On a New Result on the Ratio Exponentiated General Family of Distributions with Applications

Abstract

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.

Reliability Analysis with Wiener-Transmuted Truncated Normal Degradation Model for Linear and Non-Negative Degradation Data

“…To avoid the effect of negative drift and simultaneously relax the normality (symmetry) assumption to achieve accurate results, a transmuted truncated normal distribution is adopted herein, due to its flexibility and its attractive properties. More detail about transmuted distributions can be found in Shahbaz et al [35], Alizadeh et al [36], Bakouch et al [37], Bantan et al [38] and Muhammad et al [39], among others. Additionally, measurement errors usually occur during the observation process in practice [40]; therefore, in this study, we include the effect of the error function in order to achieve more precise life estimation results.…”

On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions

“…To go further into these limitations, new distributions, often divided into specific families of distributions, have been created. A short list of the notorious families is the following: the skew-normal family (see [1]), Marshall-Olkin-Gfamily (see [2]), exponentiated-G family (see [3]), beta-G family (see [4]), order statistics-G family (see [5]), sinh-arcsinh-G family (see [6]), transmuted-G (see [7]), gamma-G family (see [8]), Kumaraswamy-G (see [9]), Topp-Leone-G (see [10]), and ratio-exponentiated-G (see [11]). The global motivation behind them is to extend the modeling properties of a classical baseline distribution by adding one or more tuning parameters through the use of various flexible transformations (power, beta, gamma, ratio, etc.…”

Box-Cox Gamma-G Family of Distributions: Theory and Applications

“…One of the most popular approaches to define such families is by using a so-called generator. In this regard, we refer the reader to the Marshall-Olkin-G family by [1], the exp-G family by [2], the beta-G family by [3], the gamma-G family by [4], the Kumaraswamy-G family by [5], the Ristić-Balakrishnan (RB)-G family (also called gamma-G type 2) by [6], the exponentiated generalized-G family by [7], the logistic-G family by [8], the transformerX (TX)-G family by [9], the Weibull-G family by [10], the exponentiated half-logistic-G family by [11], the odd generalized exponential-G family by [12], the odd Burr III-G family by [13], the cosine-sine-G family by [14], the generalized odd gamma-G family by [15], the extended odd-G family by [16], the type II general inverse exponential family by [17], the truncated Cauchy power-G family by [18], the exponentiated power generalized Weibull power series-G family by [19], the exponentiated truncated inverse Weibull-G family by [20], the ratio exponentiated general-G family by [21] and the Topp-Leone odd Fréchet-G family by [22].…”