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Generalized linear model with functional predictors and their derivatives

Author

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  • Ahmedou, Aziza
  • Marion, Jean-Marie
  • Pumo, Besnik

Abstract

The conditional expectation E(Y|X) of a generalized functional linear model with scalar response Y is given by g{〈X,ϕ〉L2} where X and ϕ are functions defined in L2:=L2[0,1]. Let us consider that X belongs to the Sobolev space W:=W2,1[0,1] and denote X′ its derivative. In this paper we focus on an extension of the previous model where E(Y|X) is given by g{〈X,β〉W+〈X′,γ〉L2}. With a similar approach to Cardot and Sarda (2005) or Stone (1986) for generalized additive models, we propose estimators for the unknown parameters β, γ and obtain their rate of convergence. We compare numerically the prediction performance of this new model with alternative models proposed in the literature.

Suggested Citation

  • Ahmedou, Aziza & Marion, Jean-Marie & Pumo, Besnik, 2016. "Generalized linear model with functional predictors and their derivatives," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 313-324.
    • Handle: RePEc:eee:jmvana:v:146:y:2016:i:c:p:313-324
    DOI: 10.1016/j.jmva.2015.10.009
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    References listed on IDEAS

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    1. Müller, Hans-Georg & Yao, Fang, 2008. "Functional Additive Models," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1534-1544.
    2. André Mas & Besnik Pumo, 2009. "Functional linear regression with derivatives," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(1), pages 19-40.
    3. Cardot, Hervé & Sarda, Pacal, 2005. "Estimation in generalized linear models for functional data via penalized likelihood," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 24-41, January.
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    5. Ferraty, Frédéric & Vieu, Philippe, 2009. "Additive prediction and boosting for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1400-1413, February.
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    7. F. Ferraty & A. Goia & E. Salinelli & P. Vieu, 2013. "Functional projection pursuit regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 293-320, June.
    8. James, Gareth M. & Silverman, Bernard W., 2005. "Functional Adaptive Model Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 565-576, June.
    9. Gareth M. James, 2002. "Generalized linear models with functional predictors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 411-432, August.
    10. Aneiros-Pérez, Germán & Vieu, Philippe, 2006. "Semi-functional partial linear regression," Statistics & Probability Letters, Elsevier, vol. 76(11), pages 1102-1110, June.
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