, 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


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.


Bootstrap Dependent data Functional data Semi-parametric regression 

Mathematics Subject Classification

62G08 62G09 62G20 



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.


  1. Aneiros-Pérez G, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 76:1102–1110MathSciNetCrossRefGoogle Scholar
  2. Aneiros-Pérez G, Vieu P (2008) Nonparametric time series prediction: a semi-functional partial linear modeling. J Multivar Anal 99:834–857MathSciNetCrossRefGoogle Scholar
  3. Aneiros G, Vilar J, Raña P (2016) Short-term forecast of daily curves of electricity demand and price. Electr Power Energy Syst 80:96–108CrossRefGoogle Scholar
  4. Bosq D (2000) Linear processes in function spaces. Theory and applications. Lecture Notes in Statistics, vol 149. Springer, New YorkCrossRefGoogle Scholar
  5. Bücher A, Dette H, Wieczorek G (2011) Testing model assumptions in functional regression models. J Multivar Anal 102(10):1472–1488MathSciNetCrossRefGoogle Scholar
  6. Cavaliere G, Politis D, Rahbek A (2015) Recent developments in bootstrap methods for dependent data. J Time Series Anal 36(3):269–271MathSciNetCrossRefGoogle Scholar
  7. Cuevas A (2014) A partial overview of the theory of statistics with functional data. J Stat Plann Inference 147:1–23MathSciNetCrossRefGoogle Scholar
  8. Cuevas A, Febrero M, Fraiman R (2006) On the use of the bootstrap for estimating functions with functional data. Comput Stat Data Anal 51(2):1063–1074MathSciNetCrossRefGoogle Scholar
  9. Delsol L (2009) Advances on asymptotic normality in non-parametric functional time series analysis. Statistics 43:13–33MathSciNetCrossRefGoogle Scholar
  10. Efron B, Tibshirani R (1993) An introduction to the bootstrap. Monographs on statistics and applied probability, vol 57. Chapman and Hall, New YorkCrossRefGoogle Scholar
  11. Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Springer, New YorkzbMATHGoogle Scholar
  12. Ferraty F, Mas A, Vieu P (2007) Nonparametric regression on functional data: inference and practical aspects. Aust N Z J Stat 49(3):267–286MathSciNetCrossRefGoogle Scholar
  13. Ferraty F, Van Keilegom I, Vieu P (2010) On the validity of the bootstrap in non-parametric functional regression. Scand J Stat 37:286–306MathSciNetCrossRefGoogle Scholar
  14. Gneiting T, Raftery E (2007) Strictly proper scoring rules, prediction, and estimation. J Am Stat Assoc 102(477):359–378MathSciNetCrossRefGoogle Scholar
  15. Goia A, Vieu P (2014) Some advances on semi-parametric functional data modelling. Contributions in infinite-dimensional statistics and related topics, Esculapio, Bologna, pp 135–140Google Scholar
  16. Goia A, Vieu P (2016) An introduction to recent advances on high/infinite dimensional statistics. J Multivariate Anal 146:1–6MathSciNetCrossRefGoogle Scholar
  17. González-Manteiga W, Martínez-Calvo A (2011) Bootstrap in functional linear regression. J Stat Plann Inference 141:453–461MathSciNetCrossRefGoogle Scholar
  18. Hall P (1992) The bootstrap and Edgeworth expansion. Springer series in statistics. Springer, New YorkCrossRefGoogle Scholar
  19. Härdle W, Bowman A (1988) Bootstrapping in nonparametric regression: local adaptive smoothing and confidence bands. J Am Stat Assoc 83:102–110MathSciNetzbMATHGoogle Scholar
  20. Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21:1926–1947MathSciNetCrossRefGoogle Scholar
  21. Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer, New YorkCrossRefGoogle Scholar
  22. Hsing T, Eubank R (2015) Theoretical foundations of functional data analysis, with an introduction to linear operators. Wiley, ChichesterCrossRefGoogle Scholar
  23. Kreiss J, Paparoditis E (2011) Bootstrap methods for dependent data: a review. J Korean Stat Soc 40(4):357–378MathSciNetCrossRefGoogle Scholar
  24. Liang H, Härdle W, Sommerfeld V (2000) Bootstrap approximation in a partially linear regression model. J Stat Plann Inference 91:413–426MathSciNetCrossRefGoogle Scholar
  25. Mammen E (1993) Bootstrap and wild bootstrap for high-dimensional linear models. Ann Stat 21:255–285MathSciNetCrossRefGoogle Scholar
  26. McMurry T, Politis D (2011) Resampling methods for functional data. The Oxford handbook of functional data analysis. Oxford University Press, Oxford, pp 189–209Google Scholar
  27. Ramsay JO, Silverman BW (2005) Functional data analysis. Springer, New YorkCrossRefGoogle Scholar
  28. Raña P, Aneiros G, Vilar J, Vieu P (2016) Bootstrap confidence intervals in functional nonparametric regression under dependence. Electron J Stat 10(2):1973–1999MathSciNetCrossRefGoogle Scholar
  29. Shang HL (2013) Functional time series approach for forecasting very short-term electricity demand. J Appl Stat 40(1):152–168MathSciNetCrossRefGoogle Scholar
  30. Staicu AM, Lahiri S, Carroll R (2015) Significance tests for functional data with complex dependence structure. J Stat Plann Inference 156:1–13MathSciNetCrossRefGoogle Scholar
  31. You J, Zhou X (2005) Bootstrap approximation of a semiparametric partially linear model with autoregressive errors. Stat Sin 15:117–133zbMATHGoogle Scholar

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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