Estimation, Computation, and Uncertainty Quantification in Structured Regression Models

In statistical modeling, regression is the primary tool to study the relationship between a response variable and a collection of predictors. In this research project, questions related to estimation, computation, and uncertainty quantification in some structured regression models will be investigated. The imposed 'structure' refers to known features (domain knowledge) of a system and helps to reduce the complexity of the fitted statistical model/procedure. Further, imposing such structures yields interpretable (yet flexible) models. Special emphasis is given to methods applicable to multivariate data, an area that has received relatively less attention, though often necessary in performing effective data analysis. Some of the methodological development undertaken in this project will address important scientific questions arising from astronomical data. The investigator also plans to continue the tradition of mentoring undergraduate summer interns and to participate in the NYU GSTEM outreach program, a six-week summer program for high school girls.

The three main topics pursued in this project are: (i) incorporating covariate information in multiple hypothesis testing problems; (ii) convexity constrained estimation and inference in regression models; and (iii) statistical methods that are geared towards detecting piecewise constant/affine structure in a (multivariate) regression function. New methodology will be developed to address these topics along with the development of efficient algorithms for computation. Further, a systematic theoretical study of these procedures, focusing on their adaptive (risk) properties, will be undertaken, and the important issues of inference and uncertainty quantification will be addressed. The intended applications of the research are diverse, ranging from estimation of radial velocity distribution of stars in a distant galaxy (astronomy), to developing methodology for detecting interactions between pairs of neurons (neuroscience), to estimating production and utility functions (economics), and to constructing confidence intervals for parameters in a continuous multivariate piecewise affine regression function (engineering).