  Selective inference methods are developed for group lasso estimators for use
with a wide class of distributions and loss functions. The method includes the
use of exponential family distributions, as well as quasi-likelihood modeling
for overdispersed count data, for example, and allows for categorical or
grouped covariates as well as continuous covariates. A randomized
group-regularized optimization problem is studied. The added randomization
allows us to construct a post-selection likelihood which we show to be adequate
for selective inference when conditioning on the event of the selection of the
grouped covariates. This likelihood also provides a selective point estimator,
accounting for the selection by the group lasso. Confidence regions for the
regression parameters in the selected model take the form of Wald-type regions
and are shown to have bounded volume. The selective inference method for
grouped lasso is illustrated on data from the national health and nutrition
examination survey while simulations showcase its behaviour and favorable
comparison with other methods.
