Some Generalized Families of Weibull Distribution: Properties and Applications

Abstract

The Weibull distribution has been applied in various fields, especially to fit life time data. Some of these applications are limited partly due to the fact that the distribution has monotonically increasing, monotonically decreasing or constant hazard rate. This limitation undoubtedly inspired researchers to develop generalized Weibull distribution that can exhibit unimodal or bathtub hazard rate. In this article, we introduce six new families of T-Weibull{Y} distributions arising from the quantile function of a random variable Y. These six families are: The T-Weibull{uniform}, T-Weibull{exponential}, T-Weibull{log-logistic}, T-Weibull{Fré chet}, T-Weibull{logistic} and T-Weibull{extreme value}. Some properties of these families are discussed and general expressions for the quantile function, the Shannon's entropy, the non-central moments and the mean deviations are provided. Different new members of the T-Weibull{Y} families are derived and some of their properties are discussed. Two real data sets are used to illustrate the potential usefulness of the T-Weibull{Y} distributions and the results are compared with the results from some existing distributions.

A Generalization of the Weibull Distribution with Applications

“…The result follows directly by using part (iii) of Lemma 2 in Almheidat et al (2015) when the random variable T follows a Lomax distribution. The closed form quantile function in (11) makes simulating the LWD random variates straightforward.…”

“…Eugene, Lee, and Famoye (2002) introduced the beta-generated family and some properties of the family were studied by Jones (2004). Many beta-generated distributions were studied (e.g., Eugene et al, 2002;Nadarajah & Kotz, 2004;Famoye, Lee, & Eugene, 2004;Famoye, Lee, & Olumolade, 2005;Nadarajah & Kotz, 2006;Akinsete, Famoye, & Lee, 2008;Barreto-Souza, Santos, & Cordeiro, 2010;Mahmoudi, 2011;Alshawarbeh, Lee, & Famoye, 2012). For a review of betagenerated distributions and other generalizations, see Lee, Famoye, and Alzaatreh (2013).…”

“…Setting β = 1 = θ and taking T in (4) to be a Lomax random variable with CDF FT(x) = 1 -(1 + (x/θ)) -α , Almheidat et al (2015) defined the Lomax-Weibull{LL} distribution (LWD) as an example of T-Weibull{LL} family.…”

“…Almheidat, Famoye, and Lee (2015) used the T-R{Y} framework to define and study different approaches to the generalization of the Weibull distribution, the T-Weibull{Y} family. The authors defined the T-Weibull{Y} family by taking R in (2) to be a Weibull random variable with CDF   …”

“…Alzaatreh, Lee, and Famoye (2013) studied in details the case when W(F(x)) = -log(1 -F(x)). Some members of the family have been investigated, including gamma-Pareto distribution (Alzaatreh, Famoye, & Lee, 2012), WeibullPareto distribution , and gamma-normal distribution (Alzaatreh, Famoye, & Lee, 2014a). Aljarrah, Lee, and Famoye (2014) used the quantile function QY of a random variable Y to define the transformation W(.)…”