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kNN estimation in functional partial linear modeling

  • Nengxiang LingEmail author
  • Germán Aneiros
  • Philippe Vieu
Regular Article

Abstract

A statistical procedure combining the local adaptivity and the easiness of implementation of k-nearest-neighbours (kNN) estimates together with the semiparametric flexibility of partial linear modeling is developed for regression problems involving functional variable. Various asymptotic results are stated, both for the linear parameters and for the nonparametric operator involved in the model. A simulation study compares the finite sample behaviour of the kNN method with alternative estimation procedures. Finally, comparison with alternative functional regression models is carried out by means of a real curves data application which exhibits the interest both of the kNN method and of the semi-parametric modeling.

Keywords

kNN estimate Functional data analysis Partial linear regression Semi-parametrics 

Notes

Acknowledgements

This work has received financial support from the National Social Science Funds of China (NSSF 14ATJ005), the NNSF of China (NNSF 11501005), the Spanish Ministerio de Economía y Competitividad (Grant MTM2014-52876-R), the Xunta de Galicia (Centro Singular de Investigaciń 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.

References

  1. Aneiros G, Bongiorno EG, Cao R, Vieu P (2017) Functional statistics and related fields. Contributions to statistics. Springer, ChamCrossRefGoogle Scholar
  2. Aneiros G, Ling N, Vieu P (2015) Error variance estimation in semi-functional partially linear regression models. J Nonparametr Stat 27(3):316–330MathSciNetCrossRefzbMATHGoogle Scholar
  3. Aneiros-Pérez G, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 76(11):1102–1110MathSciNetCrossRefzbMATHGoogle Scholar
  4. Aneiros-Pérez G, Vieu P (2011) Automatic estimation procedure in partial linear model with functional data. Stat Pap 52(4):751–771MathSciNetCrossRefzbMATHGoogle Scholar
  5. Attouch M, Benchikh T (2012) Asymptotic distribution of robust k-nearest neighbour estimator for functional nonparametric models. Mat Vesnik 644:275–285MathSciNetzbMATHGoogle Scholar
  6. Berlinet A, Servien R (2011) Necessary and sufficient condition for the existence of a limit distribution of the nearest-neighbour density estimator. J Nonparametr Stat 23(3):633–643MathSciNetCrossRefzbMATHGoogle Scholar
  7. Biau G, Cérou F, Guyader A (2010) Rates of convergence of the functional k-nearest neighbor estimate. IEEE Trans Inf Theory 56(4):2034–2040MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bosq D, Blanke D (2007) Inference and prediction in large dimension. Wiley series in probability and statistics. Wiley, ChichesterCrossRefzbMATHGoogle Scholar
  9. Burba F, Ferraty F, Vieu P (2009) \(k\)-Nearest Neighbour method in functional nonparametric regression. J Nonparametr Stat 21(4):453–469MathSciNetCrossRefzbMATHGoogle Scholar
  10. Cerou F, Guyader A (2006) Nearest neighbor classification in infinite dimension. ESAIM Probab Stat 10:340–355MathSciNetCrossRefzbMATHGoogle Scholar
  11. Chen Z, Wang H, Wang X (2016) The consistency for the estimator of nonparametric regression model based on martingale difference errors. Stat Pap 57(2):451–469MathSciNetCrossRefzbMATHGoogle Scholar
  12. Chu B, Huynh K, Jacho-Chávez D (2013) Functionals of order statistics and their multivariate concomitants with application to semiparametric estimation by nearest neighbours. Sankhya B 75(2):238–292MathSciNetCrossRefzbMATHGoogle Scholar
  13. Collomb G (1979) Estimation de la régression par la méthode des k points les plus proches: propriétés de convergence ponctuelle (French). C R Acad Sci Paris 289(3):245–247MathSciNetzbMATHGoogle Scholar
  14. Collomb G, Hassani S, Sarda P, Vieu P (1985) Convergence uniforme d’estimateurs de la fonction de hazard pour des observations dépendantes: méthodes du noyau et des k points les plus proches. (French). C R Acad Sci Paris Sér I Math 301(12):653–656MathSciNetzbMATHGoogle Scholar
  15. Cover T-M (1968) Estimation by the nearest neighbor rule. IEEE Trans Inf Theory IT–14:50–55CrossRefzbMATHGoogle Scholar
  16. Cuevas A (2014) A partial overview of the theory of statistics with functional data. J Stat Plann Inference 147:1–23MathSciNetCrossRefzbMATHGoogle Scholar
  17. Dette H, Gefeller O (1995) The impact of different definitions of nearest neighbour distances for censored data on nearest neighbour kernel estimators of the hazard rate. J Nonparametr Stat 4(3):271–282MathSciNetCrossRefzbMATHGoogle Scholar
