Bootstrap in semi-functional partial linear regression under dependence

Abstract

This paper deals with the semi-functional partial linear regression model \[Y={{\varvec{X}}}^\mathrm{T}{\varvec{\beta }}+m({\varvec{\chi }})+\varepsilon \] under \[\alpha \]-mixing conditions. \[{\varvec{\beta }} \in \mathbb {R}^{p}\] and \[m(\cdot )\] denote an unknown vector and an unknown smooth real-valued operator, respectively. The covariates \[{{\varvec{X}}}\] and \[{\varvec{\chi }}\] are valued in \[\mathbb {R}^{p}\] and some infinite-dimensional space, respectively, and the random error \[\varepsilon \] verifies \[\mathbb {E}(\varepsilon |{{\varvec{X}}},{\varvec{\chi }})=0\]. Naïve and wild bootstrap procedures are proposed to approximate the distribution of kernel-based estimators of \[{\varvec{\beta }}\] and \[m(\chi )\], and their asymptotic validities are obtained. A simulation study shows the behavior (on finite sample sizes) of the proposed bootstrap methodology when applied to construct confidence intervals, while an application to real data concerning electricity market illustrates its usefulness in practice.