\frac{X_{ik}^{T}\theta }{\sqrt{nh}}]-\rho _{\tau _{k}}[\epsilon _{i}-c_{k}+r_{i}(u)]\right\} K_{i}(u). \end{aligned}$$

(A.2)

By Knight (1998), for any $x\ne 0$, we have

$$\begin{aligned} \rho _{\tau }(x-y)-\rho _{\tau }(x)=y[I(x<0)-\tau ]+\int _{0}^y[I(x\le t)-I(x\le 0)]dt. \end{aligned}$$

Then, we can get

$$\begin{aligned} L_{n}(\theta )=W_{n}^{T}\theta +\sum _{k=1}^q B_{n,k}(\theta ), \end{aligned}$$

where

$$\begin{aligned} W_{n}=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{\pi (Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik} \end{aligned}$$

and

$$\begin{aligned} B_{n,k}(\theta )=\sum _{i=1}^n\frac{V_{i}}{\pi (Y_{i})}\int _{0}^{\frac{X_{ik}^{T}\theta }{\sqrt{nh}}}[I(\epsilon _{i}-c_{k}+r_{i}(u)\le t)-I(\epsilon _{i}-c_{k}+r_{i}(u)\le 0)]dt K_{i}(u). \end{aligned}$$

Since $B_{n,k}(\theta )$ is a summation of i.i.d. random variables of the kernel dorm, we obtain by Lemma 1

$$\begin{aligned} B_{n,k}(\theta )=E[B_{n,k}(\theta )]+O_{p}(\delta _{n}). \end{aligned}$$

Let ${\hat{\chi }}^{0}$ be the $\sigma$-field generated by $\{X_{1}^{T}{\hat{\beta }}^{0},...,X_{n}^{T}{\hat{\beta }}^{0}\}$. Since ${\hat{\beta }}^{0}$ is a $\sqrt{n}$-consistent estimator of $\beta _{0}$, we have $f_{U}(.)=f_{U_{0}}(.)(1+o_{p}(1))$. Hence, the expectation of $\sum _{k=1}^qB_{n,k}(\theta )$ is

$$\begin{aligned} \sum _{k=1}^qE[B_{n,k}(\theta )]=\sum _{k=1}^qE\{E[B_{n,k}(\theta )|{\hat{\chi }}^{0}]\}=\frac{f_{U_{0}}(u)}{2}\theta ^{T}S\theta +O(\delta _{\beta })+O(h^{2}), \end{aligned}$$

where $S=diag(C,\mu _{2}c)$, $C=diag(f_{\epsilon }(c_{1}),...,f_{\epsilon }(c_{q}))$, $c=\sum _{k=1}^qf_{\epsilon }(c_{k})$. Hence, we have

$$\begin{aligned} L_{n}(\theta )&=W_{n}^{T}\theta +\frac{f_{U_{0}}(u)}{2}\theta ^{T}S\theta +O(\delta _{\beta })+O(h^{2})+O_{p}(\delta _{n})\nonumber \\&=W_{n}^{T}\theta +\frac{f_{U_{0}}(u)}{2}\theta ^{T}S\theta +o_{p}(1). \end{aligned}$$

(A.3)

By Lemma 2, we have

$$\begin{aligned} {\hat{\theta }}=-\frac{S^{-1}}{f_{U_{0}}(u)}W_{n}+o_{p}(1). \end{aligned}$$

(A.4)

Let $W_{n1}=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{\pi (Y_{i})}\eta _{ik}K_{i}(u)X_{ik}$, we can get $Var(W_{n}-W_{n1})=o_{p}(1)$, which implies

$$\begin{aligned} W_{n}-W_{n1}-E[W_{n}]=o_{p}(1). \end{aligned}$$

Then, we get

$$\begin{aligned} W_{n}=W_{n1}+E[W_{n}]+o_{p}(1). \end{aligned}$$

(A.5)

Combining (A.4) and (A.5), we obtain

$$\begin{aligned} {\hat{\theta }}=-\frac{S^{-1}}{f_{U_{0}}(u)}W_{n1}-\frac{S^{-1}}{f_{U_{0}}(u)}E[W_{n}]+o_{p}(1). \end{aligned}$$

(A.6)

Considering $E[W_{n}]$, we have

$$\begin{aligned} \frac{1}{\sqrt{nh}}E[W_{n}]&=-\frac{1}{nh}\sum _{k=1}^q \sum _{i=1}^nE\{f_{\epsilon }(c_{k})[g(X_{i}^{T}\beta _{0})-g(X_{i}^{T}{\hat{\beta }}^{0})\nonumber \\&\quad +r^{0}_{i}(u)]K_{i}(u)X_{ik}(1+O_{p}(n^{-1/2}+h^{2}))\}\nonumber \\&=\frac{1}{nh}\sum _{k=1}^q \sum _{i=1}^nE\{f_{\epsilon }(c_{k})[g(X_{i}^{T}{\hat{\beta }}^{0})-g(X_{i}^{T}\beta _{0})]K_{i}(u)X_{ik}\}\nonumber \\&\quad -\frac{1}{nh}\sum _{k=1}^q \sum _{i=1}^nE[f_{\epsilon }(c_{k})r^{0}_{i}(u)K_{i}(u)X_{ik}]+o_{p}(1)\nonumber \\&=I_{1}-I_{2}+o_{p}(1). \end{aligned}$$

