Nonparametric Estimation via Mixed Derivatives

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Understanding the precise relationship between a specific variable of interest and a related set of covariables is a fundamental problem arising in many areas of science and engineering. It is usually necessary to combine two different sources of information to solve this problem. The first source of information is domain knowledge or theory, which often suggests certain basic forms for the relationship between the variables. The second source of information is observed data on the values of the variables for a set of experimental conditions or subjects. Effective solutions to the problem can only be obtained by efficiently combining both sources of information. The development of such a methodology is the primary goal of this project. Assuming the function governing the relationship between the variables is smooth in a certain sense, the observed data is then brought in to find the function satisfying the assumed smoothness that best explains the observed data. This project will explore different ways of carrying out this scheme and develop novel methods and computational algorithms useful to practitioners in broad scientific areas. Most existing methods for studying these problems either make strong prior assumptions that are unrealistic and lead to wrong conclusions on the relationship between the variables, or weak prior assumptions that lead to requiring unrealistically enormous sizes of datasets for reliable conclusions. The developed approach based on "mixed partial derivative smoothness constraints" effectively compromises these two extreme approaches and will lead to methods of great practical value. The material emanating from this research will be disseminated through seminars and undergraduate as well as graduate teaching. The methods from this project will be made available to the wider statistics and scientific community through the development of software packages. This project focuses on nonparametric function estimation problems under smoothness constraints involving mixed partial derivatives. The investigator plans to expand recently developed methodology for nonparametric regression under mixed partial derivatives of first and second orders to allow for restricted interaction orders, design faster algorithms for computation, and prove theoretical accuracy results under more general design assumptions. The possibility of near parametric rates under strong sparsity settings will be explored in a more general setting involving tensor product bases (including complex exponentials and radial basis functions). Uncertainty quantification will be systematically explored using Bayesian approaches with a main focus on Cauchy priors. New mixed derivative approaches will be explored in shape-constrained regression, including those based on Entire Monotonicity with restricted interactions and total Popoviciu convexity. Mixed derivative approaches for spectral density estimation of time series and density estimation will also be studied.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.