Abstract
A new distribution called the Weibull Generalized Burr XII distribution is introduced along with simple physical motivation. The new distribution includes fourteen sub-models, seven of them are new. The new model can be used in modeling bimodal data sets. Set of its properties are derived in details. Two applications are provided to illustrate the importance of the new model. The new model is better that other competitive models via two applications. The method of maximum likelihood is used to estimate the unknown parameters.
Highlights
Introduction and physical motivationA special attention has been devoted to one of twelve models introduced by Burr (1942)
G(a,b)(x) = 1 − (1 + xa)−b, both a and b are shape parameters, when a = 1 the Burr type XII (BXII) model reduces to the Lomax (Lx) or Pareto type II (PaII) model and when b = 1 the BXII model reduces to the log-logistic (LL) model
We compare the Weibull Generalized-BXII (WGBXII) distribution, with BXII, Marshall-Olkin BXII (MOBXII), Topp Leone BXII (TLBXII), Zografos-Balakrishnan BXII (ZBBXII), Five Parameters beta BXII (FBBXII), BBXII, B exponentiated BXII (BEBXII), Five Parameters Kumaraswamy BXII (FKwBXII) and KwBXII distributions given in Afify et al (2018), Yousof et al (2018a, b), Altun et al (2018a, b) and Yousof et al (2019)
Summary
A special attention has been devoted to one of twelve models introduced by Burr (1942). Denoted by type XII (see Burr (1942), (1968) and (1973), Burr and Cislak (1968), Hatke (1949) and Rodriguez (1977)), the cumulative distribution function (CDF) of Burr type XII (BXII) is given as. Let g(a,b)(x) and G(a,b)(x) denote the PDF and the CDF of the BXII model with parameter vector ξ = (a, b). The CDF of the Weibull Generalized-BXII (WGBXII) based on Yousof et al (2018b) is defined by. The additional parameters β and θ are sought as a manner to furnish a more flexible BXII distribution (see Figure 1). The PDF of the WGBXII model can be expressed as f(x) = ∑∞r=0 υrg(a,(1+r)b)(x). Consider that the variability of this ratio of death is represented by the r.v. X and assume that it follows the Weibull model with shape γ.
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