It is easy to show that

$$\begin{aligned} I_{1}&=\left( \begin{array}{c} g^{'}(u)f_{U_{0}}(u)E(X|u)^{T}({\hat{\beta }}^{0}-\beta _{0})\sum _{k=1}^q f_{\epsilon }(c_{k})e_{k}+O(h^{2}\delta _{\beta }+\delta ^{2}_{\beta })\\ O(h\delta _{\beta })\\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} I_{2}&=\frac{1}{2}\sum _{k=1}^q f_{\epsilon }(c_{k})\{\int g^{''}(u)t^{2}h^{2}\left( \begin{array}{c} e_{k}K(t) \\ 0\\ \end{array} \right) f_{U_{0}}(x)dx\}+O(h^{3})\nonumber \\&=\frac{h^{2}}{2}g^{''}(u)f_{U_{0}}(u)\left( \begin{array}{c} \sum _{k=1}^q f_{\epsilon }(c_{k})e_{k}\mu _{2} \\ 0\\ \end{array} \right) +O(h^{3}). \end{aligned}$$

Therefore,

$$\begin{aligned}&\frac{1}{\sqrt{nh}}E[W_{n}]\nonumber \\&\quad =\left( \begin{array}{c} f_{U_{0}}(u)\sum _{k=1}^q f_{\epsilon }(c_{k})e_{k}[ g^{'}(u)E(X|u)^{T}({\hat{\beta }}^{0}-\beta _{0})-\frac{h^{2}}{2}g^{''}(u)\mu _{2}] +O(h^{2}\delta _{\beta }+\delta ^{2}_{\beta }+h^{3})\\ O(h\delta _{\beta }+h^{3})\\ \end{array} \right) . \end{aligned}$$

(A.7)

Combining (A.6) and (A.7), we complete the proof of Lemma 3. $\square$

Lemma 4

Suppose that Conditions (C1)–(C7) hold. As $h\rightarrow 0$ and $\delta _{n}\rightarrow 0$, we have

$$\begin{aligned}&\sqrt{nh}\left( \begin{array}{c} {\hat{a}}_{NWCQR1}(u,{\hat{\beta }}^{0})-g(u)-c_{1}+g^{'}(u)E(X|u)^{T}{\tilde{\beta }}-\frac{h^{2}}{2}g^{''}(u)\mu _{2}+O(\Delta _{(n,h)})\\ ... \\ {\hat{a}}_{NWCQRq}(u,{\hat{\beta }}^{0})-g(u)-c_{q}+g^{'}(u)E(X|u)^{T}{\tilde{\beta }}-\frac{h^{2}}{2}g^{''}(u)\mu _{2}+O(\Delta _{(n,h)})\\ h({\hat{g}}_{NWCQR}'(u,{\hat{\beta }}^{0})-g'(u)+O(\delta _{\beta }+h^{2}+h+1/(nh^{2}))\\ \end{array} \right) \\&\quad =-\frac{S^{-1}}{f_{U_{0}}(u)}[W_{n1}-W_{n2}]+o_{p}(1), \end{aligned}$$

where $\Delta _{(n,h)}=(h^{2}+1/(nh)+h^{2}\delta _{\beta }+\delta _{\beta }^{2}+h^{3})$, ${\tilde{\beta }}$, $\delta _{n}$ and $\delta _{\beta }$ are defined in Lemma 3.

Proof

Note that $({\hat{a}}_{NWCQR1}(u,{\hat{\beta }}^{0}),...,{\hat{a}}_{NWCQRq}(u,{\hat{\beta }}^{0}),{\hat{g}}_{NWCQR}'(u,{\hat{\beta }}^{0}))$ are obtained by minimizing the following objective function

$$\begin{aligned}&\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{{\hat{\pi }}(Y_{i})}\rho _{\tau _{k}}[Y_{i}-a_{k}-b(X_{i}^{T}{\hat{\beta }}^{0}-u)]K_{i}(u). \end{aligned}$$

(A.8)

Let ${\hat{\theta }}^{*}=\sqrt{nh}\{({\hat{a}}_{NWCQR1}(u,{\hat{\beta }}^{0})-g(u)-c_{1}),...,({\hat{a}}_{NWCQRq}(u,{\hat{\beta }}^{0})-g(u)-c_{q}),h({\hat{g}}_{NWCQR}'(u,{\hat{\beta }}^{0})-g'(u))\}$. Then, ${\hat{\theta }}^{*}$ is also the minimizer of

$$\begin{aligned} L_{n}(\theta ^{*})&=\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{{\hat{\pi }}(Y_{i})}\left\{ \rho _{\tau _{k}}[\epsilon _{i}{-}c_{k}+r_{i}(u){-}\frac{X_{ik}^{T}\theta ^{*}}{\sqrt{nh}}]{-}\rho _{\tau _{k}}[\epsilon _{i}-c_{k}+r_{i}(u)]\right\} K_{i}(u)\nonumber \\&=W_{n}^{*T}\theta ^{*}+\sum _{k=1}^q B^{*}_{n,k}(\theta ^{*}), \end{aligned}$$

(A.9)

where

$$\begin{aligned} W_{n}^{*}=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{{\hat{\pi }}(Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik} \end{aligned}$$

and

$$\begin{aligned} B^{*}_{n,k}(\theta ^{*})&=\sum _{i=1}^n\frac{V_{i}}{{\hat{\pi }}(Y_{i})}\int _{0}^{\frac{X_{ik}^{T}\theta ^{*}}{\sqrt{nh}}}[I(\epsilon _{i}-c_{k}\\&\quad +r_{i}(u)\le t)-I(\epsilon _{i}-c_{k}+r_{i}(u)\le 0)]dt K_{i}(u). \end{aligned}$$

