  This paper studies a regression model where both predictor and response
variables are random functions. We consider a functional linear model where the
conditional mean of the response variable at each time point is given by a
linear functional of the predictor variable. In this paper, we are interested
in estimation of the integral kernel $b(s,t)$ of the conditional expectation
operator, where $s$ is an output variable while $t$ is a variable that
interacts with the predictor variable. This problem is an ill-posed inverse
problem, and we consider two estimators based on the functional principal
component analysis (PCA). We show that under suitable regularity conditions, an
estimator based on the single truncation attains the convergence rate for the
integrated squared error that is characterized by smoothness of the function $b
(s,t)$ in $t$ together with the decay rate of the eigenvalues of the covariance
operator, but the rate does not depend on smoothness of $b(s,t)$ in $s$. This
rate is shown to be minimax optimal, and consequently smoothness of $b(s,t)$ in
$s$ does not affect difficulty of estimating $b$. We also consider an
alternative estimator based on the double truncation, and provide conditions
under which the alternative estimator attains the optimal rate. We conduct
simulations to verify the performance of PCA-based estimators in the finite
sample. Finally, we apply our estimators to investigate the relation between
the lifetime pattern of working hours and total income, and the relation
between the electricity spot price and the wind power infeed.
