Statistical inference for some complex high-dimensional problems

The proposed research will comprise the following two components.i) Saddlepoint approximations for stochastic differential equations.The problem of performing statistical inference for unknown parametersin models formulated through a stochastic differential equationis one that has received a great deal of attention in recent years.Models incorporating stochastic differential equations are wellestablished in finance and econometrics and are becoming increasinglyimportant in a number of other fields, particularly mathematical biology.There has recently been a surge of interest in simulation-basedapproaches to statistical inference for stochastic differential equationmodels. However, it is our belief that the best of the analyticalapproaches are very serious competitors whose potential has not beenfully exploited to date. The proposed research will exploit theremarkable numerical accuracy and excellent theoretical properties ofsaddlepoint approximations. Code written in Matlab and R forimplementing the new methods will be made available so others canbenefit from the research.ii) Pivotal bootstrap methods for high-dimensional shape dataThe development of methods for the statistical analysis of shape hasbeen rapid in the last 20 years. Shape analysis has applications inbiology, genetics, image analysis and medicine. A widely adoptedapproach is to identify landmarks on the objects of interest (e.g. aface or a skull) and then to represent the object by the coordinates ofits landmarks. Shape in this context is defined as what remains when allinformation about location, scale and orientation has been removed.Recently, new techniques (pivotal bootstrap methods) for shapes in 2Dhave been proposed. Numerical studies have shown that the newapproach performs extremely well in 2D shape analysis. However, theproperties of shape spaces for objects in 2D are very different from theproperties of shape spaces for object in 3D. We shall address the currentlyopen question of how best to develop pivotal bootstrap methods for 3Dshape analysis. We will also consider the important situationin which the number of landmarks is large, leading to the challengingopen area of high-dimensional shape analysis.