Together 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.
