Uniform consistency in number of neighbors of the kNN estimator of the conditional quantile model

Abstract

We are interested in the efficiency of the nonparametric estimation of the conditional quantile when the response variable is a scalar given a functional covariate. To do this, we adopt a technique which is based on the use of the k-Nearest Neighbors procedure to build a kernel estimator of this model. Then, we establish the uniform convergence in number of neighbors of the constructed estimator. Moreover, we discuss the optimal choices of different parameters that are involved in the model as well as the impacts of the obtained results. Finally, we show the applicability and efficiency of our methodology to investigate the fuel quality by using a Near-infrared spectroscopy dataset.

Appendix

For the sake of brevity we prove only the Lemma 2. The proof of Lemma 3 is omitted. In fact, it’s similar to that of Lemma 2 in Kara-Zaïtri et al. (2017).

Proof of Lemma 2. We write that

$$\begin{aligned}{}[t_p(x)-\delta ,\, t_p(x)+\delta ]\subset \bigcup _{j=1}^{d_n}\left( t_j-l_n, t_j+l_n\right) , \end{aligned}$$

with $\displaystyle l_n=n^{-1/2}$ and $d_n=O\left( n^{1/2 }\right)$. Furthermore, the monotonies of ${\mathrm {I\!E}}[\widehat{F }_N(x,\cdot )]$ and $\widehat{F }_N(x,\cdot )$ imply that for $1\le j\le d_n$

$$\begin{aligned} {\mathrm {I\!E}}\left[ \widehat{F }_N(x,t_j-l_n)\right]\le & {} \sup _{t\in (t_j-l_n, t_j+l_n)}{\mathrm {I\!E}}\left[ \widehat{F }_N(x,t)\right] \le {\mathrm {I\!E}}\left[ \widehat{F }_N(x,t_j+l_n)\right] ,\\ \widehat{F }_N(x,t_j-l_n)\le & {} \sup _{t\in (t_j-l_n, t_j+l_n)}\widehat{F }_N(x,t)\le \widehat{F }_N(x,t_j+l_n). \end{aligned}$$

It follows that

$$\begin{aligned}&{\sup _{t\in [t_p(x)-\delta ,\, t_p(x)+\delta ]}\left| \widehat{F }_N(x,t) -{\mathrm {I\!E}}\left[ \widehat{F }_N(x,t)\right] \right| }\\&\quad \le \max _{1\le j\le d_n} \max _{z\in \{t_j-l_n, t_j+l_n\}}\left| \widehat{F }_N(x,z)- {\mathrm {I\!E}}\left[ \widehat{F }_N(x,z)\right] \right| +2Cl_n. \end{aligned}$$

Then, it is easy to prove that $\displaystyle l_n^{}=o\left( \frac{\log n}{n\,\phi _x(a_n)}\right) ^{1/2}.$ So, it suffices to show that

$$\begin{aligned} \sup _{a_n\le h\le b_n}\max _{1\le j\le d_n}\max _{z\in \{t_j-l_n,t_j+l_n\}}\left| \widehat{F }_N(x,z)- {\mathrm {I\!E}}\left[ \widehat{F }_N(x,z)\right] \right| =O\left( \frac{\log n}{n\,\phi _x(a_n)}\right) ^{1/2}, \; a.co. \end{aligned}$$

(3)

To do that we write that

$$\begin{aligned}&{{\mathrm {I\!P}}\left( \sup _{a_n\le h\le b_n}\max _{1\le j\le d_n} \max _{z\in \{t_j-l_n,t_j+l_n\}}\left| \widehat{F }_N(x,z)- {\mathrm {I\!E}}\left[ \widehat{F }_N(x,z)\right] \right|> {\eta }\sqrt{\frac{\log n}{n\phi _x(a_n)}}\right) } \\&\quad \le \ 2d_n\max _{1\le j\le d_n}\max _{z\in \{t_j-l_n,t_j+l_n\}}{\mathrm {I\!P}}\left( \sup _{a_n\le h\le b_n} \left| \widehat{F }_N(x, z)-{\mathrm {I\!E}}[\widehat{F }_N(x, z)]\right| >{\eta }\sqrt{\frac{\log n}{n \phi _x(a_n)}}\right) . \end{aligned}$$

So, all what is left to be shown is to evaluate the following quantity

$$\begin{aligned} {\mathrm {I\!P}}\left( \sup _{a_n\le h\le b_n}\left| \widehat{F }_N(x, z)-{\mathrm {I\!E}}[\widehat{F }_N(x, z)] \right| >{\eta }\sqrt{\frac{\log n }{n\phi _x(a_n)}}\right) , \, \text{ for } \text{ all } \, z=t_j\mp l_n, \, 1\le j\le d_n. \end{aligned}$$

The proof of the latter follow similar ideas as in Einmahl and Mason (2005) which are based on the Bernstein’s inequality for empirical processes

where $K_i=K\left( h^{-1}d(x,X_{ i})\right)$. Thereafter, we get

$$\begin{aligned} {\mathrm {I\!P}}\left( \sup _{a_n\le h\le b_0} \sqrt{\frac{n\phi _x(h)}{\log n}}\left| \widehat{F }_N(x, z)-{\mathrm {I\!E}}[\widehat{F }_N(x, z)]\right| \ge \eta _0'\right) \le \log (n)n^{-C'\eta _0^2}. \end{aligned}$$

Consequently, an appropriate choice of $\eta _0$ permits to deduce (3).