  We consider the statistical analysis of data on high-dimensional spheres and
shape spaces. The work is of particular relevance to applications where
high-dimensional data are available--a commonly encountered situation in many
disciplines. First the uniform measure on the infinite-dimensional sphere is
reviewed, together with connections with Wiener measure. We then discuss
densities of Gaussian measures with respect to Wiener measure. Some nonuniform
distributions on infinite-dimensional spheres and shape spaces are introduced,
and special cases which have important practical consequences are considered.
We focus on the high-dimensional real and complex Bingham, uniform, von
Mises-Fisher, Fisher-Bingham and the real and complex Watson distributions.
Asymptotic distributions in the cases where dimension and sample size are large
are discussed. Approximations for practical maximum likelihood based inference
are considered, and in particular we discuss an application to brain shape
modeling.
