Mathematical Sciences: Parametric and Nonparametric Likelihood Studies

拟议的研究是针对基于似然的统计方法领域的一些主题。其中一组主题将涉及在参数模型领域的几种新的渐近方法的发展，重点关注在存在有害参数时改进的分布特性以及混合物中成分数量的测试方法。第二组主题围绕混合模型中的半参数和非参数推理展开，其中模型类别需要扩展，进一步的推理方法需要开发。用于得出统计结论的最广泛使用的方法被称为最大似然法，它基于一个直观概念的数学版本，即参数的真实值通常会在给定观察数据的那些看起来“最有可能”的值中找到。这个强大的工具使人们能够写下复杂现象的模型，并通过最大化可能性，得出关于产生数据的自然状态的结论。然而，似然方法还不完善，本研究计划的目的是在计算和理论上进一步发展，在几个广泛应用的领域。

The proposed research is directed towards a number of topics in the area of likelihood based statistical methods. One set of topics will involve the development of several new asymptotic methods in the area of parametric models, focussing on improved distributional properties in the presence of nuisance parameters and on testing methods for the number of components in a mixture. The second set of topics revolves around semiparametric and nonparametric inference in the mixture model, where the class of models needs to be expanded and further inferential methods need to be developed. The most widely used method for drawing statistical conclusions, called the method of maximum likelihood, is based on a mathematical version of the intuitive notion that the true value of a parameter will generally be found among those that seem "most likely" given the observed data. This powerful tool enables one to write down models for complex phenomenon and, by maximizing the likelihood, draw conclusions about the state of nature that generated the data. However, the likelihood method is not perfect, and the objective of this research proposal is to further its development, both in computation and in theory, in several areas of wide application.