A Class of Bell-Shaped Distributions with Bounded Supports

Abstract

In this article we introduce a new family of parametrized density functions that are bell-shaped and supported by a finite interval (0, 1). The corresponding random variable X assumes the structure where (W, V) is a bivariate Dirichlet random vector with parameters and is any symmetric random variable on (0, 1). We provide the formulation of our bell-shaped density function in terms of Euler’s and Appell’s integral representations of hypergeometric functions and derive closed form expressions for the density for specific parameter values. We provide closed form characterization for the moments of X. Graphical illustrations are provided for different choice of . We provide a practical simulation procedure for generating values for X. The distribution’s symmetric shape along with its bounded support, make it an attractive alternative to the normal distribution when modelling real-life phenomena observed over a finite interval. We demonstrate the distribution’s suitability for such situations by using anthropometric data. The class of bell-shaped densities constructed in this work define a rich class of kernel functions that can be used for estimation purposes in many statistical methods, such as regression, time series and density estimation.