This project focuses on some important statistical issues relating to the problem of estimating regressions curves that are known or required to satisfy certain constraints such as monotonicity, convexity, periodicity, or range bounds. These constraints arise in a variety of social, economic, biometric, and behavioral applications. Despite a vast amount of literature on function estimation, few algorithms are available for incorporating constraints. This project starts with a simple and effective method based on L1 optimization in the space of B-splines. The asymptotic properties, modeling flexibility, and computational complexity of the method are about the same as those of the unconstrained problems. The investigator will extend work on the method to include adaptive choice of dimension for the spline space, hypothesis testing on constraints, handling of binary response, semiparametric models and multivariate functions, and computation of properly ordered percentile curves. In addition to the methodological research, a user-friendly software for constrained B-spline smoothing will be developed through collaborations.
