Concave likelihood-based regression with finite-support response variables

TL;DR

This paper introduces a unified likelihood-based regression framework for finite-support response variables, providing theoretical guarantees and an efficient algorithm, applicable to various fields like survival analysis and survey modeling.

Contribution

It develops a concave log-likelihood model for finite-support responses, with asymptotic properties and a proximal Newton algorithm, extending regression methods to discrete, bounded data.

Findings

The model is asymptotically normal as sample size grows.

The proposed algorithm converges with theoretical guarantees.

Simulations and data examples demonstrate practical effectiveness.

Abstract

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.