Polarization measurement analysis II. Best estimators of polarization fraction and angle

With the forthcoming release of high precision polarization measurements, such as from the Planck satellite, it becomes critical to evaluate the performance of estimators for the polarization fraction and angle. These two physical quantities suffer from a well-known bias in the presence of measurement noise, as described in Part I of this series. In this paper, Part II of the series, we explore the extent to which various estimators may correct the bias. Traditional frequentist estimators of the polarization fraction are compared with two recent estimators: one inspired by a Bayesian analysis and a second following an asymptotic method. We investigate the sensitivity of these estimators to the asymmetry of the covariance matrix, which may vary over large datasets. We present for the first time a comparison among polarization angle estimators, and evaluate the statistical bias on the angle that appears when the covariance matrix exhibits effective ellipticity. We also address the question of the accuracy of the polarization fraction and angle uncertainty estimators. The methods linked to the credible intervals and to the variance estimates are tested against the robust confidence interval method. From this pool of polarization fraction and angle estimators, we build recipes adapted to different uses: the best estimators to build a mask, to compute large maps of the polarization fraction and angle, and to deal with low signal-to-noise data. More generally, we show that the traditional estimators suffer from discontinuous distributions at a low signal-to-noise ratio, while the asymptotic and Bayesian methods do not. Attention is given to the shape of the output distribution of the estimators, which is compared with a Gaussian distribution. In this regard, the new asymptotic method presents the best performance, while the Bayesian output distribution is shown to be strongly asymmetric with a sharp cut at a low signal-to-noise ratio. Finally, we present an optimization of the estimator derived from the Bayesian analysis using adapted priors.