Nonparametric quantile regression estimation for functional data with responses missing at random

Abstract

This paper presents the nonparametric quantile regression estimation for the regression function operator when the functional data with the responses missing at random are considered. Then, the large sample properties of the proposed estimator are established under some mild conditions. Finally, a simulation study is conducted to investigate the finite sample properties of the proposed method.

Appendix A. Proofs of main theorems

Throughout this paper, we will introduce the following notations. We use ${\mathcal {F}}_n$ to denote the set of the random variables $(X_1,\ldots , X_n)$ and $\psi (\cdot )$ denote the score function of $\rho _\tau (\cdot )$. For fixed random function $\chi$, denote $R_i=m(X_i)-m(\chi )$, $\theta (\alpha )=\sqrt{n\phi (h)}(\alpha -m(\chi ))$ and ${\hat{\theta }}=\theta ({\hat{\alpha }}).$ Then, let

$$\begin{aligned} {\hat{\eta _i}}(\theta )= & {} \frac{\delta _i}{{\hat{\pi }}(X_i)} \rho _\tau \left( \varepsilon _i+R_i-\frac{\theta }{\sqrt{n\phi (h)}}\right) K_h(d(\chi ,X_i))\\&-\frac{\delta _i}{{\hat{\pi }}(X_i)} \rho _\tau \left( \varepsilon _i+R_i\right) K_h(d(\chi ,X_i)),\\ \eta _i(\theta )= & {} \frac{\delta _i}{ \pi (X_i)} \rho _\tau \left( \varepsilon _i+R_i-\frac{\theta }{\sqrt{n\phi (h)}} \right) K_h(d(\chi ,X_i))\\&-\frac{\delta _i}{ \pi (X_i)} \rho _\tau \left( \varepsilon _i+R_i\right) K_h(d(\chi ,X_i)). \end{aligned}$$

We first introduce the following lemmas.

Lemma 1

Suppose that conditions C1–C6 hold, then

$$\begin{aligned}&\left| \frac{1}{\sqrt{n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))[\psi (\varepsilon _i) - \psi (\varepsilon _i+R_i) ]\right| =o_p(1). \end{aligned}$$

Proof

By the condition C6, we have

$$\begin{aligned}&E \left( \frac{1}{\sqrt{n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))[\psi (\varepsilon _i) - \psi (\varepsilon _i+R_i) ]\left. \right| {\mathcal {F}}_n\right) \\&\quad = \frac{1}{\sqrt{n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))f(0|X_i)(m(X_i)-m(\chi )) (1+o_p(1)) \\&\quad \le C \frac{h^\beta }{\sqrt{n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\\&\quad =O_p(1) h^\beta \sqrt{n\phi (h)}, \end{aligned}$$

where the last equality holds by Lemma 4.

On the other hand, we have

$$\begin{aligned}&\text {Var} \left( \frac{1}{\sqrt{n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))[\psi (\varepsilon _i) - \psi (\varepsilon _i+R_i) ]\left. \right| {\mathcal {F}}_n\right) \\&\quad \le \frac{1}{{n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi ^2(X_i)}K_h^2(d(\chi ,X_i))E\left( [\psi (\varepsilon _i) - \psi (\varepsilon _i+R_i) ]^2\left. \right| {\mathcal {F}}_n\right) \\&\quad = \frac{1}{ {n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi ^2(X_i)}K_h^2(d(\chi ,X_i))f(0|X_i)(m(X_i)-m(\chi )) (1+o_p(1))\\&\quad \le C \frac{h^\beta }{ {n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi ^2(X_i)}K_h^2(d(\chi ,X_i))\\&\quad =O_p(1) h^\beta . \end{aligned}$$

Then, by the condition $h^\beta \sqrt{n\phi (h)}\rightarrow 0$, we have

$$\begin{aligned}&\left| \frac{1}{\sqrt{n\phi (h)}}\sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))[\psi (\varepsilon _i) - \psi (\varepsilon _i+R_i) ]\right| \\&\quad =O_p(1) h^\beta \sqrt{n\phi (h)} +O_p(1) h^{\beta /2}\\&\quad =o_p(1). \end{aligned}$$

