  In directional statistics, the von Mises-Fisher (vMF) distribution is one of
the most basic and popular probability distributions for data on the unit
hypersphere. Recently, the spherical normal (SN) distribution was proposed as
an intrinsic counterpart to the vMF distribution by replacing the standard
Euclidean norm with the great-circle distance, which is the shortest path
joining two points on the unit sphere. We propose numerical approaches for
parameter estimation since there are no analytic formula available. We consider
the estimation problems in a general setting where non-negative weights are
assigned to observations. This leads to a more interesting contribution for
model-based clustering on the unit hypersphere by finite mixture model with SN
distributions. We validate efficiency of optimization-based estimation
procedures and effectiveness of SN mixture model using simulated and real data
examples.
