Unifying Nonparametric Regression and Optimal Design

20世纪60年代，两种重要的统计理论同时出现：非参数估计和最优设计。从本质上讲，两者都解决了同一个问题：将回归函数与观测数据进行拟合。然而，就所使用的方法而言，它们的差异如此之大，以至于其中一种的学生往往不熟悉另一种的最新发展。这两种理论都可以通过相互作用而大大受益。*提出的非参数回归和最优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)、逼近理论(最优恢复)和统计学(最优理论)这几个最发达的领域。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个层面的许多专家，并确立加拿大在统计学方面的领先地位。它还将展示现代统计学的共生本质，即使是那些不太熟悉数学的人也是如此。*尽管这两种理论都取得了巨大成功，但它们都有优缺点。非参数估计对可能的估计量没有限制，可以很好地处理无穷维函数类。然而，它的方法大多是渐近的(大数据)。最优设计主要使用非渐近工具，对数据的分布非常稳健，并研究任意的试验设计。然而，它只处理无偏估计，并且只使用有限维近似类。新提出的方法将结合这两种理论的强大特点，并消除它们的缺点。*申请人最近的出版物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。我们将使用最优恢复的方法来连接这两个理论，这是现代近似理论的一个成熟章节。*申请人最近的出版物中使用的椭圆雅可比函数将被更灵活的自同构函数取代，这可以追溯到Klein和Poincaré。这种方法将包括许多有限的变换群。类似地，将研究用于信号处理的考尔-佐洛塔雷夫椭圆滤波器的最优性。总体而言，这是一个规模很大的研究项目，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同事的现有合作关系，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. **