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.
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Acknowledgements
This work has received financial support from the Spanish Ministerio de Economía y Competitividad (Grant MTM2014-52876-R), the Xunta de Galicia (Centro Singular de Investigación de Galicia accreditation ED431G/01 2016-2019 and Grupos de Referencia Competitiva ED431C2016-015) and the European Union (European Regional Development Fund—ERDF). The authors would like to thank the Associate Editor and the two anonymous referees for their constructive and helpful comments, which have greatly improved the paper.
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Aneiros, G., Raña, P., Vieu, P. et al. Bootstrap in semi-functional partial linear regression under dependence. TEST 27, 659–679 (2018). https://doi.org/10.1007/s11749-017-0566-y
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DOI: https://doi.org/10.1007/s11749-017-0566-y