Polarization measurement analysis III. Analysis of the polarization
  angle dispersion function with high precision polarization data
  High precision polarization measurements open new opportunities for the study
of the magnetic field structure as traced by polarimetric measurements of the
interstellar dust emission. 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. 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 (S hereafter) and show that it can be positive or negative depending
on the true value. Monte Carlo simulations are performed in order 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 percent, 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 S. We also study new
estimators such as the dichotomic and the polynomial estimators. Though the
dichotomic estimator cannot be directly used to estimate S, 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 S. The method is applicable to any linear
polarization data set for which the noise covariance matrices are known.
