Some Developments in the Theory of Shape Constrained Inference

Abstract

Shape constraints enter in many statistical models. Sometimes these constraints emerge naturally from the origin of the data. In other situations, they are used to replace parametric models by more versatile models retaining qualitative shape properties of the parametric model. In this paper, we sketch a part of the history of shape constrained statistical inference in a nutshell, using landmark results obtained in this area. For this, we mainly use the prototypical problems of estimating a decreasing probability density on $[0,\infty )$ and the estimation of a distribution function based on current status data as illustrations.