Abstract
A new version of the Lomax model is introduced andstudied. The major justification for the practicality of the new model isbased on the wider use of the Lomax model. We are also motivated tointroduce the new model since the density of the new distribution exhibitsvarious important shapes such as the unimodal, the right skewed and the leftskewed. The new model can be viewed as a mixture of the exponentiated Lomaxdistribution. It can also be considered as a suitable model for fitting thesymmetric, left skewed, right skewed, and unimodal data sets. The maximumlikelihood estimation method is used to estimate the model parameters. Weprove empirically the importance and flexibility of the new model inmodeling two types of aircraft windshield lifetime data sets. The proposedlifetime model is much better than gamma Lomax, exponentiated Lomax, Lomaxand beta Lomax models so the new distribution is a good alternative to thesemodels in modeling aircraft windshield data.
Highlights
IntroductionA random variable (rv) W has the Lomax (Lx) distribution with two parameters λ and β if it has cumulative distribution function (CDF) (for w > 0) given by
A random variable W has the Lomax (Lx) distribution with two parameters λ and β if it has cumulative distribution function (CDF) given by Gλ,β (w) = − ( 1 + wβ−1)−λ (1)where λ > 0 and β > 0 are the shape and scale parameters, respectively
The proposed lifetime model is much better than gamma Lomax, exponentiated Lomax, Lomax and beta Lomax models so the new distribution is a good alternative to these models in modeling aircraft windshield data
Summary
A random variable (rv) W has the Lomax (Lx) distribution with two parameters λ and β if it has cumulative distribution function (CDF) (for w > 0) given by. Where θ is the shape parameter, g(x; ξ) and G(x; ξ) denote the PDF and CDF of the baseline model with parameter vector ξ and 1 − G(x; ξ) = G(x; ξ). We are motivated to introduce the new model since the PDF of the new distribution exhibits various important shapes such as the unimodal, the right skewed and the left skewed (see figurue 1). The new model can be viewed as a mixture of the exponentiated Lx distribution (see Subsection 2.1). It can be considered as a suitable model for fitting the symmetric, left skewed, right skewed, and unimodal data sets (see aplications Section). We prove empirically the importance and flexibility of the new model in modeling two types of aircraft windshield lifetime data sets. The reliability function (RF) (Rθ,λ,β(x)), hazard rate function (HRF) (hθ,λ,β(x)), reversed hazard rate function (RHRF) (rθ,λ,β (x)) and cumulative hazard rate function (CHRF) (Hθ,λ,β(x)) of X are given, respectively, by
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