Consistency of logistic classifier in abstract Hilbert spaces

Abstract

We study the asymptotic behavior of the logistic classifier in an abstract Hilbert space and require realistic conditions on the distribution of data for its consistency. The number $k_{n}$ of estimated parameters via maximum quasi-likelihood is allowed to diverge so that $k_{n}/n\to 0$ and $n\tau_{k_{n}}^{4}\to\infty$, where $n$ is the number of observations and $\tau_{k_{n}}$ is the variance of the last principal component of data used for estimation. This is the only result on the consistency of the logistic classifier we know so far when the data are assumed to come from a Hilbert space.