Polarization measurements analysis I. Impact of the full covariance
  matrix on polarization fraction and angle measurements
  With the forthcoming release of high precision polarization measurements,
such as from the Planck satellite, the metrology of polarization needs to
improve. In particular, it is crucial to take into account full knowledge of
the noise properties when estimating polarization fraction and angle, which
suffer from well-known biases. While strong simplifying assumptions have
usually been made in polarization analysis, we present a method for including
the full covariance matrix of the Stokes parameters in estimates for the
distributions of the polarization fraction and angle. We thereby quantify the
impact of the noise properties on the biases in the observational quantities.
We derive analytical expressions for the pdf of these quantities, taking into
account the full complexity of the covariance matrix, including the Stokes I
intensity components. We perform simulations to explore the impact of the noise
properties on the statistical variance and bias of the polarization fraction
and angle. We show that for low variations of the effective ellipticity between
the Q and U components around the symmetrical case the covariance matrix may be
simplified as is usually done, with negligible impact on the bias. For S/N on
intensity lower than 10 the uncertainty on the total intensity is shown to
drastically increase the uncertainty of the polarization fraction but not the
relative bias, while a 10\% correlation between the intensity and the polarized
components does not significantly affect the bias of the polarization fraction.
We compare estimates of the uncertainties affecting polarization measurements,
addressing limitations of estimates of the S/N, and we show how to build
conservative confidence intervals for polarization fraction and angle
simultaneously. This study is the first of a set of papers dedicated to the
analysis of polarization measurements.
