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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 1431-1451

\vspace{3pc} \normalsize \centerline{\large\bf STATISTICAL
INFERENCES FOR LINEAR MODELS } \vspace{2pt} \centerline{\large\bf
WITH FUNCTIONAL RESPONSES} \vspace{.25cm}
\author{Jin-Ting Zhang}
\vspace{.4cm} \centerline{\it National University of Singapore}
\vspace{.55cm} \sz09
\begin{quotation}
\noindent {\it 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.\par
 \vspace{9pt} \noindent {\it Key words and phrases:}
Functional data, functional hypothesis test, F-type test, Gaussian
process, root-$n$ consistency, $\chi^2$-type mixtures,
$\chi^2$-approximation. \par
\end{quotation}

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