  We propose a generalized functional linear regression model for a regression
situation where the response variable is a scalar and the predictor is a random
function. A linear predictor is obtained by forming the scalar product of the
predictor function with a smooth parameter function, and the expected value of
the response is related to this linear predictor via a link function. If, in
addition, a variance function is specified, this leads to a functional
estimating equation which corresponds to maximizing a functional
quasi-likelihood. This general approach includes the special cases of the
functional linear model, as well as functional Poisson regression and
functional binomial regression. The latter leads to procedures for
classification and discrimination of stochastic processes and functional data.
We also consider the situation where the link and variance functions are
unknown and are estimated nonparametrically from the data, using a
semiparametric quasi-likelihood procedure. An essential step in our proposal is
dimension reduction by approximating the predictor processes with a truncated
Karhunen-Loeve expansion. We develop asymptotic inference for the proposed
class of generalized regression models. In the proposed asymptotic approach,
the truncation parameter increases with sample size, and a martingale central
limit theorem is applied to establish the resulting increasing dimension
asymptotics. We establish asymptotic normality for a properly scaled distance
between estimated and true functions that corresponds to a suitable L^2 metric
and is defined through a generalized covariance operator.
