Testing for lack of dependence in the functional linear model

Abstract

The authors consider the linear model Y, = *X, + E, relating a functional response with explanatory variables. They propose a simple test of the nullity of \k based on the principal component decomposition. The limiting distribution of their test statistic is chi-squared, but this distribution is also an excellent approximation in finite samples. The authors illustrate their method using data from terrestrial magnetic observatories.Un test d'absence de dependance dans un modele fonctionnel lineaire R 6 s u d : Les auteurs s'intkressent au modkle lintaire Y, = Q X , + E, liant une variable dponse fonctionnelle h des variables explicatives. 11s proposent un test simple de nullit6.de * fond6 sur la dkomposition en composantes principales. L a loi limite de leur statistique est une khi-deux, mais cette loi foumit aussi une excellente approximation h taille finie. Les auteurs illustrent leur m6thode au moyen de donnks provenant d'observatoires du champ magnttique terrestre.

The effect of the spatial domain in FANOVA models with ARH(1) error term

“…In the case (P 2 ,a,C 2 ), a generation of a functional value (surface) of the response is represented in Figure 1. The statistics (40) and (42) are evaluated in all the cases displayed in Table 1 (see Tables 2 and 3 for the statistics L ∞ β and L ∞ Y , respectively).…”

“…. , n, (38) is defined in equations (15) The empirical functional mean quadratic errors (see equations (40) and (42)) are displayed in Table 7, for the estimation of the functional parameter vector β, and in Table 8 for the estimation of the response Y. It can be observed, as in the rectangular domain, that the order of magnitude of the empirical functional mean quadratic errors associated with the estimation of β is of order 10 −3 , and for the estimation of the response is 10 −2 .…”

“…We particularly refer to the functional linear regression context (see, for example, [10]; [11]; [12]; [18]; [13]; [16]; [26]; [40], among others).…”

A test of linearity in partial functional linear regression

“…Cardot et al (2003) considered the problem of testing a simple hypothesis in the case where the response is scalar and the predictor is a random function, while Mas (2007) investigated a test for the mean of random curves. Recently, Xu et al (2011) discussed an F-test in a linear longitudinal data model, and Kokoszka et al (2008) tested for lack of dependence in a functional linear model where both the response and the predictor are curves. Delsol et al (2011) proposed a theoretical framework for structural testing procedures adapted to functional regression.…”

Testing for equality of means of a Hilbert space valued random variable

“…The book by Ramsay and Silverman [14] gives a comprehensive introduction to this field of statistics, whereas nonparametric approaches are tackled in [9]. Several recent papers deal with statistical testing procedures for functional variables, mainly in the context of functional linear regression models for which tests for no effect are considered (see, e.g., [3,4,11]). However, a few papers propose tests on the means of functional variables.…”