A statistical formalism for alignment analysis
  The detection of anisotropies with respect to a given direction in a vector
field is a common problem in astronomy. Several methods have been proposed that
rely on the distribution of the acute angles between the data and a reference
direction. Different approaches use Monte Carlo methods to quantify the
statistical significance of a signal, although often lacking an analytical
framework. Here we present two methods to detect and quantify alignment signals
and test their statistical robustness. The first method considers the deviance
of the relative fraction of vector components in the plane perpendicular to a
reference direction with respect to an isotropic distribution. We also derive
the statistical properties and stability of the resulting estimator, and
therefore does not rely on Monte Carlo simulations to assess its statistical
significance. The second method is based on a fit over the residuals of the
empirical cumulative distribution function with respect to that expected for a
uniform distribution, using a small set of harmonic orthogonal functions, which
does not rely on any binning scheme. We compare these methods with others
commonly used in the literature, using Monte Carlo simulations, finding that
the proposed statistics allow the detection of alignment signals with greater
significance.
