Abstract
This paper develops the exponentiated Mfamily of continuous distributions, aiming to provide new statistical models for data fitting purposes. It stands out from the other families, as it depends on two baseline distributions, with the use of ratio and power transforms in the definition of the main cumulative distribution function. Thanks to the joint action of the possibly different baseline distributions, flexible statistical models can be created, motivating a complete study in this regard. Thus, we discuss the theoretical properties of the new family, with emphasis on those of potential interest to the overall probability and statistics. Then, a new three-parameter lifetime distribution is derived, with the choices of the inverse exponential and exponential distributions as baselines. After pointing out the great flexibility of the related model, we apply it to analyze an actual dataset of current interest: the daily COVID-19 cases observed in Pakistan from 21 March to 29 May 2020 (inclusive). As notable results, we demonstrate that the proposed model is the best among the 15 top ranked models in the literature, including the inverse exponential and exponential models, several modern extensions of them depending on more parameters, and the “unexponentiated” version of the proposed model as well. As future perspectives, the proposed model can be of interest to analyze data on COVID-19 cases in other countries, for possible comparison studies.
Highlights
The modeling and analysis of real-life data are essential to understand important features of random phenomena and to draw suitable conclusions as well. This requires the choice of statistical models based on probability distributions, whose adequateness against the observations will strongly influence the pertinence of the outputs
We derived a natural extension of the M family, called the exponentiated M (EM)
We investigated its main mathematical properties and discussed its ability in terms of statistical modeling
Summary
The modeling and analysis of real-life data are essential to understand important features of random phenomena and to draw suitable conclusions as well. It was proven that the related model had a better fit to the exponentiated exponential, Weibull, and gamma models, for the failure times of the air conditioning system data from [13]. This nice result validated the entry of the M family on the short list. (iii) We consider F1 ( x; ξ 1 ) and F2 ( x; ξ 2 ) of different natures, i.e., exponential and inverse exponential, respectively, to create a new promising (three-parameter lifetime) distribution, which demonstrates a high modeling ability for data fitting; versatile shapes are observed for the main functions.
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