Polarization measurement analysis III. Analysis of the polarization angle dispersion function with high precision polarization data

High precision polarization measurements, such as those from the Planck satellite, open new opportunities for the study of the magnetic field structure as traced by polarimetric measurements of the interstellar dust emission. The polarization parameters suffer from bias in the presence of measurement noise. It is critical to take into account all the information available in the data in order to accurately derive these parameters. In our previous work, we studied the bias on polarization fraction and angle, various estimators of these quantities, and their associated uncertainties. The goal of this paper is to characterize the bias on the polarization angle dispersion function that is used to study the spatial coherence of the polarization angle. We characterize for the first time the bias on the conventional estimator of the polarization angle dispersion function and show that it can be positive or negative depending on the true value. Monte Carlo simulations were performed to explore the impact of the noise properties of the polarization data, as well as the impact of the distribution of the true polarization angles on the bias. We show that in the case where the ellipticity of the noise in (Q, U) varies by less than 10%, one can use simplified, diagonal approximation of the noise covariance matrix. In other cases, the shape of the noise covariance matrix should be taken into account in the estimation of the polarization angle dispersion function. We also study new estimators such as the dichotomic and the polynomial estimators. Though the dichotomic estimator cannot be directly used to estimate the polarization angle dispersion function, we show that, on the one hand, it can serve as an indicator of the accuracy of the conventional estimator and, on the other hand, it can be used for deriving the polynomial estimator. We propose a method for determining the upper limit of the bias on the conventional estimator of the polarization angle dispersion function. The method is applicable to any linear polarization data set for which the noise covariance matrices are known.