**Yuval Nardi and Alessandro Rinaldo**

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 unique solution of a convex penalized likelihood program
using the group lasso penalty. We illustrate how it is possible to
construct, in a straightforward way, an estimator of the underlying
log-linear model based on the blocks of non-negative coefficients
recovered by the group lasso procedure.

We investigate the asymptotic properties of the group lasso estimator
and of the associated model selection criterion 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 norm consistent and that the group lasso model
selection is a consistent procedure, in the sense that, with
overwhelming probability as the sample size increases, it will
correctly identify all the sets of non-zero 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.

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