Generalized Linear Models

主要研究者：Peter McCullagh职务：广义线性模型摘要PI与学生和同事一起研究各种主题，所有这些主题都直接或间接地与参数统计模型有关。 主要主题是建立一个框架，用于构建逻辑上一致的统计模型，其特点是可扩展性或范围。 PI建议将这一概念发展到各种设置，例如聚类分析模型，其中出现的对象不是向量空间，而是聚类意义上的分区，或者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。 共形不变性在农业中是一个激进的假设，具有特定的可检验的后果。 迄今为止的证据似乎支持共形不变性，但空间参数不容易准确估计，因此这种情况并不牢固。 PI旨在研究共形不变性的含义，并在尽可能广泛的农业数据上测试假设。 能够在常规农业应用中拟合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，有希望为统计计算带来可观的回报。 用于估计积分的标准蒙特-卡罗设计实现了标准的收敛速度，但是对于特定积分，超高效设计可以实现更快的速度。 PI和他的同事研究了超效率现象，以及它何时以及如何发生，看看它是否可以在常规统计应用中被利用。 作为一个例子，特征值过程有可能被用作积分的正交点在适当的情况下，首要的目标是实现更好地理解统计模型的过程，最初在数学意义上，如果可能的话，在机械或物理意义上。 在农业田间试验中，“只有局部空间特性才重要”的概念被转化为数学，成为一个具有可检验后果的特定假设。 共形不变性不太可能有经济或农业的影响，但如果得到证实，它需要重新思考目前的想法有关的性质和原因的空间变化在陆地过程。 关于共形不变性的工作是专门针对农业领域的试验，但也可能适用于其他地方，例如天球（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适形模型而产生的。 最后一个主题涉及蒙特-卡罗积分，它已经在统计计算中产生了很大的影响。 本提案指出了可以扩大这些方法并使其更加有效的途径。

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.