  Coronavirus case-count data has influenced government policies and drives
most epidemiological forecasts. Limited testing is cited as the key driver
behind minimal information on the COVID-19 pandemic. While expanded testing is
laudable, measurement error and selection bias are the two greatest problems
limiting our understanding of the COVID-19 pandemic; neither can be fully
addressed by increased testing capacity. In this paper, we demonstrate their
impact on estimation of point prevalence and the effective reproduction number.
We show that estimates based on the millions of molecular tests in the US has
the same mean square error as a small simple random sample. To address this, a
procedure is presented that combines case-count data and random samples over
time to estimate selection propensities based on key covariate information. We
then combine these selection propensities with epidemiological forecast models
to construct a \emph{doubly robust} estimation method that accounts for both
measurement-error and selection bias. This method is then applied to estimate
Indiana's active infection prevalence using case-count, hospitalization, and
death data with demographic information, a statewide random molecular sample
collected from April 25--29th, and Delphi's COVID-19 Trends and Impact Survey.
We end with a series of recommendations based on the proposed methodology.
