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, Volume 27, Issue 3, pp 659–679 | Cite as

Bootstrap in semi-functional partial linear regression under dependence

  • Germán AneirosEmail author
  • Paula Raña
  • Philippe Vieu
  • Juan Vilar
Original Paper

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.

Keywords

Bootstrap Dependent data Functional data Semi-parametric regression 

Mathematics Subject Classification

62G08 62G09 62G20 

Notes

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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Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidade da CoruñaA CoruñaSpain
  2. 2.Institut de MathématiquesUniversité Paul SabatierToulouseFrance

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