  Analysis of high-dimensional data, where the number of covariates is larger
than the sample size, is a topic of current interest. In such settings, an
important goal is to estimate the signal level $\tau^2$ and noise level
$\sigma^2$, i.e., to quantify how much variation in the response variable can
be explained by the covariates, versus how much of the variation is left
unexplained. This thesis considers the estimation of these quantities in a
semi-supervised setting, where for many observations only the vector of
covariates $X$ is given with no responses $Y$. Our main research question is:
how can one use the unlabeled data to better estimate $\tau^2$ and $\sigma^2$?
We consider two frameworks: a linear regression model and a linear projection
model in which linearity is not assumed. In the first framework, while linear
regression is used, no sparsity assumptions on the coefficients are made. In
the second framework, the linearity assumption is also relaxed and we aim to
estimate the signal and noise levels defined by the linear projection. We first
propose a naive estimator which is unbiased and consistent, under some
assumptions, in both frameworks. We then show how the naive estimator can be
improved by using zero-estimators, where a zero-estimator is a statistic
arising from the unlabeled data, whose expected value is zero. In the first
framework, we calculate the optimal zero-estimator improvement and discuss ways
to approximate the optimal improvement. In the second framework, such
optimality does no longer hold and we suggest two zero-estimators that improve
the naive estimator although not necessarily optimally. Furthermore, we show
that our approach reduces the variance for general initial estimators and we
present an algorithm that potentially improves any initial estimator. Lastly,
we consider four datasets and study the performance of our suggested methods.
