Mathematical Sciences: Sampling Plans, Asymptotic Results, Resampling Algorithms, & Applications to Random Processes on the 3-dimensional Sphere

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To: "Alan J. Izenman" aizenman@nsf.gov From: aizenman@nsf.gov (Alan J. Izenman) Subject: Brown Abstract Cc: Bcc: X-Attachments: 9404130 Brown Research on an isotropic random field on the sphere has mainly focused on the representation of the random field, although, there has been some research on proving a central limit theorem (CLT) for a continuously indexed random field on the sphere. From a practical point of view, however, it is impossible to sample continuously throughout the sphere and a finite global sampling plan needs to be found in order to investigate statistical relations associated with global data, in particular a CLT and resampling algorithms. Work by Brown 1993 lays the groundwork for future research in this area and he has applied his research to global land-area and coastline data. His work addresses the above statistical issues in a nonparametric setting using weak general conditions as the radius of the sphere grows without bound with the sample size. More recently, in the parametric setting there has been work done on modeling the entire random process on the sphere. This research will consider both nonparametric and parametric cases and asymptotic results for situations where the radius of the sphere remains fixed while the sampling plan gets more dense on the sphere and where the radius of the sphere grows without bound along with the sample size. Specifically, this research will study the following areas: finite global sampling plans, asymptotic results for different sampling plans and general statistics, resampling mechanisms of the data for various sampling plans, and applications to this new research. Typically, we only have an opportunity to sample data at a finite number of points on the globe, comprising a global sampling plan. Once a sampling plan is established, data is gathered at the points and quantities of interest are calculated. In order to make decisions about the quantity of interest, some of its characteristics need to be known or estimated. Since gathering data via the sampling process is expensive, some methods for reusing (resampling) the original data need to be developed as well. This research will study the following areas: global sampling plans, characteristics for various quantities of interest, resampling mechanisms for global data, and applications to this new research.