The Opone family of distributions. Beyond the power continuous Bernoulli distribution

Recent developments in applied statistics have given rise to the continuous Bernoulli distribution, a one-parameter distribution with support of [0, 1]. In this paper, we use it for a more general purpose: the creation of a family of distributions. We thus exploit the flexible functionalities of the continuous Bernoulli distribution to enhance the modeling properties of well-referenced distributions. We first focus on the theory of this new family, including the quantiles, expansion of important functions, and moments. Then we exemplify it by considering a special baseline: the Topp–Leone distribution. Thanks to the functional structure of the continuous Bernoulli distribution, we create a new two-parameter distribution with support for [0, 1] that possesses versatile shape capacities. In particular, the corresponding probability density function has left-skewed, N-type and decreasing shapes, and the corresponding hazard rate function has increasing and bathtub shapes, beyond the possibilities of the corresponding functions of the Topp–Leone distribution. Its quantile and moment properties are also examined. We then use our modified Topp–Leone distribution from a statistical perspective. The two parameters are supposed to be unknown and then estimated from proportional-type data with the maximum likelihood method. Two different data sets are considered, and reveal that the modified Topp–Leone distribution can fit them better than popular rival distributions, including the unit-Weibull, unit-Gompertz, and log-weighted exponential distributions. It also outperforms the Topp–Leone and continuous Bernoulli distributions.