  We consider the linear regression problem under semi-supervised settings
wherein the available data typically consists of: (i) a small or moderate sized
'labeled' data, and (ii) a much larger sized 'unlabeled' data. Such data arises
naturally from settings where the outcome, unlike the covariates, is expensive
to obtain, a frequent scenario in modern studies involving large databases like
electronic medical records (EMR). Supervised estimators like the ordinary least
squares (OLS) estimator utilize only the labeled data. It is often of interest
to investigate if and when the unlabeled data can be exploited to improve
estimation of the regression parameter in the adopted linear model.
  In this paper, we propose a class of 'Efficient and Adaptive Semi-Supervised
Estimators' (EASE) to improve estimation efficiency. The EASE are two-step
estimators adaptive to model mis-specification, leading to improved (optimal in
some cases) efficiency under model mis-specification, and equal (optimal)
efficiency under a linear model. This adaptive property, often unaddressed in
the existing literature, is crucial for advocating 'safe' use of the unlabeled
data. The construction of EASE primarily involves a flexible
'semi-non-parametric' imputation, including a smoothing step that works well
even when the number of covariates is not small; and a follow up 'refitting'
step along with a cross-validation (CV) strategy both of which have useful
practical as well as theoretical implications towards addressing two important
issues: under-smoothing and over-fitting. We establish asymptotic results
including consistency, asymptotic normality and the adaptive properties of
EASE. We also provide influence function expansions and a 'double' CV strategy
for inference. The results are further validated through extensive simulations,
followed by application to an EMR study on auto-immunity.
