Abstract
In this article, we introduce a new probability distribution generator called the Lambert-F generator. For any continuous baseline distribution F, with positive support, the corresponding Lambert-F version is generated by using the new generator. The result is a new class of distributions with one extra parameter that generalizes the baseline distribution and whose quantile function can be expressed in closed form in terms of the Lambert W function. The hazard rate function of a Lambert-F distribution corresponds to a modification of the baseline hazard rate function, greatly increasing or decreasing the baseline hazard rate for earlier times. Herein, we study the main structural properties of the new class of distributions. Special attention is given to two particular cases that can be understood as two-parameter extensions of the well-known exponential and Rayleigh distributions. We discuss parameter estimation for the proposed models considering the moments and maximum likelihood methods. Finally, two applications were developed to illustrate the usefulness of the proposed distributions in the analysis of data from different real settings.
Highlights
In recent years, towards more flexibility, many studies have been developed in which new methods are proposed to add one or more parameters to a baseline probability distribution
The scale σ in both models is inherited from the respective baseline distribution, while the shape parameter α arises from the application of the Lambert transformation to the baseline distribution
Note that the asymmetry and kurtosis values of the baseline distributions were extended to a range of values in the respective Lambert versions, showing greater flexibility of these latter distributions
Summary
Towards more flexibility, many studies have been developed in which new methods are proposed to add one or more parameters to a baseline probability distribution. These methods have given way to the generation of models with more complex parametric structures and more flexibility in aspects such as the shapes of the density and hazard rate functions and the asymmetry or kurtosis of the distribution. We will show that the hazard rate function (hrf) of a Lambert-F distribution corresponds to a modification of the baseline hrf, greatly increasing or decreasing the baseline hrf for the lower values of X (earlier times).
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