Unifying Nonparametric Regression and Optimal Design

大约在60年代出现了两个重要的统计理论：非参数估计和最优设计。本质上，两者都处理相同的问题：将回归函数拟合到观察到的数据中。然而，就所使用的方法而言，它们差异如此之大，以至于其中一门课程的学生往往不熟悉另一门课程的最新发展。这两种理论都达到了可以通过相互作用而大大受益的地步。***提出的非参数回归和优化设计的统一将需要同时使用微积分（自同构函数）、代数（有限变换群）、近似理论（最优恢复）和统计学（最优性理论）中最发达的领域。这将创造一个全新的研究领域，具有巨大的未来增长潜力，吸引包括年轻研究人员在内的各级专家，并建立加拿大在统计学方面的领先地位。它还将向那些不太熟悉数学的人展示现代统计学的共生本质。尽管两种理论都取得了巨大的成功，但它们各有优缺点。非参数估计对可能的估计量没有限制，能很好地处理无穷维的函数类。然而，它的方法大多是渐进的（大数据）。最优设计主要使用非渐近工具，对数据的分布具有很强的鲁棒性，并研究任意的实验设计。然而，它只处理无偏估计量，并且只使用有限维近似类。新提出的方法将结合两种理论的优点，消除它们的缺点。在申请人最近的出版物中已经开始统一这些理论。基于这种方法，提出的研究的主要目标是在优化设计中采用无限维回归函数类。这将需要全新的方法。最优恢复方法是现代近似理论中发展良好的一章，我们将用它来建立连接这两种理论的桥梁。***申请人最近发表的文章中使用的椭圆Jacobi函数将被更灵活的自同构函数所取代，这可以追溯到Klein和poincarcar<s:1>。这种方法将包括许多有限的变换群。在类似的情况下，将研究用于信号处理的Cauer-Zolotarev椭圆滤波器的最优性。总的来说，这是一个具有重大意义的研究项目，其目的是提高现有统计方法的准确性和多样性。在其中，我计划进一步加强与马赛和巴伊兰大学的同事们现有的合作关系，并吸引一大批研究生。**

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. **