  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.
