  Dynamic treatment regimes are of growing interest across the clinical
sciences as these regimes provide one way to operationalize and thus inform
sequential personalized clinical decision making. A dynamic treatment regime is
a sequence of decision rules, with a decision rule per stage of clinical
intervention; each decision rule maps up-to-date patient information to a
recommended treatment. We briefly review a variety of approaches for using data
to construct the decision rules. We then review an interesting challenge, that
of nonregularity that often arises in this area. By nonregularity, we mean the
parameters indexing the optimal dynamic treatment regime are nonsmooth
functionals of the underlying generative distribution.
  A consequence is that no regular or asymptotically unbiased estimator of
these parameters exists. Nonregularity arises in inference for parameters in
the optimal dynamic treatment regime; we illustrate the effect of nonregularity
on asymptotic bias and via sensitivity of asymptotic, limiting, distributions
to local perturbations. We propose and evaluate a locally consistent Adaptive
Confidence Interval (ACI) for the parameters of the optimal dynamic treatment
regime. We use data from the Adaptive Interventions for Children with ADHD
study as an illustrative example. We conclude by highlighting and discussing
emerging theoretical problems in this area.
