  The main ideas behind the classical multivariate logistic regression model
make sense when translated to the functional setting, where the explanatory
variable $X$ is a function and the response $Y$ is binary. However, some
important technical issues appear (or are aggravated with respect to those of
the multivariate case) due to the functional nature of the explanatory
variable. First, the mere definition of the model can be questioned: while most
approaches so far proposed rely on the $L_2$-based model, we suggest an
alternative (in some sense, more general) approach, based on the theory of
Reproducing Kernel Hilbert Spaces (RKHS). The validity conditions of such
RKHS-based model, as well as its relation with the $L_2$-based one are
investigated and made explicit in two formal results. Some relevant particular
cases are considered as well. Second we show that, under very general
conditions, the maximum likelihood (ML) of the logistic model parameters fail
to exist in the functional case. Third, on a more positive side, we suggest an
RKHS-based restricted version of the ML estimator. This is a methodological
paper, aimed at a better understanding of the functional logistic model, rather
than focusing on numerical and practical issues.
