# On functional logistic regression: some conceptual issues

## TL;DR

This paper critically examines the conceptual foundations of functional logistic regression, proposing an RKHS-based model, analyzing its validity, and addressing issues related to the existence of maximum likelihood estimators in the functional setting.

## Contribution

It introduces an RKHS-based approach to functional logistic regression, compares it with traditional $L_2$ models, and explores the conditions for the existence of ML estimators.

## Key findings

- RKHS-based model validity conditions derived
- ML estimators often do not exist in the functional case
- Proposes a restricted RKHS-based ML estimator

## Abstract

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.