Flexible cdf-quantile distributions on the closed unit interval
For some time modeling doubly-bounded random variables has been hindered by a scarcity of applicable distributions with finite densities at the bounds. Most of the useful distributions for these variables have finite density only on (0,1) or, at best, inconsistently on [0,1]. This talk presents a flexible family of 2- and 3-parameter distributions whose support is the closed interval [0,1] in the sense that they always have finite nonzero densities at 0 and at 1. These distributions have explicit density, cumulative density, and quantile functions, so they are well-suited for quantile regression. The densities at the boundaries are determined by dispersion and skew parameters, and a third parameter influences location. These distributions have a single mode in (0,1) but also can simultaneously have modes at 0 or at 1, or they can be U- or J-shaped. Some of them include the uniform distribution as a special case. Their location, dispersion, and skew parameters are easy to interpret and each of them can have a submodel with its own predictors. They have been implemented in packages for R and Stata.
Can you apply your approach to the beta distribution described At the beginning of the talk? If so could you provide some additional details? If not, what are the challenges for attempting such an application.