Notice that

$$\begin{aligned}&B^{*}_{n,k}(\theta ^{*})\\&\quad =\sum _{i=1}^n\frac{V_{i}}{\pi (Y_{i})}\int _{0}^{\frac{X_{ik}^{T}\theta ^{*}}{\sqrt{nh}}}[I(\epsilon _{i}-c_{k}+r_{i}(u)\le t)-I(\epsilon _{i}-c_{k}+r_{i}(u)\le 0)]dt K_{i}(u)\\&\qquad -\sum _{i=1}^n\frac{V_{i}[{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi (Y_{i}){\hat{\pi }}(Y_{i})}\int _{0}^{\frac{X_{ik}^{T}\theta ^{*}}{\sqrt{nh}}}[I(\epsilon _{i}-c_{k}+r_{i}(u) \le t)\\&\qquad -I(\epsilon _{i}-c_{k}+r_{i}(u)\le 0)]dt K_{i}(u).\\&\quad =I_{3}-I_{4}, \end{aligned}$$

where $I_{3}$ is the first summation, $I_{4}$ is the second one. From the proof of Lemma 3, we have $I_{3}=\frac{f_{U_{0}}(u)}{2}\theta ^{*T}S\theta ^{*}+o_{p}(1)=O_{p}(1)$. Considering the fact $sup_{y}|{\hat{\pi }}(y)-\pi (y)|=o_{p}(1)$, we have $I_{4}=o_{p}(1)$. Then, it follows that

$$\begin{aligned} \sum _{k=1}^qB^{*}_{n,k}(\theta ^{*})=\frac{f_{U_{0}}(u)}{2}\theta ^{*T}S\theta ^{*}+o_{p}(1). \end{aligned}$$

(A.10)

Recalling the definition of $W_{n}^{*}$, we have

$$\begin{aligned} W_{n}^{*}&=W_{n}-\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}[{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi ^{2}(Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik}+o_{p}(1)\nonumber \\&=W_{n}-\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{[V_{i}-\pi (Y_{i})][{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi ^{2}(Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik}\nonumber \\&\quad -\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{[{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi (Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik}+o_{p}(1)\nonumber \\&=W_{n1}+E[W_{n}]-I_{5}-I_{6}+o_{p}(1), \end{aligned}$$

(A.11)

where

$$\begin{aligned} I_{5}&=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{[V_{i}-\pi (Y_{i})][{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi ^{2}(Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik},\\ I_{6}&=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{[{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi (Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik}, \end{aligned}$$

$W_{n}$ and $W_{n1}$ are given in Lemma 3.

Recalling the definition of ${\hat{\pi }}(y)$, we have

$$\begin{aligned} {\hat{\pi }}(y)-\pi (y)&=\frac{\sum _{j=1}^n(V_{j}-\pi (Y_{j}))L_{h}(Y_{j}-y)}{\sum _{j=1}^nL_{h}(Y_{j}-y)}\\&\quad +\frac{\sum _{j=1}^n(\pi (Y_{j})-\pi (y))L_{h}(Y_{j}-y)}{\sum _{j=1}^nL_{h}(Y_{j}-y)}\\&=\frac{1}{nf_{Y}(y)}\sum _{j=1}^n(V_{j}-\pi (Y_{j}))L_{h}(Y_{j}-y)\\&\quad +\frac{\sum _{j=1}^n(\pi (Y_{j})-\pi (y))L_{h}(Y_{j}-y)}{nf_{Y}(y)}\\&=\frac{1}{nf_{Y}(y)}\sum _{j=1}^n(V_{j}-\pi (Y_{j}))L_{h}(Y_{j}-y)\\&\quad +O(h^{2})+O_{p}\left( 1/\sqrt{nh^{-1}}\right) , \end{aligned}$$

where $O(h^{2})$ and $O_{p}(1/\sqrt{nh^{-1}})$ don’t depend on the $V_{j}$, $(j=1,...,n)$.

Noting that $E\sum _{k=1}^q[\eta _{jk}(u)K_{j}(u)X_{jk}|Y_{j}]=O_{p}(h)$, we have

$$\begin{aligned} I_{5}&=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\sum _{j=1}^n\frac{[V_{i}-\pi (Y_{i})][V_{j}-\pi (Y_{j})]}{nf_{Y}(Y_{i})\pi ^{2}(Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik}L_{h}(Y_{j}-Y_{i})\\&\quad +O(h^{2})+O_{p}\left( 1/\sqrt{nh^{-1}}\right) \\&=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i\ne j}^n\frac{[V_{i}-\pi (Y_{i})][V_{j}-\pi (Y_{j})]}{nf_{Y}(Y_{i})\pi ^{2}(Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik}L_{h}(Y_{j}-Y_{i})\\&\quad +\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{[V_{i}-\pi (Y_{i})]^2}{nf_{Y}(Y_{i})\pi ^{2}(Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik}L_{h}(0)+O(h^{2})\\&\quad +\,O_{p}\left( 1/\sqrt{nh^{-1}}\right) \\&=I^{1}_{5}+I^{2}_{5}+O(h^{2})+O_{p}\left( 1/\sqrt{nh^{-1}}\right) . \end{aligned}$$

By calculating the mean and the variance, we can get

$$\begin{aligned} I^{1}_{5}=O_{p}\left( 1/\sqrt{nh}\right) ,\quad I^{2}_{5}=O_{p}\left( 1/\sqrt{nh}\right) . \end{aligned}$$

Then, we have

$$\begin{aligned} I_{5}=O_{p}\left( 1/\sqrt{nh}\right) +O(h^{2}). \end{aligned}$$

(A.12)