  18. Devroye L, Györfi L, Krzyzak A, Lugosi G (1994) On the strong universal consistency of nearest neighbor regression function estimates. Ann Stat 22:1371–1385MathSciNetCrossRefzbMATHGoogle Scholar
  19. Devroye L, Wagner T (1977) The strong uniform consistency of nearest neighbor density estimates. Ann Stat 5(3):536–540MathSciNetCrossRefzbMATHGoogle Scholar
  20. Devroye L, Wagner T (1982) Nearest neighbor methods in discrimination. In: Classification, pattern recognition and reduction of dimensionality. Handbook of statistics, vol 2. North-Holland, Amsterdam. pp 193–197Google Scholar
  21. Engle RF, Granger CWJ, Rice J, Weiss A (1986) Semiparametric estimates of the relation between weather and electricity sales. J Am Stat Assoc 81:310–320CrossRefGoogle Scholar
  22. Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Theory and practice. Springer, New YorkzbMATHGoogle Scholar
  23. Goia A, Vieu P (2014) Some advances in semiparametric functional data modelling. In: Contributions in infinite-dimensional statistics and related topics. Esculapio, Bologna, pp 135–141Google Scholar
  24. Goia A, Vieu P (2016) An introduction to recent advances in high/infinite dimensional statistics. J Multivar Anal 146:1–6MathSciNetCrossRefzbMATHGoogle Scholar
  25. Györfi L, Kohler M, Krzyzak A, Walk H (2002) A distribution-free theory of nonparametric regression. Springer series in statisics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  26. Härdle W, Liang H, Gao J (2000) Partially linear models. Physica-Verlag, HeidelbergCrossRefzbMATHGoogle Scholar
  27. Hong S (1992) Estimation theory of a class of semiparametric regression models. Sci China Ser A 35(6):657–674MathSciNetzbMATHGoogle Scholar
  28. Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer series in statistics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  29. Hsing T, Eubank R (2015) Theoretical foundations to functional data analysis with an introduction to linear operators. Wiley series in probability and statistics. Wiley, ChichesterCrossRefzbMATHGoogle Scholar
  30. Kara-Zaitri L, Laksaci A, Rachdi M, Vieu P (2017a) Data-driven kNN estimation in nonparametric functional data-analysis. J Multivar Anal 153:176–188CrossRefzbMATHGoogle Scholar
  31. Kara-Zaitri L, Laksaci A, Rachdi M, Vieu P (2017b) Uniform in bandwidth consistency for various kernel estimators involving functional data. J Nonparametr Stat 29(1):85–107MathSciNetCrossRefzbMATHGoogle Scholar
  32. Kudraszow N, Vieu P (2013) Uniform consistency of \(k\)NN regressors for functional variables. Stat Probab Lett 83(8):1863–1870MathSciNetCrossRefzbMATHGoogle Scholar
  33. Laloë T (2008) A k-nearest approach for functional regression. Stat Prob Lett 10:1189–1193MathSciNetCrossRefzbMATHGoogle Scholar
  34. Li H, Li Q, Liu R (2016) Consistent model specification tests based on k-nearest-neighbor estimation methods. J Econom 194(1):187–202MathSciNetCrossRefzbMATHGoogle Scholar
  35. Lian H (2011) Functional partial linear model. J Nonparametr Stat 23(1):115–128MathSciNetCrossRefzbMATHGoogle Scholar
  36. Ouadah S (2013) Uniform-in-bandwidth nearest-neighbor density estimation. Stat Probab Lett 83(8):1835–1843MathSciNetCrossRefzbMATHGoogle Scholar
  37. Paindaveine D, Van Bever G (2015) Nonparametrically consistent depth-based classifiers. Bernoulli 21(1):69–82MathSciNetCrossRefzbMATHGoogle Scholar
  38. Ramsay J, Silverman B (2005) Functional data analysis. Springer series in statistics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  39. Robinson P (1995) Nearest-neighbour estimation of semiparametric regression models. J Nonparametr Stat 5(1):33–41MathSciNetCrossRefzbMATHGoogle Scholar
  40. Sancetta A (2010) Nearest neighbor conditional estimation for Harris recurrent Markov chains. J Multivar Anal 100(10):2224–2236MathSciNetCrossRefzbMATHGoogle Scholar
  41. Shang H (2014) Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density. Comput Stat 29(3–4):829–848MathSciNetCrossRefzbMATHGoogle Scholar
  42. Zhang J (2013) Analysis of variance for functional data. Monographs on statistics and applied probability. Chapman & Hall/CRC, Boca RatonGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Nengxiang Ling
    • 1
    Email author
  • Germán Aneiros
    • 2
  • Philippe Vieu
    • 3
  1. 1.School of MathematicsHefei University of TechnologyHefeiChina
  2. 2.Departamento de MatemáticasUniversidade da CoruñaA CoruñaSpain
  3. 3.Institut de MathématiquesUniversité Paul SabatierToulouseFrance

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