Michael Smithson

Dr. Yiyun Shou

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.