In this paper, two new general families of distributions supported on the unit interval are introduced. The proposed families include several known models as special cases and define at least twenty (each one) new special models. Since the list of well-being indicators may include several double bounded random variables, the applicability for modeling those is the major practical motivation for introducing the distributions on those families. We propose a parametrization of the new families in terms of the median and develop a shiny application to provide interactive density shape illustrations for some special cases. Various properties of the introduced families are studied. Some special models in the new families are discussed. In particular, the complementary unit Weibull distribution is studied in some detail. The method of maximum likelihood for estimating the model parameters is discussed. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Applications to the literacy rate in Brazilian and Colombian municipalities illustrate the usefulness of the two new families for modeling well-being indicators.

We propose a new lifetime model called the exponentiated power generalized Weibull (EPGW) distribution, which is obtained from the exponentiated family applied to the power generalized Weibull (PGW) distribution. It can also be derived from a power transform on an exponentiated Nadarajah-Haghighi random variable. Since several structural properties of the PGW distribution have not been studied, they can be obtained from those of the EPGW distribution. The model is very flexible for modeling all common types of hazard rate functions. It is a very competitive model to the well-known Weibull, exponentiated exponential and exponentiated Weibull distributions, among others. We also give a physical motivation for the new distribution if the power parameter is an integer. Some of its mathematical properties are investigated. We discuss estimation of the model parameters by maximum likelihood and provide two applications to real data. A simulation study is performed in order to examine the accuracy of the maximum likelihood estimators of the model parameters.

We study a five-parameter model called the Weibull Burr XII (WBXII) distribution, which extends several models, including new ones. This model is quite flexible in terms of the hazard function, which exhibits increasing, decreasing, upside-down bathtub, and bathtub shapes. Its density function allows different forms such as left-skewed, right-skewed, reversed-J, and bimodal. We aim to provide some general mathematical quantities for the proposed distribution, which can be useful to real data analysis. We develop a shiny application to provide interactive illustrations of the WBXII density and hazard functions. We estimate the model parameters using maximum likelihood and derive a profile log-likelihood for all members of the Weibull-G family. The survival analysis application reveals that the WBXII model is suitable to accommodate left-skewed tails, which are very common when the variable of interest is the time to failure of a product. The income application is related to player salaries within a professional sports league and it is peculiar because the mean of the player's salaries is much higher than for most professions. Both applications illustrate that the new distribution provides much better fits than other models with the same and less number of parameters.

We propose a new family of distributions, so-called the unit ratio-extended Weibull family ( U R E W ). It is derived from ratio transformation in an extended Weibull random variable. The use of this transformation is a novelty of the work since it has been less explored than the exponential and has not yet been studied within the extended Weibull class. Moreover, we offer a valuable alternative to model double-bounded variables on the unit interval. Five U R E W special models are studied in detail, namely the: i) unit ratio-Gompertz; ii) unit ratio-Burr XII; iii) unit ratio-Lomax; v) unit ratio-Rayleigh, and vi) unit ratio-Weibull distributions. We propose a quantile-parameterization for the new family. The maximum likelihood estimators (MLEs) are presented. A Monte Carlo study is performed to evaluate the behavior of the MLEs of unit ratio-Gompertz and unit ratio-Rayleigh distributions. This last model has closed-form and approximately unbiased MLE for small sample sizes. Further, the U R E W submodels are adjusted to the dropout rate in Brazilian undergraduate courses. We focus on the areas of civil engineering, economics, computer sciences, and control engineering. The applications show that the new family is suitable for modeling educational data and may provide effective alternatives compared to other usual unit models, such as the Beta, Kumaraswamy, and unit gamma distributions. They can also outperform some recent contributions in the unit distribution literature. Thus, the U R E W family can provide competitive alternatives when those models are unsuitable.