For $I_{6}$, we can get

$$\begin{aligned} I_{6}&=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{1}{\pi (Y_{i})}\eta _{ik}(u)K_{i}(u)X_{ik}\frac{1}{nf_{Y}(Y_{i})}\sum _{j=1}^n(V_{j}-\pi (Y_{j}))L_{h}(Y_{j}-Y_{i})\\&\quad +O(h^{2})\sqrt{nh}+O_{p}(h) \end{aligned}$$

Denote $D_{ij}=\sum _{k=1}^q \frac{1}{nf_{Y}(Y_{i})\pi (Y_{i})}L_{h}(Y_{j}-Y_{i})\eta _{ik}(u)K_{i}(u)X_{ik}$. For given $Y_{j}$, $D_{ij}$ is independent with $D_{lj}$, $i\ne l$. Hence, we have

$$\begin{aligned} \sum _{i=1}^nD_{ij}=\frac{1}{\pi (Y_{j})}E\left[ \sum _{k=1}^q\eta _{jk}(u)K_{j}(u)X_{jk}|Y_{j}\right] +O_{p}(n^{-1/2}). \end{aligned}$$

Then, $I_{6}$ can be rewritten as

$$\begin{aligned} I_{6}&=\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{j=1}^n\frac{(V_{j}-\pi (Y_{j}))}{\pi (Y_{j})}E[\eta _{jk}(u)K_{j}(u)X_{jk}|Y_{j}]+O_{p}\left( 1/\sqrt{nh}\right) \nonumber \\&\quad +O(h^{2})\sqrt{nh}+O_{p}(h)\nonumber \\&=W_{n2}+o_{p}(\sqrt{h})+O_{p}\left( 1/\sqrt{nh}\right) +O(h^{2})\sqrt{nh}+O_{p}(h), \end{aligned}$$

(A.13)

where

$$\begin{aligned} W_{n2}=\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}-\pi (Y_{j})}{\pi (Y_{j})}E[\eta _{jk}K_{j}(u)X_{jk}|Y_{j}]. \end{aligned}$$

Combining (A.11)–(A.13), we have

$$\begin{aligned} W^{*}_{n}=W_{1n}-W_{2n}+E[W_{n}]+O_{p}\left( 1/\sqrt{nh}\right) +O(h^{2})\sqrt{nh}+O_{p}(h). \end{aligned}$$

By Lemma 2, we complete the proof of Lemma 4. $\square$

Proof of Theorem 3.1

When the selection probability is known, we denote ${\hat{\beta }}_{WCQR}^{1}$ and ${\hat{c}}_{WCQRk}^{1}$, $(k=1,...,q)$ to be the estimators of $\beta _{0}$ and $c_{k}$, $(k=1,...,q)$ which are obtained in 1th iteration in the algorithm proposed in Sect. 2.1. Then, ${\hat{\beta }}_{WCQR}^{1}$ and ${\hat{c}}_{WCQRk}^{1}$, $(k=1,...,q)$ are the minimizer of the following loss function:

$$\begin{aligned}&\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}\rho _{\tau _{k}}\Big [Y_{i}-c_{k}-{\hat{g}}_{WCQR}(X^{T}_{i}{\hat{\beta }}^{0},{\hat{\beta }}^{0})\nonumber \\&\quad -{\hat{g}}'_{WCQR}(X_{i}^{T}{\hat{\beta }}^{0},{\hat{\beta }}^{0})(X_{i}^{T}\beta -X_{i}^{T}{\hat{\beta }}^{0})\Big ]. \end{aligned}$$

(A.14)

Let ${\hat{\zeta }}_{WCQR}=\sqrt{n}({\hat{\beta }}_{WCQR}^{1}-\beta _{0})$, ${\hat{v}}_{WCQRk}=\sqrt{n}({\hat{c}}_{WCQRk}^{1}-c_{k})$, $r_{ik}=c_{k}+{\hat{g}}_{WCQR}(X^{T}_{i}{\hat{\beta }}^{0},{\hat{\beta }}^{0})-g(X^{T}_{i}\beta _{0})+{\hat{g}}'_{WCQR}(X_{i}^{T}{\hat{\beta }}^{0},{\hat{\beta }}^{0})(X_{i}^{T}\beta _{0}-X_{i}^{T}{\hat{\beta }}^{0})$ and $N_{i}={\hat{g}}'_{WCQR}(X_{i}^{T}{\hat{\beta }}^{0},{\hat{\beta }}^{0})X_{i}$. Then, ${\hat{\zeta }}_{WCQR}$ and ${\hat{v}}_{WCQRk}$, $(k=1,...,q)$ minimize the following:

$$\begin{aligned}&Q_{n}(v_{1},...,v_{q},\zeta ,\pi (Y_{i}))\nonumber \\&\quad =\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}\rho _{\tau _{k}}(\epsilon _{i}-r_{ik}-\frac{1}{\sqrt{n}}(N^{T}_{i}\zeta +v_{k}))\nonumber \\&\quad \quad -\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}\rho _{\tau _{k}}(\epsilon _{i}-r_{ik})\nonumber \\&\quad =\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}[I(\epsilon _{i}< r_{ik})-\tau _{k}]N^{T}_{i}\zeta \nonumber \\&\quad \quad +\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}[I(\epsilon _{i}< r_{ik})-\tau _{k}]v_{k} \nonumber \\&\quad \quad +\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{\pi (Y_{i})}\int _{0}^{\frac{1}{\sqrt{n}}(N^{T}_{i}\zeta +v_{k})}[I(\epsilon _{i}\le r_{ik}+t)-I(\epsilon _{i}\le r_{ik})]dt\nonumber \\&\quad =M^{T}_{n}\zeta +\frac{1}{2}\zeta ^{T}\Gamma \zeta +\sum _{k=1}^qz_{nk}v_{k}+\frac{1}{2}\sum _{k=1}^qf_{\epsilon }(c_{k})v_{k}^{2}+o_{p}(1), \end{aligned}$$

