Some theoretical results regarding the polygonal distribution

Polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. We demonstrate that the densities of polygonal distributions are dense in the class of continuous and concave densities with bounded second derivatives. Furthermore, we prove that polygonal density functions provide O(g^{? 2}) approximations (where g is the number of triangular distribution components), in the supremum distance, to any density function from the hypothesized class. Parametric consistency and Hellinger consistency results for the maximum likelihood (ML) estimator are obtained. A result regarding model selection via penalized ML estimation is proved.