  Personalized optimal decision making, finding the optimal decision rule (ODR)
based on individual characteristics, has attracted increasing attention
recently in many fields, such as education, economics, and medicine. Current
ODR methods usually require the primary outcome of interest in samples for
assessing treatment effects, namely the experimental sample. However, in many
studies, treatments may have a long-term effect, and as such the primary
outcome of interest cannot be observed in the experimental sample due to the
limited duration of experiments, which makes the estimation of ODR impossible.
This paper is inspired to address this challenge by making use of an auxiliary
sample to facilitate the estimation of ODR in the experimental sample. We
propose an auGmented inverse propensity weighted Experimental and Auxiliary
sample-based decision Rule (GEAR) by maximizing the augmented inverse
propensity weighted value estimator over a class of decision rules using the
experimental sample, with the primary outcome being imputed based on the
auxiliary sample. The asymptotic properties of the proposed GEAR estimators and
their associated value estimators are established. Simulation studies are
conducted to demonstrate its empirical validity with a real AIDS application.
