Putting the Precision in Precision Cosmology: How accurate should your
  data covariance matrix be?
  Cosmological parameter estimation requires that the likelihood function of
the data is accurately known. Assuming that cosmological large-scale structure
power spectra data are multivariate Gaussian-distributed, we show the accuracy
of parameter estimation is limited by the accuracy of the inverse data
covariance matrix - the precision matrix. If the data covariance and precision
matrices are estimated by sampling independent realisations of the data, their
statistical properties are described by the Wishart and Inverse-Wishart
distributions, respectively. Independent of any details of the survey, we show
that the fractional error on a parameter variance, or a Figure-of-Merit, is
equal to the fractional variance of the precision matrix. In addition, for the
only unbiased estimator of the precision matrix, we find that the fractional
accuracy of the parameter error depends only on the difference between the
number of independent realisations and the number of data points, and so can
easily diverge. For a 5% error on a parameter error and N_D << 100 data-points,
a minimum of 200 realisations of the survey are needed, with 10% accuracy for
the data covariance. If the number of data-points N_D >>100 we need N_S > N_D
realisations and a fractional accuracy of <sqrt[2/N_D] in the data covariance.
As the number of power spectra data points grows to N_D>10^4 -10^6 this
approach will be problematic. We discuss possible ways to relax these
conditions: improved theoretical modelling; shrinkage methods;
data-compression; simulation and data resampling methods.
