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Best estimation of functional linear models

Author

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  • Aletti, Giacomo
  • May, Caterina
  • Tommasi, Chiara

Abstract

Observations that are realizations of some continuous process are frequently found in science, engineering, economics, and other fields. In this paper, we consider linear models with possible random effects and where the responses are random functions in a suitable Sobolev space. In particular, the processes cannot be observed directly. By using smoothing procedures on the original data, both the response curves and their derivatives can be reconstructed, both as an ensemble and separately. From these reconstructed functions, one representative sample is obtained to estimate the vector of functional parameters. A simulation study shows the benefits of this approach over the common method of using information either on curves or derivatives. The main theoretical result is a strong functional version of the Gauss–Markov theorem. This ensures that the proposed functional estimator is more efficient than the best linear unbiased estimator (BLUE) based only on curves or derivatives.

Suggested Citation

  • Aletti, Giacomo & May, Caterina & Tommasi, Chiara, 2016. "Best estimation of functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 54-68.
    • Handle: RePEc:eee:jmvana:v:151:y:2016:i:c:p:54-68
    DOI: 10.1016/j.jmva.2016.07.005
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    References listed on IDEAS

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    1. Laura M. Sangalli & Piercesare Secchi & Simone Vantini & Alessandro Veneziani, 2009. "Efficient estimation of three‐dimensional curves and their derivatives by free‐knot regression splines, applied to the analysis of inner carotid artery centrelines," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 58(3), pages 285-306, July.
    2. Stefano Baraldo & Francesca Ieva & Anna Maria Paganoni & Valeria Vitelli, 2013. "Outcome Prediction for Heart Failure Telemonitoring Via Generalized Linear Models with Functional Covariates," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(3), pages 403-416, September.
    3. Pigoli, Davide & Sangalli, Laura M., 2012. "Wavelets in functional data analysis: Estimation of multidimensional curves and their derivatives," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1482-1498.
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