New methodologies for the real Watson distribution

Abstract

Spherical data are output in various research lines. These data may be categorized as directional (when such line is directed) and axial (otherwise). Directional data can be understood as points on a sphere; while, axial data are pairs of antipodal points (i.e., opposite points) on a sphere. The Watson (W) model is often used for describing axial data. The W distribution has two parameters: the mean axis and the concentration parameter. It is known making inference under lower concentration is a hard task. First, to outperform this gap, for the W parameters, we provide an improved maximum likelihoodbased estimation procedure for the W concentration parameter. In particular, we present a closed-form expression for the second-order bias according to the Cox-Snell methodology. Further, an approximated expression for the Fisher information matrix is derived as well. To quantify the performance of the our proposal, a Monte Carlo study is made. Results indicate that our estimation procedure is suitable to obtain more accurate estimates for the W concentration parameter. Second, aims to study a natural extension of the minimum distance estimators discussed by Cao et al. (1994).More precisely, under the assumption of Watson directional distribution, to produce hypotheses and point statistical inference procedures, as well as goodness, and to propose mathematical antecedents for the statistical method based on the minimum distance of L² for the Watson model. Third, based on Renyi divergence, we propose two hypothesis tests to verify whether two samples come from populations with the same concentration parameter. The results of synthetic and real data indicate that the proposed tests can produce good performance on Watson data. The small sample behavior of the proposed estimators is using Monte Carlo simulations. An application is illustrated with real data sets.