$\square$

Lemma 2

Suppose that conditions 1–6 hold, then for any $L>0$, it has

$$\begin{aligned}&\sup \limits _{\Vert { \theta }\Vert \le L} \left| \sum _{i=1}^n \left[ \eta _i( \theta ) + \frac{\theta }{\sqrt{n\phi (h)}}\frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\psi (\varepsilon _i)\right] \right. \\&\quad \left. -E\left( \sum _{i=1}^n \eta _i( \theta ) \right) \right| =o_p(1). \end{aligned}$$

Proof

By the Knight’s identity, we have

$$\begin{aligned}&\eta _i(\theta )+ \frac{\theta }{\sqrt{n\phi (h)}}\frac{\delta _i}{ \pi (X_i)}\psi (\varepsilon _i)\\&\quad =\frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\int _0^{\frac{\theta }{\sqrt{n\phi (h)}}}[I_{\{\varepsilon _i+R_i<s\}}-I_{\{\varepsilon _i<0\}}]ds. \end{aligned}$$

Then, by direct calculation, we have

$$\begin{aligned} \text {E} \left( \frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\int _0^{\frac{\theta }{\sqrt{n\phi (h)}}}[I_{\{\varepsilon _i+R_i<s\}}-I_{\{\varepsilon _i<0\}}]ds \right) ^2=O\left( (n\phi (h))^{-3/2} \right) . \end{aligned}$$

Denote $\zeta _i(\theta )=\eta _i(\theta ) + \frac{\theta }{\sqrt{n\phi (h)}}\frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\psi (\varepsilon _i+R_i).$ Invoking Lemma 19.24 of Van der Vaart (1998), we have

$$\begin{aligned} \sup \limits _{\Vert { \theta }\Vert \le L} \left| \sum _{i=1}^n \zeta _i( \theta ) - \sum _{i=1}^n E\left[ \zeta _i( \theta ) \right] \right| =o_p(1). \end{aligned}$$

(A.1)

Then, according to Lemma 1, one has

$$\begin{aligned}&\sup \limits _{|{ \theta }|\le L} \left| \sum _{i=1}^n \zeta _i( \theta ) - \sum _{i=1}^n \eta _i( \theta ) \right| =o_p(1). \end{aligned}$$

Combining this with Eq. (A.1), we can obtain

$$\begin{aligned}&\sup \limits _{|{ \theta }|\le L} \left| \sum _{i=1}^n \left[ \eta _i( \theta ) + \frac{\theta }{\sqrt{n\phi (h)}}\frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\psi (\varepsilon _i)\right] \right. \\&\quad \left. -E\left( \sum _{i=1}^n \eta _i( \theta ) \right) \right| =o_p(1). \end{aligned}$$

$\square$

Lemma 3

Suppose that conditions 1–6 hold, then for any $L>0$, it has that

$$\begin{aligned}&\sup \limits _{|{ \theta }|\le L} \left| \sum _{i=1}^n \left[ \hat{\eta }_i( \theta ) + \frac{\theta }{\sqrt{n\phi (h)}}\frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\psi (\varepsilon _i)\right] \right. \\&\quad \left. -E\left( \sum _{i=1}^n {\hat{\eta }}_i(\varvec{\theta }) \right) \right| =o_p(1). \end{aligned}$$

Proof

To prove Lemma 3, it is sufficient to show that

$$\begin{aligned} \sup \limits _{|{ \theta }|\le L} \left| \sum _{i=1}^n \left[ \hat{\eta }_i( \theta )- \eta _i( \theta ) \right] \right| =o_p(1). \end{aligned}$$

