CLT in functional linear regression models

By Hervé Cardot, André Mas, Pascal Sarda. 2005

141 citations recorded by the source corpus.

We propose in this work to derive a CLT in the functional linear regression model. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f143cb597646560a26b0bf0de4292e79b8172b01

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Citations in the corpus

Presmoothing in Functional Linear Regression (2012, methodology): “Second methodology is based on functional principal components analysis (FPCA), as developed and analyzed by Cardot, Ferraty, and Sarda (1999) Cardot, Ferraty, and Sarda (2003), Cai and Hall (2006), Hall and Hosseini-Nasab (2006), Hall and Horowitz (2007), and Cardot, Mas, and Sarda (2007).”
Adaptive functional linear regression (2011, methodology): “…estimation procedures of the slope function studied in the statistical literature are based on principal components regression (see e.g. Bosq [2000], Frank and Friedman [1993], Cardot et al. [1999], Cardot et al. [2007] or Müller and Stadtmüller [2005] in the context of generalized linear models).”
Asymptotics of prediction in functional linear regression with functional outputs (2009, methodology): “Besides, as can be seen from Lemma 4 in Cardot, Mas and Sarda [7] [I +(zI − )−1/2( − n)(zI − )−1/2]Rn(z) = I .”
Locally Sparse Estimator for Functional Linear Regression Models (2017, background): “Convergence rates of some estimators in the weak topology of L2([0,T ]) were treated in Cardot, Mas, and Sarda (2007).”
Smoothing splines estimators for functional linear regression (2009, methodology): “The distance between α̂ and α is evaluated with respect to L2 semi-norms induced by the operator Γ, ‖u‖2Γ = 〈Γu,u〉 with 〈u, v〉 = ∫I u(t)v(t)dt, or its discretized or empirical versions (see, e.g., Cardot, Ferraty and Sarda [7] or Müller and Stadtmüller [23] for similar setups).”
Theory for high-order bounds in functional principal components analysis (2009, background): “Cardot et al. (2007) have given a central limit theorem in the context of functional linear regression.”
Truncated linear models for functional data (2014, unknown): “…data analysis; Cai and Hall (2006) and Apanasovich and Goldstein (2008) addressed mean-squared prediction error in functional linear regression; and Cardot et al. (2007), Hall and Horowitz (2007), Li and Hsing (2007), Cai and Yuan (2012) and Johannes and Schenk (2013) discussed convergence rates…”
Minimax adaptive tests for the Functional Linear model (2012, methodology): “A second class of procedures is based on the functional principal components analysis (PCA) of X [9, 21].”