(A.15)

where

$$\begin{aligned} M_{n}&=\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}[I(\epsilon _{i}< r_{ik})-\tau _{k}]N_{i},\\ z_{nk}&=\frac{1}{\sqrt{n}}\sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}[I(\epsilon _{i}< r_{ik})-\tau _{k}] \end{aligned}$$

and

$$\begin{aligned} \Gamma =cE[g'(X^{T}\beta _{0})^{2}XX^{T}]. \end{aligned}$$

Letting $M_{n1}=\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}[I(\epsilon _{i}< c_{k})-\tau _{k}]N_{i}$, we can get $Var(M_{n}-M_{n1}|{\hat{\chi }}^{0},X)=o_{p}(1)$. Then,

$$\begin{aligned} M_{n}&=M_{n1}+E[M_{n}|{\hat{\chi }}^{0},X]+o_{p}(1)\nonumber \\&=M_{n1}+\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{i=1}^nf_{\epsilon }(c_{k})(r_{ik}-c_{k})N_{i}+o_{p}(1). \end{aligned}$$

(A.16)

With the help of Lemma 3 and Condition (C9), we have

$$\begin{aligned} r_{ik}&=g(X^{T}_{i}{\hat{\beta }}^{0})+c_{k}-g'(X^{T}_{i}{\hat{\beta }}^{0})E(X_{i}|X^{T}_{i}{\hat{\beta }}^{0})({\hat{\beta }}^{0}-\beta _{0})+\frac{h^{2}}{2}g''(X^{T}_{i}{\hat{\beta }}^{0})\mu _{2}\\&\quad -\frac{1}{q}\sum _{k=1}^q\frac{1}{nhf_{U_{0}}(X^{T}_{i}{\hat{\beta }}^{0})f_{\epsilon }(c_{k})}\sum _{j=1}^n \frac{V_{j}}{\pi (Y_{j})}\eta _{jk}K_{j}(X^{T}_{i}{\hat{\beta }}^{0})\\&\quad +O(h^{2}\delta _{\beta }+\delta ^{2}_{\beta }+h^{3})-g(X^{T}_{i}\beta _{0})\\&\quad +[g'(X_{i}^{T}{\hat{\beta }}^{0})-O(\delta _{n}/h)+O(\delta _{\beta }+h^{2})](X_{i}^{T}\beta _{0}-X_{i}^{T}{\hat{\beta }}^{0})\\&=c_{k}-g'(X^{T}_{i}{\hat{\beta }}^{0})E(X_{i}|X^{T}_{i}{\hat{\beta }}^{0})({\hat{\beta }}^{0}-\beta _{0})\\&\quad -\frac{1}{q}\sum _{k=1}^q\frac{1}{nhf_{U_{0}}(X^{T}_{i}{\hat{\beta }}^{0})f_{\epsilon }(c_{k})}\sum _{j=1}^n \frac{V_{j}}{\pi (Y_{j})}\eta _{jk}K_{j}(X^{T}_{i}{\hat{\beta }}^{0})+o_{p}(n^{-1/2}). \end{aligned}$$

Thus, we have

$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{i=1}^nf_{\epsilon }(c_{k})(r_{ik}-c_{k})N_{i}\nonumber \\&=\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{i=1}^nf_{\epsilon }(c_{k})[-g'(X^{T}_{i}{\hat{\beta }}^{0})E(X_{i}|X^{T}_{i}{\hat{\beta }}^{0})({\hat{\beta }}^{0}-\beta _{0})\nonumber \\&\quad -\frac{1}{nhf_{U_{0}}(X^{T}_{i}{\hat{\beta }}^{0})f_{\epsilon }(c_{k})}\sum _{j=1}^n \frac{V_{j}}{\pi (Y_{j})}\eta _{jk}K_{j}(X^{T}_{i}{\hat{\beta }}^{0})]N_{i}\nonumber \\&=-\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{i=1}^nf_{\epsilon }(c_{k})N_{i}g'(X^{T}_{i}{\hat{\beta }}^{0})E(X_{i}|X^{T}_{i}{\hat{\beta }}^{0})({\hat{\beta }}^{0}-\beta _{0})\nonumber \\&\quad -\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{i=1}^n\sum _{j=1}^n \frac{N_{i}}{nhf_{U_{0}}(X^{T}_{i}{\hat{\beta }}^{0})} \frac{V_{j}}{\pi (Y_{j})}\eta _{jk}K_{j}(X^{T}_{i}{\hat{\beta }}^{0})+o_{p}(1)\nonumber \\&=-I_{7}-I_{8}+o_{p}(1). \end{aligned}$$

(A.17)

For $I_{7}$, we have

$$\begin{aligned}&I_{7}=\sqrt{n}C_{0}({\hat{\beta }}^{0}-\beta _{0})+o_{p}(1), \end{aligned}$$

(A.18)

where $C_{0}=cE[g'(X^{T}\beta _{0})^2E(X|X^{T}\beta _{0})E(X|X^{T}\beta _{0})^{T}]$ and c defined in Lemma 3.

