  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.
