  We provide a high-dimensional semi-supervised inference framework focused on
the mean and variance of the response. Our data are comprised of an extensive
set of observations regarding the covariate vectors and a much smaller set of
labeled observations where we observe both the response as well as the
covariates. We allow the size of the covariates to be much larger than the
sample size and impose weak conditions on a statistical form of the data. We
provide new estimators of the mean and variance of the response that extend
some of the recent results presented in low-dimensional models. In particular,
at times we will not necessitate consistent estimation of the functional form
of the data. Together with estimation of the population mean and variance, we
provide their asymptotic distribution and confidence intervals where we
showcase gains in efficiency compared to the sample mean and variance. Our
procedure, with minor modifications, is then presented to make important
contributions regarding inference about average treatment effects. We also
investigate the robustness of estimation and coverage and showcase widespread
applicability and generality of the proposed method.
