# Some Theoretical Results Regarding the Polygonal Distribution

## TL;DR

This paper introduces the polygonal distribution class, explores its theoretical properties, including density density and consistency of estimators, and compares it as an alternative to beta distributions.

## Contribution

It provides new theoretical insights into polygonal distributions, including density density, consistency results, and model selection theorems.

## Key findings

- Polygonal densities are dense in continuous concave densities.
- Maximum likelihood estimators are pointwise and Hellinger consistent.
- A related distribution via squared polygonal densities is analyzed.

## Abstract

The polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. The class includes the uniform and trapezoidal distributions, and is an alternative to the beta distribution. We demonstrate that the polygonal densities are dense in the class of continuous and concave densities with bounded second derivatives. Pointwise consistency and Hellinger consistency results for the maximum likelihood (ML) estimator are obtained. A useful model selection theorem is stated as well as results for a related distribution that is obtained via the pointwise square of polygonal density functions.