Statistical inference for some complex high-dimensional problems

拟议的研究将包括以下两个部分。随机微分方程的鞍点近似。对随机微分方程模型中未知参数进行统计推断是近年来备受关注的一个问题。包含随机微分方程的模型在金融和计量经济学中建立得很好，并且在许多其他领域，特别是数学生物学中变得越来越重要。最近，人们对随机微分方程模型的统计推断的模拟方法产生了浓厚的兴趣。然而，我们相信，最好的分析方法是非常严重的竞争对手，其潜力迄今尚未得到充分利用。所提出的研究将利用鞍点近似的显著数值精度和优良的理论性质。将提供用Matlab和R编写的实现新方法的代码，以便其他人可以从研究中受益。在过去的20年里，形状统计分析方法的发展非常迅速。形状分析在生物学、遗传学、图像分析和医学中都有应用。一种被广泛采用的方法是识别感兴趣对象（例如脸或头骨）上的地标，然后用地标的坐标表示对象。在这种情况下，形状被定义为当所有关于位置、规模和方向的信息都被删除后所保留的东西。最近，人们提出了一种新的二维图形处理技术（关键引导方法）。数值研究表明，该方法在二维形状分析中表现优异。然而，二维物体的形状空间属性与三维物体的形状空间属性有很大的不同。我们将解决当前开放的问题，即如何最好地开发用于三维形状分析的关键引导方法。我们还将考虑地标数量较大的重要情况，这导致高维形状分析的开放区域具有挑战性。

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.