Estimation and simulation in directional and statistical shape models

This thesis is concerned with problems in two related areas of statistical shape analysis in two dimensional landmarks data and directional statistics in various sample spaces. Directional observations can be regarded as points on the circumference of a circle of unit radius in two dimensions or on the surface of a sphere in three dimensions. Special directional methods and models are required which take into account the structure of these sample spaces. Shape analysis involves methods for the study of the shape of objects where location, scale and orientation are removed. Specifically, we consider the situation where the objects are summarized by points on the object called landmarks. The non-Euclidean nature of the shape space causes several problems when defining a distribution on it. Any distribution which could be considered needs to be tractable and a realistic model for landmark data. One aim of this thesis is to investigate the saddlepoint approximations for the normalizing constants of some directional and shape distributions. In particular, we consider the normalizing constant of the CBQ distribution which can be expressed as a one dimensional integral of normalizing constants for Bingham distributions. Two new methods are explored to evaluate this normalizing constant based on saddlepoint approximations namely the Integrated Saddlepoint (ISP) approximation and the Saddlepoint-Integration (SPI) approximation. Another objective of this thesis is to develop new simulation methods for some directional and shape models. We propose an efficient acceptance-rejection simulation algorithm for the Bingham distribution on unit sphere using an angular central Gaussian (ACG) density as an envelope. This envelope is justified using inequalities based on concave functions. An immediate consequence is a method to simulate 3 x 3 matrix Fisher rotation matrices. In addition, a new accept-reject algorithm is developed to generate samples from the complex Bingham quartic (CBQ) distribution. The last objective of this thesis is to develop a new moment method to estimate the parameters of the wrapped normal torus distribution based on the sample sine and cosine moments.