Unimodal Regression Using Bernstein-Schoenberg Splines and Penalties

Research in the field of nonparametric shape constrained regression has been intensive. However, only few publications explicitly deal with unimodality although there is need for such methods in applications, for example, in dose-response analysis. In this article, we propose unimodal spline regression methods that make use of Bernstein-Schoenberg splines and their shape preservation property. To achieve unimodal and smooth solutions we use penalized splines, and extend the penalized spline approach toward penalizing against general parametric functions, instead of using just difference penalties. For tuning parameter selection under a unimodality constraint a restricted maximum likelihood and an alternative Bayesian approach for unimodal regression are developed. We compare the proposed methodologies to other common approaches in a simulation study and apply it to a dose-response data set. All results suggest that the unimodality constraint or the combination of unimodality and a penalty can substantially improve estimation of the functional relationship.