Statistica Sinica 21 (2011), 1431-1451
doi:10.5705/ss.2009.302

STATISTICAL INFERENCES FOR LINEAR MODELS

WITH FUNCTIONAL RESPONSES

Jin-Ting Zhang

National University of Singapore

09

Abstract: With modern technology development, functional responses are observed frequently in fields such as biology, meteorology, and ergonomics, among others. Consider statistical inferences for functional linear models in which the response functions depend on a few time-independent covariates, but the covariate effects are functions of time. Of interest is a test of a general linear hypothesis about the covariate effects. Existing test procedures include the $L^2$-norm based test proposed by Zhang and Chen (2007) and the $F$-type test proposed by Shen and Faraway (2004), among others. However, the asymptotic powers of these testing procedures have not been studied, and the null distributions of the test statistics are approximated using a naive method. In this paper, we investigate the $F$-type test for the general linear hypothesis and derive its asymptotic power. We show that the $F$-type test is root-$n$ consistent. In addition, we propose a bias-reduced method to approximate the null distribution of the $F$-type test. A simulation study demonstrates that the bias-reduced method and the naive method perform similarly when the data are highly or moderately correlated, but the former outperforms the latter significantly when the data are nearly uncorrelated. The $F$-type test with the bias-reduced method is illustrated via applications to a functional data set collected in ergonomics.

Key words and phrases: Functional data, functional hypothesis test, F-type test, Gaussian process, root-n consistency, χ²-type mixtures, χ²approximation.