By the Knight’s identity, one has

$$\begin{aligned}&\sum _{i=1}^n \left[ {\hat{\eta }}_i( \theta )- \eta _i( \theta ) \right] \\&\quad = \sum _{i=1}^n \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) \rho _\tau \left( \varepsilon _i+R_i-\frac{\theta }{\sqrt{n\phi (h)}}\right) K_h(d(\chi ,X_i))\\&\quad \quad -\sum _{i=1}^n \left( \frac{\delta _i}{{\hat{\pi }}(X_i)} -\frac{\delta _i}{ \pi (X_i)} \right) \rho _\tau \left( \varepsilon _i+R_i\right) K_h(d(\chi ,X_i))\\&\quad =\sum _{i=1}^nK_h(d(\chi ,X_i)) \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) \\&\quad \quad \times \,\left[ \rho _\tau \left( \varepsilon _i+R_i-\frac{\theta }{\sqrt{n\phi (h)}}\right) - \rho _\tau \left( \varepsilon _i+R_i\right) \right] \\&\quad = \sum _{i=1}^nK_h(d(\chi ,X_i)) \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) \frac{ \theta }{\sqrt{n\phi (h)}}\left[ I_{\{\varepsilon _i+R_i<0\}}-\tau \right] \\&\quad \quad + \sum _{i=1}^nK_h(d(\chi ,X_i)) \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) \int _0^{\frac{\theta }{\sqrt{n\phi (h)}} }\left[ I_{\{\varepsilon _i+R_i<s\}}-I_{\{\varepsilon _i+R_i<0\}} \right] ds\\&\quad =I_{n1}+I_{n2}, \end{aligned}$$

where the definitions of $I_{n1}$ and $I_{n2}$ are clear from the context.

First, we consider $I_{n1}.$

$$\begin{aligned} I_{n1}&= \sum _{i=1}^nK_h(d(\chi ,X_i)) \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) \frac{\theta }{\sqrt{n\phi (h)}} \left[ I_{\{\varepsilon _i+R_i<0\}}-\tau \right] \\&\quad =\sum _{i=1}^nK_h(d(\chi ,X_i)) \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) \frac{ \theta }{\sqrt{n\phi (h)}} \left[ I_{\{\varepsilon _i<0\}}-\tau \right] \\&\quad \quad + \sum _{i=1}^nK_h(d(\chi ,X_i)) \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) \frac{ \theta }{\sqrt{n\phi (h)}}\left[ I_{\{\varepsilon _i+R_i<0\}}-I_{\{\varepsilon _i<0\}} \right] \\&\quad =I_{n11}+I_{n12}. \end{aligned}$$

It is easy to show $E(I_{n11})=0.$ By conditions C3-C4 and Proposition 2 of Laib and Louani (2010), one has

$$\begin{aligned} E(I_{n11}^2)&= \frac{ \theta ^2}{ n\phi (h)} \tau (1-\tau )\sum _{i=1}^nE\left[ K_h^2(d(\chi ,X_i)) \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) ^2\right] \\&= \frac{ \theta ^2}{ n\phi (h)} \tau (1-\tau )\sum _{i=1}^nE\left[ K_h^2(d(\chi ,X_i)) \left( \frac{1}{{\hat{\pi }}(X_i)}-\frac{1}{ \pi (X_i)} \right) ^2 \pi (X_i)\right] \\&= \frac{ \theta ^2}{ n\phi (h)} \tau (1-\tau )\sum _{i=1}^nE\left[ K_h^2(d(\chi ,X_i)) \frac{({\hat{\pi }}(X_i)- \pi (X_i))^2}{ \pi (X_i)} \right] (1+o(1))\\&\le \frac{(1+o(1))\log \log (n)}{n^2\phi ^2(h)c_0^2} \theta ^2\tau (1-\tau )\sum _{i=1}^nE\left[ K_h(d(\chi ,X_i)) \right] \\&= \frac{(1+o(1))\log \log (n)}{n^2\phi (h)c_0^2} \theta ^2\tau (1-\tau )\sum _{i=1}^n \left[ M_1f_{i,1}(\chi ) +O_{a.s}(g_{i,\chi }(h)) \right] \\&= O\left( \frac{\log \log (n)}{n\phi (h)}\right) . \end{aligned}$$

Thus, one has $I_{n11}=O_p\left( \sqrt{\frac{\log \log (n)}{n\phi (h)}}\right) =o_p(1)$ by condition C3. Similar to $I_{n11},$ we have $E(I_{n12})=O\left( \frac{h^\beta }{\phi (h)}\sqrt{{\log \log (n)}}\right)$ and $E(I^2_{n12})=O\left( \frac{h^\beta \log \log (n)}{n\phi (h)}\right) .$ Therefore, we have $I_{n12}=O_p\left( \frac{h^\beta }{\phi (h)}\sqrt{{\log \log (n)}}\right) .$ Thus $I_{n1}=o_p(1).$

