The Weibull-G Family of Probability Distributions
TL;DR: The Weibull distribution is the most important distribution for problems in reliability as discussed by the authors, and it has been studied extensively in the literature, including in the context of the wider Weibbull-G family of distributions.
Abstract: The Weibull distribution is the most important distribution for problems in reliability. We study some mathematical properties of the new wider Weibull-G family of distributions. Some special models in the new family are discussed. The properties derived hold to any distribution in this family. We obtain general explicit expressions for the quantile function, ordinary and incomplete moments, generating function and order statistics. We discuss the estimation of the model parameters by maximum likelihood and illustrate the potentiality of the extended family with two applications to real data.
Citations
More filters
TL;DR: A new model called the Weibull-Lomax distribution is introduced which extends the LomAX distribution and has increasing and decreasing shapes for the hazard rate function and outperforms the McDonald- lomax, Kumaraswamy-Lamax, gamma-Lmax, beta-LMax, exponentiated Lmax and Lomax models.
Abstract: We introduce a new model called the Weibull-Lomax distribution which extends the Lomax distribution and has increasing and decreasing shapes for the hazard rate function. Various structural properties of the new distribution are derived including explicit expressions for the moments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, probability weighted moments, generating and quantile function. The Rényi and q entropies are also obtained. We provide the density function of the order statistics and their moments. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The potentiality of the new model is illustrated by means of two real life data sets. For these data, the new model outperforms the McDonald-Lomax, Kumaraswamy-Lomax, gamma-Lomax, beta-Lomax, exponentiated Lomax and Lomax models. 2000 AMS Classi cation: 60E05; 62N05.
149 citations
Cites methods from "The Weibull-G Family of Probability..."
...[20] proposed the Weibull-G class in uenced by the Zografos-Balakrishnan-G class....
[...]
TL;DR: In this paper, the authors introduce a new family of continuous distributions generated from a logistic random variable called the logistic-X family, which can be expressed as a linear combination of exponentiated densities based on the same baseline distribution.
Abstract: The logistic distribution has a prominent role in the theory and practice of statistics. We introduce a new family of continuous distributions generated from a logistic random variable called the logistic-X family. Its density function can be symmetrical, left-skewed, right-skewed, and reversed-J shaped, and can have increasing, decreasing, bathtub, and upside-down bathtub hazard rates shaped. Further, it can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon entropy, and order statistics. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. We also investigate the properties of one special model, the logistic-Frechet distribution, and illustrate its importance by means of two applications to real data sets.
129 citations
TL;DR: A new family of continuous distributions called the odd generalized exponential family, whose hazard rate could be increasing, decreasing, J, reversed-J, bathtub and upside-down bathtub is proposed, which includes as a special case the widely known exponentiated-Weibull distribution.
Abstract: We propose a new family of continuous distributions called the odd generalized exponential family, whose hazard rate could be increasing, decreasing, J, reversed-J, bathtub and upside-down bathtub. It includes as a special case the widely known exponentiated-Weibull distribution. We present and discuss three special models in the family. Its density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon and Renyi entropies and order statistics. For the first time, we obtain the generating function of the Frechet distribution. Two useful characterizations of the family are also proposed. The parameters of the new family are estimated by the method of maximum likelihood. Its usefulness is illustrated by means of two real lifetime data sets.
AMS Subject Classification Primary 60E05; secondary 62N05; 62F10
125 citations
Cites background from "The Weibull-G Family of Probability..."
...…transformed-transformer (T-X) (Weibull-X and gamma-X) by Alzaatreh et al. (2013), exponentiated T-X by Alzaghal et al. (2013), odd Weibull-G by Bourguignon et al. (2014), exponentiated half-logistic by Cordeiro et al. (2014a), T-X{Y}-quantile based approach by Aljarrah et al. (2014) and…...
[...]
...Consider a system formed by α independent components following the odd exponential-G class (Bourguignon et al. 2014) given by H(x; λ, ξ) = 1 − e−λ G(x;ξ) G(x;ξ) ....
[...]
TL;DR: A new class of distributions called the Lomax generator with two extra positive parameters to generalize any continuous baseline distribution is proposed, and the estimation of the model parameters by maximum likelihood is discussed.
