Estimating Cosmological Parameter Covariance
  We investigate the bias and error in estimates of the cosmological parameter
covariance matrix, due to sampling or modelling the data covariance matrix, for
likelihood width and peak scatter estimators. We show that these estimators do
not coincide unless the data covariance is exactly known. For sampled data
covariances, with Gaussian distributed data and parameters, the parameter
covariance matrix estimated from the width of the likelihood has a Wishart
distribution, from which we derive the mean and covariance. This mean is biased
and we propose an unbiased estimator of the parameter covariance matrix.
Comparing our analytic results to a numerical Wishart sampler of the data
covariance matrix we find excellent agreement. An accurate ansatz for the mean
parameter covariance for the peak scatter estimator is found, and we fit its
covariance to our numerical analysis. The mean is again biased and we propose
an unbiased estimator for the peak parameter covariance. For sampled data
covariances the width estimator is more accurate than the peak scatter
estimator. We investigate modelling the data covariance, or equivalently data
compression, and shown that the peak scatter estimator is less sensitive to
biases in the model data covariance matrix than the width estimator, but
requires independent realisations of the data to reduce the statistical error.
If the model bias on the peak estimator is sufficiently low this is promising,
otherwise the sampled width estimator is preferable.