Next, we consider $I_{n2}.$ By straightforward algebra, one has

$$\begin{aligned} E(I_{n 2})&= \sum _{i=1}^nE\left( K_h(d(\chi ,X_i)) \left( \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right) \right. \\&\quad \left. \times \,\int _0^{\frac{ \theta }{\sqrt{n\phi (h)}}}\left[ I_{\{\varepsilon _i+R_i<s\}}-I_{\{\varepsilon _i+R_i<0\}} \right] ds \right) \\&\le C \frac{1}{ {n\phi (h)}} \sum _{i=1}^nE\left( K_h(d(\chi ,X_i)) \left| \frac{\delta _i}{{\hat{\pi }}(X_i)}-\frac{\delta _i}{ \pi (X_i)} \right| \right) \\&\le C\sqrt{\frac{\log \log (n)}{n\phi (h)}} \frac{1}{ {n\phi (h)}} \sum _{i=1}^nE\left( K_h(d(\chi ,X_i)) \right) \\&\le C\sqrt{\frac{\log \log (n)}{n\phi (h)}}. \end{aligned}$$

Combining the above arguments with condition C5, we have

$$\begin{aligned} \sup \limits _{|{ \theta }|\le L} \left| \sum _{i=1}^n \left[ \hat{\eta }_i( \theta )- \eta _i( \theta ) \right] \right| =O_p\left( \frac{h^\beta \sqrt{{\log \log (n)}}}{\phi (h) }\right) =o_p(1). \end{aligned}$$

$\square$

Lemma 4

Suppose that conditions 1–6 hold, then we have

$$\begin{aligned} E\left( \frac{1}{ \pi (X_i)}K^k_h(d(\chi ,X_i))\right)&=\phi (h)\left( M_k\frac{f_{1}(\chi )}{\pi (\chi )}+O_{a.s}(g_{i,\chi }(h)/\pi (\chi )) \right) , k=1,2, \\ E\left( \frac{\delta _i}{ \pi (X_i)}K^k_h(d(\chi ,X_i))\right)&=\phi (h)\left( M_kf_{1}(\chi )+o(1) \right) , k=1,2. \end{aligned}$$

and

$$\begin{aligned} \frac{1}{ {n\phi (h)}}\sum _{i=1}^n E\left( K_h(d(\chi ,X_i)) f(0|X_i) \right) =M_1 f(0|\chi )+o(1). \end{aligned}$$

The proof of Lemma 4 is similar to Lemma 1 of Laib and Louani (2010). By Lemma 4 and the Lindeberg–Feller central limit theorem, we have the following Lemma 5.

Lemma 5

Suppose that conditions 1–6 hold, then we have

$$\begin{aligned} \frac{1}{ \sqrt{n\phi (h)}} \sum \limits _{i=1}^n \frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\psi (\varepsilon _i)\rightarrow N(0,\sigma ^2(\chi )), \end{aligned}$$

where $\sigma ^2(\chi )=\tau (1-\tau )M_2\frac{f_1(\chi )}{\pi (\chi )}.$

Lemma 6

Suppose that conditions 1–6 hold, then we have that

$$\begin{aligned} E\left( \sum _{i=1}^n \eta _i( \theta ) \right) =\theta ^2M_1 f(0|\chi )+o(1). \end{aligned}$$

Proof

By the Knight’s identity, we have

$$\begin{aligned} \sum _{i=1}^n \eta _i(\theta )= & {} \sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)} K_h(d(\chi ,X_i)) \left( \rho _\tau \left( \varepsilon _i+R_i-\frac{\theta }{\sqrt{n\phi (h)}} \right) - \rho _\tau \left( \varepsilon _i+R_i\right) \right) \\ \quad= & {} \sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)} K_h(d(\chi ,X_i)) \frac{\theta }{\sqrt{n\phi (h)}} (I_{\{\varepsilon _i<0\}}-\tau ) \\ \quad \quad+ & {} \sum _{i=1}^n \frac{\delta _i}{ \pi (X_i)} K_h(d(\chi ,X_i)) \int _0^{ \frac{\theta }{\sqrt{n\phi (h)}} } (I_{\{\varepsilon _i+R_i<s\}}-I_{\{\varepsilon _i<0\}})ds. \end{aligned}$$

