Testing Hypotheses in the Functional Linear Model

Authors: Cardot, Hervé¹; Ferraty, Frédéric²; Mas, André³; Sarda, Pascal⁴

Source: Scandinavian Journal of Statistics, Volume 30, Number 1, March 2003 , pp. 241-255(15)

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Abstract:

The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of <openface>R</openface> and the response is scalar. The response is modelled as Y=Ψ(X)+ɛ, where Ψ is some linear continuous operator defined on the space of square integrable functions and valued in <openface>R</openface>. The random input X is independent from the noise ɛ. In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of Ψ restricted to the Hilbert space generated by the random variable X. We introduce two test statistics based on the norm of the empirical cross-covariance operator of (X,Y). The first test statistic relies on a χ² approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X. The test procedures can be applied to check a given relationship between X and Y. The method is illustrated through a simulation study.

Keywords: asymptotic normality; functional linear model; Hilbert space valued random variables; splines; tests

Document Type: Research article

DOI: http://dx.doi.org/10.1111/1467-9469.00329

Affiliations: 1: Unité Biométrie et Intelligence Artificielle, INRA Toulouse. 2: GRIMM, Université Toulouse Le Mirail. 3: CREST-INSEE et Laboratoire de Statistique et Probabilités, Université Paul Sabatier. 4: Laboratoire de Statistique et Probabilités, Université Paul Sabatier.

Publication date: 2003-03-01