  We study a high-dimensional regression setting under the assumption of known
covariate distribution. We aim at estimating the amount of explained variation
in the response by the best linear function of the covariates (the signal
level). In our setting, neither sparsity of the coefficient vector, nor
normality of the covariates or linearity of the conditional expectation are
assumed. We present an unbiased and consistent estimator and then improve it by
using a zero-estimator approach, where a zero-estimator is a statistic whose
expected value is zero.
  More generally, we present an algorithm based on the zero estimator approach
that in principle can improve any given estimator. We study some asymptotic
properties of the proposed estimators and demonstrate their finite sample
performance in a simulation study.
