An elliptically symmetric angular Gaussian distribution

We define a distribution on the unit sphere Sd−1 called the elliptically symmetric angular Gaussian distribution. This distribution, which to our knowledge has not been studied before, is a subfamily of the angular Gaussian distribution closely analogous to the Kent subfamily of the general Fisher–Bingham distribution. Like the Kent distribution, it has elliptical contours, enabling modelling of rotational asymmetry about the mean direction, but it has the additional advantages of being simple and fast to simulate from, and having a density and hence likelihood that is easy and very quick to compute exactly. These advantages are especially beneficial for computationally intensive statistical methods, one example of which is a parametric bootstrap procedure for inference for the directional mean that we describe

Bayesian linear size-and-shape regression with applications to face data

Regression models for size-and-shape analysis are developed, where the model is specified in the Euclidean space of the landmark coordinates. Statistical models in this space (which is known as the top space or ambient space) are often easier for practitioners to understand than alternative models in the quotient space of size-and-shapes. We consider a Bayesian linear size-and-shape regression model in which the response variable is given by labelled configuration matrix, and the covariates represent quantities such as gender and age. It is important to parameterize the model so that it is identifiable, and we use the LQ decomposition in the intercept term in the model for this purpose. Gamma priors for the inverse variance of the error term, matrix Fisher priors for the random rotation matrix, and flat priors for the regression coefficients are used. Markov chain Monte Carlo algorithms are used for sampling from the posterior distribution, in particular by using combinations of Metropolis-Hastings updates and a Gibbs sampler.The proposed Bayesian methodology is illustrated with an application to forensic facial data in three dimensions, where we investigate the main changes in growth by describing relative movements of landmarks for each gender over time