While non-parametric methods like kernel density estimation are flexible and less restrictive on the underlying data-generating process, often they require tuning parameters. Shape constraint methods do not rely on tuning parameters and impose conditions mainly on the shape of certain functions. Also, shape restrictions like unimodality or log-concavity arise naturally and apply to a large class of continuous densities.
The traditional approach for estimating the location parameters in semiparametric models involves the use of various tuning parameters. In "Adaptive estimation in symmetric location model under log-concavity constraint", I show that shape restriction of log-concavity leads to adaptive estimation with virtually no tuning in symmetric location models. The method can be implemented using the R package log.location.
The class of s-concave densities includes many common continuous densities, such as the class of log-concave densities (s=0) and t-densities. Although this is a rich class, s-concave densities are necessarily unimodal, thereby excluding many types of mixture densities which naturally arise in many fields like speech recognition, pattern recognition, climatology, to name a few. In our paper "Bi-s*-concave distributions", we introduce a new shape-constrained class of distribution functions, the bi-s*-concave class, which permits distributions with bimodal and multimodal densities. Below is the image of bi-s*-concave bands for the distribution function of a t-distribution.