Considering $I_{8}$, we have

$$\begin{aligned} I_{8}&=\frac{1}{\sqrt{n}nh}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}}{\pi (Y_{j})}\eta _{jk} \sum _{i=1}^n\frac{N_{i}}{f_{U_{0}}(X^{T}_{i}{\hat{\beta }}^{0})} K_{j}(X^{T}_{i}{\hat{\beta }}^{0})\nonumber \\&=\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}}{\pi (Y_{j})}\eta _{jk}{\hat{g}}'(X_{j}^{T}{\hat{\beta }}^{0})E[X_{j}|X^{T}_{j}\beta _{0}]+o_{p}(1)\nonumber \\&=C_{n}+o_{p}(1), \end{aligned}$$

(A.19)

where $C_{n}=\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}}{\pi (Y_{j})}\eta _{jk}{\hat{g}}'(X_{j}^{T}{\hat{\beta }}^{0},{\hat{\beta }}^{0})E[X_{j}|X^{T}_{j}\beta _{0}]$.

Combining (A.15)–(A.19), we obtain

$$\begin{aligned} Q_{n}(v_{1},...,v_{q},\zeta ,\pi (Y_{i}))&=[M_{n1}-C_{n}-\sqrt{n}C_{0}({\hat{\beta }}^{0}-\beta _{0})]\zeta +\frac{1}{2}\zeta ^{T}\Gamma \zeta \\&\quad +\sum _{k=1}^qz_{nk}v_{k}+\frac{1}{2}\sum _{k=1}^qf_{\epsilon }(c_{k})v_{k}^{2}+o_{p}(1). \end{aligned}$$

By Lemma 2, we have

$$\begin{aligned} {\hat{\zeta }}_{WCQR}=-\Gamma ^{-1}[M_{n1}-C_{n}]+\sqrt{n}\Gamma ^{-1}C_{0}({\hat{\beta }}^{0}-\beta _{0})+o_{p}(1). \end{aligned}$$

Recalling the definition of ${\hat{\zeta }}_{WCQR}$, we have

$$\begin{aligned}&({\hat{\beta }}_{WCQR}^{1}-\beta _{0})=-\frac{1}{\sqrt{n}}\Gamma ^{-1}[M_{n1}-C_{n}]+\Gamma ^{-1}C_{0}({\hat{\beta }}^{0}-\beta _{0})+o_{p}\left( 1/\sqrt{n}\right) . \end{aligned}$$

(A.20)

Denote ${\tilde{\Gamma }}=\Gamma ^{-\frac{1}{2}} C_{0}\Gamma ^{-\frac{1}{2}}$ and ${\hat{\beta }}_{WCQR}^{k}$ to be the estimator of $\beta _{0}$ gained in the kth iteration. For k, replace ${\hat{\beta }}_{WCQR}^{1}$ and ${\hat{\beta }}^{0}$ by ${\hat{\beta }}_{WCQR}^{k+1}$ and ${\hat{\beta }}_{WCQR}^{k}$, respectively, Equation (A.20) still holds. Letting $\vartheta ^{k}=\Gamma ^{\frac{1}{2}}({\hat{\beta }}_{WCQR}^{k}-\beta _{0})$, we have

$$\begin{aligned}&\vartheta ^{k+1}=-\frac{1}{\sqrt{n}}\Gamma ^{-\frac{1}{2}}[M_{n1}-C_{n}]+{\tilde{\Gamma }}\vartheta ^{k}+o_{p}\left( 1/\sqrt{n}\right) . \end{aligned}$$

Similar to Xia and Härdle (2006), we can get the convergence of our algorithm. For sufficiently large k, we have

$$\begin{aligned}&\Gamma ^{\frac{1}{2}}({\hat{\beta }}_{WCQR}-\beta _{0})=-\frac{1}{\sqrt{n}}\Gamma ^{-\frac{1}{2}}[M_{n1}-C_{n}]+{\tilde{\Gamma }}\Gamma ^{\frac{1}{2}}({\hat{\beta }}_{WCQR}-\beta _{0})+o_{p}\left( 1/\sqrt{n}\right) . \end{aligned}$$

Then,

$$\begin{aligned}&(\Gamma -\Gamma ^{\frac{1}{2}}{\tilde{\Gamma }}\Gamma ^{\frac{1}{2}})({\hat{\beta }}_{WCQR}-\beta _{0})=-\frac{1}{\sqrt{n}}[M_{n1}-C_{n}]+o_{p}(1/\sqrt{n}). \end{aligned}$$

(A.21)

Recalling the definition of ${\tilde{\Gamma }}$, (A.21) can be rewritten as

$$\begin{aligned}&\sqrt{n}D({\hat{\beta }}_{WCQR}-\beta _{0})=-[M_{n1}-C_{n}]+o_{p}(1), \end{aligned}$$

(A.22)

where $D=\Gamma -C_{0}$.

By Central limit theorem, we know

$$\begin{aligned}&-[M_{n1}-C_{n}]\xrightarrow {d}N(\mathbf 0 , D_{1}), \end{aligned}$$

(A.23)

where $D_{1}$ is given in Theorem 3.1.

Combining Equations (A.22) and (A.23), we obtain the asymptotic normality of ${\hat{\beta }}_{WCQR}$. $\square$

Proof of Theorem 3.2

Denote ${\hat{\beta }}_{NWCQR}^{1}$ and ${\hat{c}}_{NWCQRk}^{1}$, $(k=1,...,q)$ to be the calculation results of the 1th iteration when using the estimated selection probability. ${\hat{\beta }}_{NWCQR}^{1}$ and ${\hat{c}}_{NWCQRk}^{1}$, $(k=1,...,q)$ are obtained by minimizing the following objective function:

$$\begin{aligned}&\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{{\hat{\pi }}(Y_{i})}\rho _{\tau _{k}}{[}Y_{i}-c_{k}-{\hat{g}}_{NWCQR}(X^{T}_{i}{\hat{\beta }}^{0},{\hat{\beta }}^{0})\nonumber \\&\qquad -{\hat{g}}'_{NWCQR}(X_{i}^{T}{\hat{\beta }}^{0},{\hat{\beta }}^{0})(X_{i}^{T}\beta -X_{i}^{T}{\hat{\beta }}^{0})]. \end{aligned}$$

