The local linear functional kNN estimator of the conditional expectile: uniform consistency in number of neighbors

Abstract

The main purpose of the present paper is to investigate the problem of the nonparametric estimation of the expectile regression in which the response variable is scalar while the covariate is a random function. More precisely, an estimator is constructed by using the local linear k Nearest Neighbor procedures (kNN). The main contribution of this study is the establishment of the Uniform consistency in Number of Neighbors of the constructed estimators. These results are established under fairly general structural conditions on the classes of functions and the underlying models. The usefulness of our result for the smoothing parameter automatic selection is discussed. Some simulation studies are carried out to show the finite sample performances of the kNN estimator. The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in functional data analysis.

Appendix

This appendix contains supplementary lemmas that are essential parts of providing a more comprehensive understanding of the paper.

Lemma 5

(Theorem 2.14.1 in van der Vaart and Wellner (1996), page 239) Let $Z_1, Z_2,\ldots , Z_n$ be independent and identically distributed ${{{\mathcal {F}}}}$-valued random variables and consider $\mathcal{C}$ a pointwise measurable class of functions g :  $\mathcal{F}\rightarrow \mathbb {R}$ with envelope function F, then

$$\begin{aligned} \Vert \Vert \alpha _n(g) \Vert _{{{{\mathcal {C}}}}}\Vert _p\le C J(1, {{{\mathcal {C}}}})\Vert F\Vert _{p\vee 2}, \end{aligned}$$

where

$$\begin{aligned} \alpha _n(g)= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n(g(Z_i)-{\mathrm {I\!E}}g(Z_i)), \\ J(1, {{{\mathcal {C}}}})= & {} \displaystyle \sup _Q \int _0^1 \sqrt{1+\log {{{\mathcal {N}}}} (\epsilon \Vert F\Vert _{Q,2},{{{\mathcal {C}}}},d_Q)}d\epsilon , \\ \Vert \alpha _n(g) \Vert _{{{{\mathcal {C}}}}}= & {} \sup _{g\in {{\mathcal {C}}}}|\alpha _n(g) |, \end{aligned}$$

with $\Vert \cdot \Vert _p={\mathrm {I\!E}}^{1/p}[\cdot ]^p$ and $s\vee t$ denoting the maximum of s and t.

Lemma 6

(Theorem 3.1 in Dony and Einmahl 2009) Let ${{{\mathcal {C}}}}$ be a a pointwise measurable class of functions g :  $\mathcal{F}\rightarrow \mathbb {R}$ satisfying:

$$\begin{aligned} {\mathrm {I\!E}}\Vert \alpha _n(g) \Vert _{{{{\mathcal {C}}}}}\le C \Vert F\Vert _{2}, \end{aligned}$$

with F is an envelope function of ${{{\mathcal {C}}}}$. Then, for any $A\in {{{\mathcal {A}}}}$, we have:

$$\begin{aligned} {\mathrm {I\!E}}\Vert \alpha _n(g.\mathbb {1}_A) \Vert _{{{{\mathcal {C}}}}}\le 2C \Vert F.\mathbb {1}_A\Vert _{ 2}. \end{aligned}$$

Lemma 7

(Theorem 4.1 in Dony and Einmahl 2009) Let $Z_1, Z_2,\ldots , Z_n$ be independent and identically distributed ${{{\mathcal {F}}}}$-valued random variables and consider ${{{\mathcal {C}}}}$ a pointwise measurable class of functions g :  ${{{\mathcal {F}}}}\rightarrow \mathbb {R}$ with envelope function F. Assume that for some $H>0$:

$$\begin{aligned} {\mathrm {I\!E}}[F^p(Z)]\le \frac{p!}{2}\sigma ^2H^{p-2} \hbox { where } \sigma ^2\ge {\mathrm {I\!E}}[F^2(Z)]. \end{aligned}$$

Then, for $\beta _n={\mathrm {I\!E}}[\Vert \sqrt{n}\alpha _n(g) \Vert _{{{{\mathcal {C}}}}}]$, we have for any $t>0$:

$$\begin{aligned} \textrm{I}\!\textrm{P}\left\{ \max _{1\le k\le n}\Vert \alpha _n(g) \Vert _{{{{\mathcal {C}}}}}\ge \beta _n+t\right\} \le \exp \left( -\frac{t^2}{2n\sigma ^2+2tH}\right) . \end{aligned}$$