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In this paper, we first show a new probability result which can be concisely formulated as follows: the function 2 G β / ( 1 + G α ) , where G denotes a baseline cumulative distribution function of a continuous distribution, can have the properties of a cumulative distribution function beyond the standard assumptions on α and β (possibly different and negative, among others). Then, we provide a complete mathematical treatment of the corresponding family of distributions, called the ratio exponentiated general family. To link it with the existing literature, it constitutes a natural extension of the type II half logistic-G family or, from another point of view, a compromise between the so-called exponentiated-G and Marshall-Olkin-G families. We show that it possesses tractable probability functions, desirable stochastic ordering properties and simple analytical expressions for the moments, among others. Also, it reaches high levels of flexibility in a wide statistical sense, mainly thanks to the wide ranges of possible values for α and β and thus, can be used quite effectively for the real data analysis. We illustrate this last point by considering the Weibull distribution as baseline and three practical data sets, with estimation of the model parameters by the maximum likelihood method.
Bantan et al. (Wed,) studied this question.