Generalized Linear Models

与学生和同事一起，PI研究各种主题，所有这些主题都直接或间接地与参数统计模型有关。主要主题是开发一个框架，用于构建逻辑上一致的统计模型，其特点是可扩充性或范围。PI建议将这一概念发展到各种环境中，例如集群分析模型，其中产生的对象不是向量空间，而是集群意义上的分区，或递归分区意义上的树。第二个应用是农业田间试验中的空间变异。农业中的共形不变性是一个具有特定可检验结果的基本假设。到目前为止，现有的证据似乎支持共形不变性，但空间参数并不容易准确估计，因此这种情况并不牢固。PI的目的是研究共形不变性的含义，并在尽可能广泛的农业数据上检验这一假设。能够在日常农业应用中适应这些模型的软件是一个重要的副产品。蒙特卡罗积分是一个古老的话题，但统计模型和蒙特卡罗积分之间的相互作用是一个相对较新的发展，有望为统计计算带来实质性的回报。积分估计的标准蒙特卡罗设计可以达到标准的收敛速度，而超高效设计可以实现特定积分的更快收敛速度。PI和他的同事们研究了超效率现象，研究了它何时以及如何发生，看看它是否可以在常规统计应用中加以利用。例如，本征值过程有可能在适当的条件下用作积分的积分点。最重要的目标是更好地理解统计模型作为过程，最初在数学意义上，如果可能的话，从机械或物理意义上理解。在农业田间试验中，“只有局部空间属性才重要”的概念被转化为数学，成为一个具有可检验结果的特定假说。共形不变性不太可能有经济或农业方面的影响，但如果得到证实，就需要重新思考目前关于陆地过程空间差异的性质和原因的想法。关于共形不变性的工作是专门针对农业田间试验的，但很可能适用于其他地方，例如在天球(天空)上的过程。作为副产品，统计方法和软件是专门为拟合和测试保形模型而产生的。最后一个话题涉及蒙特卡罗积分，它已经在统计计算中产生了重大影响。本提案指出了可扩展这些方法并使其更加有效的方法。

PI: Peter McCullaghTitle: Generalized Linear ModelsABSTRACTTogether with students and colleagues, the PI investigates various topics, all bearing directly or indirectly on parametric statistical models. The main theme is the development of a framework for constructing logically consistent statistical models, whose hallmark is extendibility or scope. The PI proposes to develop this notion to a variety of settings, such as cluster-analysis models, where the objects that arise are not vector spaces but partitions in the sense of clusters, or trees in the sense of recursive partitions. A second application is to spatial variation in agricultural field trials. Conformal invariance in agriculture is a radical hypothesis with specific testable consequences. The available evidence to date seems to favor conformal invariance, but spatial parameters are not easy to estimate accurately, so the case is not firmly established. The PI aims to study the implications of conformal invariance and to test the hypothesis on as wide a range of agricultural data as possible. Software capable of fitting these models in routine agricultural applications is an important by-product. Monte Carlo integration is a venerable topic, but the interplay between statistical models and Monte Carlo integration is a relatively new development with promise of substantial payoff for statistical computation. Standard Monte-Carlo designs for the estimation of integrals achieve the standard rate of convergence, but super-efficient designs can achieve faster rates for specific integrals. The PI and his colleagues study the phenomenon of super-efficiency, when and how it occurs, to see whether it can be exploited in routine statistical applications. As an example, eigenvalue processes have the potential to be used as quadrature points for integration in suitable circumstances.The over-riding goal is to achieve a better understanding of statistical models as processes, initially in the mathematical sense and if possible in a mechanistic or physical sense. The notion in agricultural field experiments that "only local spatial properties matter" is translated into mathematics, emerging as a specific hypothesis with testable consequences. Conformal invariance is unlikely to have economic or agricultural implications, but if confirmed it demands a re-thinking of current ideas concerning the nature and causes of spatial variation in terrestrial processes. The work on conformal invariance is aimed specifically at agricultural field trials, but might well be applicable elsewhere, for example processes on the celestial sphere (sky). As a by-product, statistical methods and software are produced specifically for fitting and testing conformal models. The last topic concerns Monte-Carlo integration, which has already had a big impact in statistical computation. The present proposal indicates ways in which those methods might be extended and made more effective.