Bayesian Local Extrema Splines

Abstract

We consider the problem of shape restricted nonparametric regression on a closed set X ?\in R; where it is reasonable to assume the function has no more than H local extrema interior to X: Following a Bayesian approach we develop a nonparametric prior over a novel class of local extrema splines. This approach is shown to be consistent when modeling any continuously differentiable function within the class of functions considered, and is used to develop methods for hypothesis testing on the shape of the curve. Sampling algorithms are developed, and the method is applied in simulation studies and data examples where the shape of the curve is of interest.