  We provide a collection of results on covariance expressions between Monte
Carlo based multi-output mean, variance, and Sobol main effect variance
estimators from an ensemble of models. These covariances can be used within
multi-fidelity uncertainty quantification strategies that seek to reduce the
estimator variance of high-fidelity Monte Carlo estimators with an ensemble of
low-fidelity models. Such covariance expressions are required within approaches
like the approximate control variate and multi-level best linear unbiased
estimator. While the literature provides these expressions for some
single-output cases such as mean and variance, our results are relevant to both
multiple function outputs and multiple statistics across any sampling strategy.
Following the description of these results, we use them within an approximate
control variate scheme to show that leveraging multiple outputs can
dramatically reduce estimator variance compared to single-output approaches.
Synthetic examples are used to highlight the effects of optimal sample
allocation and pilot sample estimation. A flight-trajectory simulation of
entry, descent, and landing is used to demonstrate multi-output estimation in
practical applications.