113 citations
Cites background from "The Weibull-G Family of Probability..."
...[6] and exponentiated half-logistic-G by Cordeiro et al....
[...]
TL;DR: In this article, a new four-parameter lifetime model called the Weibull Frechet distribution is defined and studied, which can serve as an alternative model to other lifetime distributions in the existing literature.
Abstract: A new four-parameter lifetime model called the Weibull Frechet distribution is defined and studied. Various of its structural properties including ordinary and incomplete moments, quantile and generating functions, probability weighted moments, Renyi and δ-entropies and order statistics are investigated. The new density function can be expressed as a linear mixture of Frechet densities. The maximum likelihood method is used to estimate the model parameters. The new distribution is applied to two real data sets to prove empirically its flexibility. It can serve as an alternative model to other lifetime distributions in the existing literature for modeling positive real data in many areas.
103 citations
Cites methods from "The Weibull-G Family of Probability..."
...[5] replaced the argument x by G(x; θ)/Ḡ(x; θ), where Ḡ(x; θ) = 1 − G(x; θ), and defined the cdf of the Weibull-G family by...
[...]
...[5], we construct the four-parameter WFr model and give a comprehensive description of some of its mathematical properties....
[...]
References
More filters
Journal Article•
TL;DR: Copyright (©) 1999–2012 R Foundation for Statistical Computing; permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and permission notice are preserved on all copies.
Abstract: Copyright (©) 1999–2012 R Foundation for Statistical Computing. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that this permission notice may be stated in a translation approved by the R Core Team.
272,030 citations
TL;DR: The frequent opportunities I have had of receiving pleasure from your writings and conversation, have induced me to prefer offering to the Royal Society through your medium, this Paper on Life Contingencies, which forms part of a continuation of my original paper on the same subject, published among the valuable papers of the Society, as by passing through your hands it may receive the advantage of your judgment.
Abstract: Dear Sir, The frequent opportunities I have had of receiving pleasure from your writings and conversation, have induced me to prefer offering to the Royal Society through your medium, this Paper on Life Contingencies, which forms part of a continuation of my original paper on the same subject, published among the valuable papers of the Society, as by passing through your hands it may receive the advantage of your judgment.
3,257 citations
TL;DR: In this article, a three-parameter generalized exponential distribution (GED) was used for analysis of lifetime data, which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar et al.
Abstract: Summary The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.
1,084 citations
"The Weibull-G Family of Probability..." refers methods in this paper
...Its fit is also compared with the widely known exponentiated Weibull (EW) (Mudholkar and Srivastava, 1993) and exponentiated exponential (EE) (Gupta and Kundu, 1999) models with corresponding densities: EW : fEW(x;α, β, λ) = αβ λ β xβ−1e−(λx) β ( 1− e−(λx)β )α−1 , x > 0, EE : fEE(x;α, λ) = αλ e −λx…...
[...]
...Its fit is also compared with the widely known exponentiated Weibull (EW) (Mudholkar and Srivastava, 1993) and exponentiated exponential (EE) (Gupta and Kundu, 1999) models with corresponding densities:...
[...]
TL;DR: In this article, a simple generalization of the Weibull distribution is presented, which is well suited for modeling bathtub failure rate lifetime data and for testing goodness-of-fit of the weibull and negative exponential models as subhypotheses.
Abstract: A simple generalization of the Weibull distribution is presented. The distribution is well suited for modeling bathtub failure rate lifetime data and for testing goodness-of-fit of the Weibull and negative exponential models as subhypotheses. >
1,028 citations
"The Weibull-G Family of Probability..." refers methods in this paper
...Its fit is also compared with the widely known exponentiated Weibull (EW) (Mudholkar and Srivastava, 1993) and exponentiated exponential (EE) (Gupta and Kundu, 1999) models with corresponding densities: EW : fEW(x;α, β, λ) = αβ λ β xβ−1e−(λx) β ( 1− e−(λx)β )α−1 , x > 0, EE : fEE(x;α, λ) = αλ e −λx…...
[...]
...Its fit is also compared with the widely known exponentiated Weibull (EW) (Mudholkar and Srivastava, 1993) and exponentiated exponential (EE) (Gupta and Kundu, 1999) models with corresponding densities:...
[...]