  Beta regression is a popular methodology when the outcome variable $y$ is on
the open interval $(0,1)$. When $y$ is in the closed interval $[0,1]$, it is
commonly accepted that beta regression is inapplicable. Instead, common
solutions are to use augmented beta regression or censoring models or else to
subjectively rescale the endpoints to allow beta regression. We provide an
attractive new approach with a family of models that treats the entirety of
$y\in[0,1]$ in a single model without rescaling or the need for the
complications of augmentation or censoring. This family provides the
interpretational convenience of a single straightforward model for the
expectation of $y \in [0,1]$ over its entirety. We establish the conditions for
the existence of a unique MLE and then examine this new family of models from
both maximum-likelihood and Bayesian perspectives. We successfully apply the
models to employment data in which augmented beta regression was difficult due
to data separation. We also apply the models to healthcare panel data that were
originally examined by way of rescaling.
