  In order to identify important variables that are involved in making optimal
treatment decision, Lu et al. (2013) proposed a penalized least squared
regression framework for a fixed number of predictors, which is robust against
the misspecification of the conditional mean model. Two problems arise: (i) in
a world of explosively big data, effective methods are needed to handle
ultra-high dimensional data set, for example, with the dimension of predictors
is of the non-polynomial (NP) order of the sample size; (ii) both the
propensity score and conditional mean models need to be estimated from data
under NP dimensionality.
  In this paper, we propose a two-step estimation procedure for deriving the
optimal treatment regime under NP dimensionality. In both steps, penalized
regressions are employed with the non-concave penalty function, where the
conditional mean model of the response given predictors may be misspecified.
The asymptotic properties, such as weak oracle properties, selection
consistency and oracle distributions, of the proposed estimators are
investigated. In addition, we study the limiting distribution of the estimated
value function for the obtained optimal treatment regime. The empirical
performance of the proposed estimation method is evaluated by simulations and
an application to a depression dataset from the STAR*D study.