(A.24)

Let ${\hat{\zeta }}_{NWCQR}=\sqrt{n}({\hat{\beta }}_{NWCQR}^{1}-\beta _{0})$, ${\hat{v}}_{NWCQRk}=\sqrt{n}({\hat{c}}_{NWCQRk}^{1}-c_{k})$, $N^{*}_{i}={\hat{g}}'_{NWCQR}(X_{i}^{T}{\hat{\beta }}^{0},{\hat{\beta }}^{0})X_{i}$ and $r^{*}_{ik}=c_{k}+{\hat{g}}_{NWCQR}(X^{T}_{i}{\hat{\beta }}^{0},{\hat{\beta }}^{0})-g(X^{T}_{i}\beta _{0})+{\hat{g}}'_{NWCQR}(X_{i}^{T}{\hat{\beta }}^{0},{\hat{\beta }}^{0})(X_{i}^{T}\beta _{0}-X_{i}^{T}{\hat{\beta }}^{0})$ , $(k=1,...,q)$. Then, ${\hat{\zeta }}_{NWCQR}$, ${\hat{v}}_{NWCQRk}$, $(k=1,...,q)$ minimizes the following

$$\begin{aligned}&Q_{n}(v_{1},...,v_{q},\zeta ,{\hat{\pi }}(Y_{i}))\nonumber \\&\quad =\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{{\hat{\pi }}(Y_{i})}\rho _{\tau _{k}}(\epsilon _{i}-r^{*}_{ik}-\frac{1}{\sqrt{n}}(N^{*T}_{i}\zeta +v_{k}))\nonumber \\&\quad \quad -\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{{\hat{\pi }}(Y_{i})}\rho _{\tau _{k}}(\epsilon _{i}-r^{*}_{ik})\nonumber \\&\quad =\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{{\hat{\pi }}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*T}_{i}\zeta \nonumber \\&\quad \quad +\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{{\hat{\pi }}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]v_{k} \nonumber \\&\quad \quad +\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{{\hat{\pi }}(Y_{i})}\int _{0}^{\frac{1}{\sqrt{n}}(N^{*T}_{i}\zeta +v_{k})}[I(\epsilon _{i}\le r^{*}_{ik}+t)-I(\epsilon _{i}\le r^{*}_{ik})]dt\nonumber \\&\quad ={\hat{M}}^{*T}_{n}\zeta +\frac{1}{2}\zeta ^{T}\Gamma \zeta +\sum _{k=1}^q{\hat{z}}^{*}_{nk}v_{k}+\frac{1}{2}\sum _{k=1}^qf_{\epsilon }(c_{k})v_{k}^{2}+o_{p}(1), \end{aligned}$$

(A.25)

where

$$\begin{aligned} {\hat{M}}^{*}_{n}&=\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{{\hat{\pi }}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i},\\ {\hat{z}}^{*}_{nk}&=\frac{1}{\sqrt{n}}\sum _{i=1}^n \frac{V_{i}}{{\hat{\pi }}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}] \end{aligned}$$

and $\Gamma$ is given in the proof of Theorem 3.1. Note that

$$\begin{aligned} {\hat{M}}^{*}_{n}&=M^{*}_{n}-\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}[{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi ^{2}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i}+o_{p}(1) \nonumber \\&=M^{*}_{n}-\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{[V_{i}-\pi (Y_{i})][{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi ^{2}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i} \nonumber \\&\quad -\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{[{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi (Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i}+o_{p}(1) \nonumber \\&=M^{*}_{n}-I_{9}-I_{10}+o_{p}(1), \end{aligned}$$

(A.26)

where

$$\begin{aligned} M^{*}_{n}&=\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}}{\pi (Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i},\\ I_{9}&=\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{[V_{i}-\pi (Y_{i})][{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi ^{2}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i}, \end{aligned}$$

and

$$\begin{aligned}&I_{10}=\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{[{\hat{\pi }}(Y_{i})-\pi (Y_{i})]}{\pi (Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i}. \end{aligned}$$

Using Lemma 4 and Conditions (C8)–(C9), and following the proof of Theorem 3.1, we have

$$\begin{aligned}&M^{*}_{n}=M_{n1}-C_{n}+C_{n1}-\sqrt{n}C_{0}({\hat{\beta }}^{0}-\beta _{0})+o_{p}(1), \end{aligned}$$

(A.27)

where

$$\begin{aligned}&C_{n1}=\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}-\pi (Y_{j})}{\pi (Y_{j})}E\{\eta _{jk}g^{'}(X^{T}_{j}\beta _{0})E[X_{j}|X^{T}_{j}\beta _{0}]|Y_{j}\}. \end{aligned}$$

