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\noindent Statistica Sinica {\bf 13}(2003), 571-591

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\centerline{\large\bf SPLINE ESTIMATORS FOR THE FUNCTIONAL}
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\centerline{\large\bf LINEAR MODEL}
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\author{Herv\'e Cardot$^\dagger$, Fr\'ed\'eric
Ferraty$^\ddagger$ and Pascal Sarda$^\ddagger$}
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\centerline{\it
$^\dagger$INRA, Toulouse and $^\ddagger$Universit\'e Paul Sabatier, Toulouse}
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\begin{quotation}
\noindent {\it Abstract:}
We consider a regression setting where the response is a scalar and
the predictor is a random function defined on a compact set of
${\Bbb R}$.
Many fields of applications are concerned with this kind of data,
for instance chemometrics when the predictor is a signal digitized in
many points. Then, people have mainly considered the multivariate
linear model and have adapted the least squares procedure to take
care of highly correlated predictors.
  Another point of view is to introduce a continuous version of this
model, i.e., the functional linear model with scalar response.
We are then faced with the estimation of a functional coefficient or,
equivalently, of a linear functional.
We first study an estimator based on a B-splines expansion of the
functional coefficient which in some way generalizes ridge regression.
We derive an upper bound for the $L^2$ rate of convergence of this estimator.
As an alternative we also introduce a smooth version of functional
principal components regression for which $L^2$ convergence is
achieved. Finally both methods are compared by means of a simulation
study.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Convergence, functional linear model, Hilbert space valued random variables,
principal components regression, regularization, splines.\par
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