The log-linear group-lasso estimator and its asymptotic properties

TL;DR

This paper introduces a group-lasso estimator for hierarchical log-linear models, analyzing its asymptotic properties and demonstrating its effectiveness in model selection and variable interaction recovery in high-dimensional settings.

Contribution

It develops a convex penalized likelihood estimator for natural parameters in hierarchical log-linear models and establishes its asymptotic model selection consistency.

Findings

Estimator correctly identifies nonzero interactions with high probability

Recovery is possible even when model complexity exceeds sample size

Central limit results for the estimator are derived

Abstract

We define the group-lasso estimator for the natural parameters of the exponential families of distributions representing hierarchical log-linear models under multinomial sampling scheme. Such estimator arises as the solution of a convex penalized likelihood optimization problem based on the group-lasso penalty. We illustrate how it is possible to construct an estimator of the underlying log-linear model using the blocks of nonzero coefficients recovered by the group-lasso procedure. We investigate the asymptotic properties of the group-lasso estimator as a model selection method in a double-asymptotic framework, in which both the sample size and the model complexity grow simultaneously. We provide conditions guaranteeing that the group-lasso estimator is model selection consistent, in the sense that, with overwhelming probability as the sample size increases, it correctly identifies all the sets of nonzero interactions among the variables. Provided the sequences of true underlying models is sparse enough, recovery is possible even if the number of cells grows larger than the sample size. Finally, we derive some central limit type of results for the log-linear group-lasso estimator.