EPSCoR Research Fellows: NSF: Nonparametric Estimation for Long Memory Linear Random Fields with Innovation in the Domain of Attraction of Stable Laws

Linear random fields are important stochastic models that can help data analysts understand complex systems, quantify risk and uncertainty, and find optimal solutions. The principal investigator (PI) will develop new methods to study long-memory linear random fields. This will be the first research program on random fields and spatial data analysis at the University of Mississippi (UM) and within the state of Mississippi. This fellowship will advance and broaden the scope of the PI’s research program beyond UM and the state. The PI will travel with a UM Ph.D. student to Michigan State University (MSU) to work closely with MSU Foundation Professor Yimin Xiao, an expert in stochastic processes and random fields. The PI and Dr. Xiao will organize invited sessions on topics in this field of research at national and international conferences. Dr. Xiao and other experts in this field will visit UM to deliver research talks on the proposed research. Additionally, the PI will develop a graduate course in spatial data analysis at UM. This fellowship will have a lasting impact on the PI’s career, strengthen the probability and statistics group at UM, and enhance graduate and undergraduate education. This research will result in collaborative research projects in nonparametric estimation for long-memory linear random fields, focusing on innovation in the domain of attraction of stable laws. The PI and Dr. Xiao will primarily investigate the unbiasedness and limit theorems of the kernel and wavelet estimators for the density function and the quadratic entropy functional. During the first year of this fellowship, the PI will collaborate with Dr. Xiao to explore kernel estimators for density and quadratic entropy and derive the limit theorems for these estimators. In the second year, the PI and Dr. Xiao will explore wavelet estimators for density and quadratic entropy, obtaining the corresponding limit theorems. Fourier transform and orthogonal projection methods will serve as important tools in this study. The investigation of kernel and wavelet density and entropy estimation for linear random fields aims to provide a comprehensive understanding of the sampling requirements and conditions on the coefficients and innovations of long-memory linear random fields, establishing ideal limit theorems for the estimators. Through this research, the PI will gain insights into the advantages and disadvantages of these estimation methods. This study will illuminate data analysis when observations exhibit heavy tails and long-range dependence. Through this series of collaborations, the PI will expand into a new research area and form long-lasting partnerships with Dr. Xiao and other experts in probability and statistics across the country. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.