The entropy based goodness of fit tests for generalized von Mises-Fisher distributions and beyond

TL;DR

This paper introduces new unimodal rotational invariant directional distributions generalizing von Mises-Fisher, develops goodness of fit tests based on entropy estimators, and applies them to fiber direction data.

Contribution

It proposes novel distribution classes, formulates entropy-based goodness of fit tests, and demonstrates their effectiveness on simulated and real fiber orientation data.

Findings

New distribution classes for directional data

Effective entropy-based goodness of fit tests

Successful application to fiber direction analysis

Abstract

We introduce some new classes of unimodal rotational invariant directional distributions, which generalize von Mises-Fisher distribution. We propose three types of distributions, one of which represents axial data. For each new type we provide formulae and short computational study of parameter estimators by the method of moments and the method of maximum likelihood. The main goal of the paper is to develop the goodness of fit test to detect that sample entries follow one of the introduced generalized von Mises--Fisher distribution based on the maximum entropy principle. We use $k$th nearest neighbour distances estimator of Shannon entropy and prove its $L^2$-consistency. We examine the behaviour of the test statistics, find critical values and compute power of the test on simulated samples. We apply the goodness of fit test to local fiber directions in a glass fibre reinforced composite material and detect the samples which follow axial generalized von Mises--Fisher distribution.