  We propose a unified framework for likelihood-based regression modeling when
the response variable has finite support. Our work is motivated by the fact
that, in practice, observed data are discrete and bounded. The proposed methods
assume a model which includes models previously considered for
interval-censored variables with log-concave distributions as special cases.
The resulting log-likelihood is concave, which we use to establish asymptotic
normality of its maximizer as the number of observations $n$ tends to infinity
with the number of parameters $d$ fixed, and rates of convergence of
$L_1$-regularized estimators when the true parameter vector is sparse and $d$
and $n$ both tend to infinity with $\log(d) / n \to 0$. We consider an inexact
proximal Newton algorithm for computing estimates and give theoretical
guarantees for its convergence. The range of possible applications is wide,
including but not limited to survival analysis in discrete time, the modeling
of outcomes on scored surveys and questionnaires, and, more generally,
interval-censored regression. The applicability and usefulness of the proposed
methods are illustrated in simulations and data examples.
