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Measures of influence for the functional linear model with scalar response

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  • Febrero-Bande, Manuel
  • Galeano, Pedro
  • González-Manteiga, Wenceslao

Abstract

This paper studies how to identify influential observations in the functional linear model in which the predictor is functional and the response is scalar. Measurement of the effects of a single observation on estimation and prediction when the model is estimated by the principal components method is undertaken. For that, three statistics are introduced for measuring the influence of each observation on estimation and prediction of the functional linear model with scalar response that are generalizations of the measures proposed for the standard regression model by [D.R. Cook, Detection of influential observations in linear regression, Technometrics 19 (1977) 15-18; D. Peña, A new statistic for influence in linear regression, Technometrics 47 (2005) 1-12] respectively. A smoothed bootstrap method is proposed to estimate the quantiles of the influence measures, which allows us to point out which observations have the larger influence on estimation and prediction. The behavior of the three statistics and the quantile estimation bootstrap based method is analyzed via a simulation study. Finally, the practical use of the proposed statistics is illustrated by the analysis of a real data example, which show that the proposed measures are useful for detecting heterogeneity in the functional linear model with scalar response.

Suggested Citation

  • Febrero-Bande, Manuel & Galeano, Pedro & González-Manteiga, Wenceslao, 2010. "Measures of influence for the functional linear model with scalar response," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 327-339, February.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:2:p:327-339
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    References listed on IDEAS

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    1. Chiou, Jeng-Min & Muller, Hans-Georg, 2007. "Diagnostics for functional regression via residual processes," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4849-4863, June.
    2. Bing Li, 2008. "Comments on: Augmenting the bootstrap to analyze high dimensional genomic data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(1), pages 19-21, May.
    3. Cardot, Hervé & Ferraty, Frédéric & Sarda, Pascal, 1999. "Functional linear model," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 11-22, October.
    4. Hervé Cardot & Frédéric Ferraty & André Mas & Pascal Sarda, 2003. "Testing Hypotheses in the Functional Linear Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 241-255, March.
    5. Svitlana Tyekucheva & Francesca Chiaromonte, 2008. "Rejoinder on: Augmenting the bootstrap to analyze high dimensional genomic data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(1), pages 47-55, May.
    6. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2006. "On the use of the bootstrap for estimating functions with functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 1063-1074, November.
    7. Peter Hall & Mohammad Hosseini‐Nasab, 2006. "On properties of functional principal components analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 109-126, February.
    8. Svitlana Tyekucheva & Francesca Chiaromonte, 2008. "Augmenting the bootstrap to analyze high dimensional genomic data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(1), pages 1-18, May.
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    Cited by:

    1. Siegfried Hörmann & Łukasz Kidziński, 2015. "A Note on Estimation in Hilbertian Linear Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(1), pages 43-62, March.
    2. Febrero-Bande, Manuel & de la Fuente, Manuel Oviedo, 2012. "Statistical Computing in Functional Data Analysis: The R Package fda.usc," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 51(i04).
    3. Manuel Febrero-Bande & Pedro Galeano & Wenceslao González-Manteiga, 2017. "Functional Principal Component Regression and Functional Partial Least-squares Regression: An Overview and a Comparative Study," International Statistical Review, International Statistical Institute, vol. 85(1), pages 61-83, April.

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