In several areas of the sciences, traditional parametric regression models (e.g., polynomials) have been seen as not flexible enough for modeling unknown nonlinear dependencies. Therefore, methods of nonparametric regression (and density estimation) are being used more and more routinely in data analysis (applied statistics). Most widely used are local polynomial regression estimators, in particular kernel smoothers (generalized running weighted means), smoothing splines and local regression.
Our project is a novel semi-parametric approach to curve estimation. The number of inflection points is taken as `model order' of the class of smooth functions in consideration. More mathematically, we maximize the penalized likelihood as in the approach of smoothing splines. Our `Wp' approach however penalizes relative change of curvature whereas a measure of total curvature is penalized for splines. The practical solution of the corresponding variational problem entails numerical integration of a boundary value ordinary differential equation.
For density estimation, the model order is the number of modes, or equivalently, local extrema. Our semi-parametric density estimator therefore parametrizes the location of modes (maxima) and anti-modes (minima) and is completely nonparametric otherwise.