Considering $I_{9}$, we have

$$\begin{aligned} I_{9}&=\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{V_{i}-\pi (Y_{i})}{\pi ^{2}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i}\nonumber \\&\qquad \left\{ \frac{1}{nf_{Z}(Z_{i})}\sum _{j=1}^n[V_{j}-\pi (Y_{j})]L_{h}(Y_{j}-Y_{i})+O(h^{2})+O_{p}\left( 1/\sqrt{nh^{-1}}\right) \right\} \nonumber \\&\quad =\frac{1}{\sqrt{n}n}\sum _{k=1}^q \sum _{i\ne j}^n \frac{[V_{i}-\pi (Y_{i})][V_{j}-\pi (Y_{j})]}{\pi ^{2}(Y_{i})f_{Y}(Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i}L_{h}(Y_{j}-Y_{i})\nonumber \\&\qquad +\frac{1}{\sqrt{n}n}\sum _{k=1}^q \sum _{i=1}^n \frac{[V_{i}-\pi (Y_{i})]^2}{\pi ^{2}(Y_{i})f_{Y}(Y_{i})}\left[ I\left( \epsilon _{i}< r^{*}_{ik}\right) -\tau _{k}\right] \nonumber \\&\qquad N^{*}_{i}L_{h}(0)+O(h^{2})+O_{p}\left( 1/\sqrt{nh^{-1}}\right) \nonumber \\&\quad =O_{p}\left( 1/\sqrt{nh}\right) +O_{p}\left( 1/\sqrt{nh^{2}}\right) +O(h^{2})+O_{p}\left( 1/\sqrt{nh^{-1}}\right) =o_{p}(1). \end{aligned}$$

(A.28)

Using Conditions (C8)–(C9), we have

$$\begin{aligned} I_{10}&=\frac{1}{\sqrt{n}}\sum _{k=1}^q \sum _{i=1}^n \frac{1}{\pi (Y_{i})}[I(\epsilon _{i}< r^{*}_{ik})-\tau _{k}]N^{*}_{i}\nonumber \\&\qquad \left\{ \frac{1}{nf_{Y}(Y_{i})}\sum _{j=1}^n[V_{j}-\pi (Y_{j})]L_{h}(Y_{j}-Y_{i})+O(h^{2})+O_{p}\left( 1/\sqrt{nh^{-1}}\right) \right\} \nonumber \\&=\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}-\pi (Y_{j})}{\pi (Y_{j})}E\left\{ [I(\epsilon _{j}< r^{*}_{jk})-\tau _{k}]N^{*}_{j}|Y_{j}\right\} +O\left( \sqrt{n}h^{2}\right) +o_{p}(1)\nonumber \\&=\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}-\pi (Y_{j})}{\pi (Y_{j})}E\{\eta _{jk}g^{'}(X^{T}_{j}\beta _{0})X_{j}|Y_{j}\}+O\left( \sqrt{n}h^{2}\right) +o_{p}(1)\nonumber \\&=M_{n2}+o_{p}(1), \end{aligned}$$

(A.29)

where

$$\begin{aligned}&M_{n2}=\frac{1}{\sqrt{n}}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}-\pi (Y_{j})}{\pi (Y_{j})}E\{\eta _{jk}g^{'}(X^{T}_{j}\beta _{0})X_{j}|Y_{j}\}. \end{aligned}$$

Combining (A.26)–(A.29), we have

$$\begin{aligned}&{\hat{M}}^{*}_{n}=M_{n1}-C_{n}-M_{n2}+C_{n1}-\sqrt{n}C_{0}({\hat{\beta }}^{0}-\beta _{0})+o_{p}(1). \end{aligned}$$

(A.30)

By Central limit theorem, we have

$$\begin{aligned}&-[M_{n1}-C_{n}-M_{n2}+C_{n1}]\rightarrow _{d}N(\mathbf 0 ,D_{2}), \end{aligned}$$

(A.31)

where $D_{2}$ is given in Theorem 3.2. Following the proof of Theorem 3.1, we obtain the asymptotic normality of ${\hat{\beta }}_{NWCQR}$. $\square$

Proof of Theorem 3.3

By Theorem 3.1, we can see that ${\hat{\beta }}_{WCQR}$ is a $\sqrt{n}$-consistent estimator of $\beta _{0}$. Following the proof of Lemma 3, the asymptotic normality of ${\hat{g}}(u,{\hat{\beta }}_{WCQR})$ is obtained. $\square$

Proof of Theorem 3.4

By Theorem 3.2, we can see that ${\hat{\beta }}_{NWCQR}$ is a $\sqrt{n}$-consistent estimator of $\beta _{0}$. Suppose that Conditions (C1)–(C7) and (C10) hold, by Lemma 4, we have

$$\begin{aligned}&\sqrt{nh}\left( \begin{array}{c} {\hat{a}}_{1}(u,{\hat{\beta }}_{NWCQR})-g(u)-c_{1}-\frac{h^{2}}{2}g''(u)\mu _{2}+O(h^{2})\\ ... \\ {\hat{a}}_{q}(u,{\hat{\beta }}_{NWCQR})-g(u)-c_{q}-\frac{h^{2}}{2}g''(u)\mu _{2}+O(h^{2})\\ \end{array} \right) \\&\quad =-\frac{C^{-1}}{f_{U_{0}}(u)}[W_{n1*}-W_{n2*}]+o_{p}(1), \end{aligned}$$

where C is defined in Lemma 3,

$$\begin{aligned}&W_{n1*}=\frac{1}{\sqrt{nh}}\sum _{k=1}^q \sum _{i=1}^n\frac{V_{i}}{\pi (Y_{i})}\eta _{ik}K\left( \frac{X^{T}_{i}{\hat{\beta }}_{NWCQR}-u}{h}\right) e_{k} \end{aligned}$$

and

$$\begin{aligned}&W_{n2*}=\frac{1}{\sqrt{nh}}\sum _{k=1}^q\sum _{j=1}^n\frac{V_{j}-\pi (Y_{j})}{\pi (Y_{j})}E[\eta _{jk}K\left( \frac{X^{T}_{j}{\hat{\beta }}_{NWCQR}-u}{h}\right) e_{k} |Y_{j}]. \end{aligned}$$

By calculating the expectation and variance, we obtain the asymptotic normality of ${\hat{g}}(u,{\hat{\beta }}_{NWCQR})$. $\square$