Invoking the fact $EI_{\{\varepsilon _i<0\}}=\tau ,$ we have

$$\begin{aligned}&E\left( \sum _{i=1}^n \eta _i( \theta ) \right) \\&\quad =\sum _{i=1}^n E\left( \frac{\delta _i}{ \pi (X_i)} K_h(d(\chi ,X_i)) \int _0^{ \frac{\theta }{\sqrt{n\phi (h)}} } (I_{\{\varepsilon _i+R_i<s\}}-I_{\{\varepsilon _i<0\}})ds \right) \\&\quad =\sum _{i=1}^n E\left( \frac{\delta _i}{ \pi (X_i)} K_h(d(\chi ,X_i)) \int _0^{ \frac{\theta }{\sqrt{n\phi (h)}} } f(0|X_i)s(1+o_p(1))ds \right) \\&\quad =\sum _{i=1}^n E\left( \frac{\delta _i}{ \pi (X_i)} K_h(d(\chi ,X_i)) f(0|X_i)\frac{\theta ^2}{ {2n\phi (h)}} (1+o_p(1)) \right) \\&\quad =\sum _{i=1}^n E\left( K_h(d(\chi ,X_i)) f(0|X_i)\frac{\theta ^2}{ {2n\phi (h)}} (1+o_p(1)) \right) \\&\quad =\frac{1}{2}\theta ^2M_1 f_1(\chi )f(0|\chi )+o(1), \end{aligned}$$

where the last equality holds by Lemma 4. $\square$

Proof of Theorem 1

By Lemmas 3 and 6, we have

$$\begin{aligned}&\sum \limits _{i=1}^n \frac{\delta _i}{{\hat{\pi }}(X_i)} \rho _\tau \left( Y_i-m(\chi )\right) K_h(d(\chi ,X_i))-\sum \limits _{i=1}^n \frac{\delta _i}{{\hat{\pi }}(X_i)} \rho _\tau \left( \varepsilon _i+R_i\right) K_h(d(\chi ,X_i))\\&\quad =\sum \limits _{i=1}^n \frac{\delta _i}{{\hat{\pi }}(X_i)} \rho _\tau \left( \varepsilon _i+R_i-\frac{\theta }{\sqrt{n\phi (h)}}\right) K_h(d(\chi ,X_i))\\&\quad \quad -\sum \limits _{i=1}^n \frac{\delta _i}{{\hat{\pi }}(X_i)} \rho _\tau \left( \varepsilon _i+R_i\right) K_h(d(\chi ,X_i))\\&\quad = \sum \limits _{i=1}^n \frac{\theta }{\sqrt{n\phi (h)}}\frac{\delta _i}{ \pi (X_i)}K_h(d(\chi ,X_i))\psi (\varepsilon _i)+\theta ^2M_1 f(0|\chi )+o(1)\\&\quad =\theta W_n+\frac{1}{2}\theta ^2M_1f_1(\chi ) f(0|\chi )+o(1), \end{aligned}$$

where the last equality holds by Lemma 5, $W_n$ is normal random variable with zero mean and variance $\sigma ^2(\chi )=\tau (1-\tau )M_2\frac{f_1(\chi )}{\pi (\chi )}.$ Invoking the epiconvergence results of Geyer (1994), we have

$$\begin{aligned} {\hat{\theta }}=(M_1f_1(\chi ) f(0|\chi ))^{-1}W_n+o_p(1). \end{aligned}$$

Combining this with Lemma 5, we have

$$\begin{aligned} \sqrt{n\phi (h)}({\hat{m}}_n(\chi ) -m(\chi )) \rightarrow N\left( 0, \varpi (\chi )\right) , \end{aligned}$$

where $\varpi (\chi )= \tau (1-\tau )\frac{M_2}{M_1^2}\frac{1}{ f_1(\chi )\pi (\chi ) f^2(0|\chi )}.$$\square$