Unifying Nonparametric Regression and Optimal Design

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Two important statistical theories emerged around the same time in the 60s: Nonparametric Estimation and Optimal Design. Essentially, both dealt with the same problem: fitting a regression function to the observed data. However, with respect to the methods used, they differed so significantly that students in one of them were often unfamiliar with recent developments in the other. Both theories came to a point where they can greatly benefit by interacting with each other. The proposed unification of Nonparametric Regression and Optimal Design will require simultaneous use of most developed areas of Calculus (automorphic functions), Algebra (finite transformation groups), Approximation Theory (optimal recovery), and Statistics (optimality theory). This will create a completely new area of research with a vast potential for future growth, attract many specialists at all levels, including young researchers, and establish Canada’s leading position in Statistics. It will also demonstrate the symbiotic nature of modern Statistics even to those not closely familiar with mathematics. Despite their great successes, both theories have their pluses and minuses. Nonparametric Estimation poses no restriction on possible estimators and handles well infinitely dimensional classes of functions. However, its approach is mostly asymptotic (large data). Optimal Design uses mainly non-asymptotic tools, is very robust to the distribution of the data and studies arbitrary experimental designs. However, it deals exclusively with unbiased estimators and uses only finite dimensional approximating classes. The newly proposed approach will combine the strong features of both theories and eliminate their shortcomings. Unifying these theories has already begun in the applicant’s recent publications. Based on this approach, the main goal of the proposed research is to adopt infinitely dimensional classes of regression functions in Optimal Design. This will require radically new methods. A bridge connecting the two theories will be built using methods of Optimal Recovery, a well-developed chapter of modern Approximation Theory. The elliptic Jacobi functions, used in the applicant’s recent publications, will be replaced by more flexible automorphic functions, going back to Klein and Poincaré. This approach will include numerous finite groups of transformations. In a similar vein, the optimality of the Cauer-Zolotarev elliptic filter, used in signal processing, will be studied. Overall this is a research project of significant scope with the aim is to increase accuracy and diversity of existing statistical methods. Within it, I am planning to strengthen further the existing collaboration ties with my colleagues from Marseilles and Bar Ilan University, and attract a large group of graduate students.