Asymptotic equivalence theory has emerged as a growing new area of mathematical statistics research. The purpose of this project is to describe the asymptotic behavior of some nonparametric curve estimation models, which have distributions that depend on a smooth function f(x). For each of these nonparametric models, the investigator derives a limiting model that can be used as an asymptotic approximation, such that inference in the original experiment can be performed under the limiting experiment without loss of information. Many regular nonparametric models have been shown to be approximable in the limit by a Gaussian white-noise-with-drift experiment which observes a Brownian motion plus a mean that depends on f(x). In this project, the investigator considers extensions of the regular nonparametric problem to include nuisance parameters such as an unknown variance or an unknown distribution of sample points. The limiting experiments in these cases contain a secondary component that describes the nuisance parameter, and this demonstrates that the estimation of f(x) can be separated from other unknowns in the problems. The investigator also approximates nonparametric problems with two-dimensional sample spaces by a Brownian sheet process plus a mean. These asymptotic approximations allow the non-standard problems to be solved using techniques from the simpler Gaussian models.
In many areas of technology today, large sets of data that do not follow a typical pattern are difficult for scientists to analyze. This problem comes up in tasks such as interpreting medical images and signals, describing distributions of plant species, and analyzing financial series. In this project, the investigator transforms the problem into a form that can be solved by already existing statistical tools. As a result, existing methods of analysis can be applied to a wider range of technological problems. Thus the techniques developed by the investigator may be used to improve the efficiency of medical imaging, signal interpretation, financial analysis, and other applications.
