Unimodal Bayesian Nonparametric Quantile Regression With Bernstein-Schoenberg Splines

Nonparametric shape constrained regression has been widely applied in medical research. This article considers the unimodality in applications, such as the dose-response analysis. We propose a nonparametric unimodal regression method that make use of Bernstein-Schoenberg splines with their shape preservation property with quantile regressions. Quantile regression permits covariates to affect the response differently at varying joint quantile levels in the follow-up dose, thus providing a comprehensive study of the response distribution. To account for the small sample cases in most applications, the Bayesian splines analysis are drawn in. We intro-duce the Bernstein-Schoenberg spline, construct a joint likelihood, and develope the Bayesian inference. One of the key distinguishing features of the proposed estimator is that the shape con-straint corresponding to the unimodality can be embedded into the prior by a suitable selection. The limited sample performance of the proposed method is evaluated by simulation studies and real dose-response data analysis.