Optimal estimation in functional linear regression for sparse noise‐contaminated data

• Autores: Behdad Mostafaiy, Mohammad Reza-Faridrohani, Shojaeddin Chenouri
• Localización: Canadian Journal of Statistics = Revue Canadienne de Statistique, ISSN 0319-5724, Vol. 47, Nº. 4, 2019, págs. 524-559
• Idioma: inglés
• DOI: 10.1002/cjs.11511
• Enlaces
  • Texto completo
• Resumen

  • In this article, we propose a novel approach to fit a functional linear regression in which both the response and the predictor are functions. We consider the case where the response and the predictor processes are both sparsely sampled at random time points and are contaminated with random errors. In addition, the random times are allowed to be different for the measurements of the predictor and the response functions. The aforementioned situation often occurs in longitudinal data settings. To estimate the covariance and the cross‐covariance functions, we use a regularization method over a reproducing kernel Hilbert space. The estimate of the cross‐covariance function is used to obtain estimates of the regression coefficient function and of the functional singular components. We derive the convergence rates of the proposed cross‐covariance, the regression coefficient, and the singular component function estimators. Furthermore, we show that, under some regularity conditions, the estimator of the coefficient function has a minimax optimal rate. We conduct a simulation study and demonstrate merits of the proposed method by comparing it to some other existing methods in the literature. We illustrate the method by an example of an application to a real‐world air quality dataset. The Canadian Journal of Statistics 47: 524–559; 2019 © 2019 Statistical Society of Canada