Abstract
A new G family of probability distributions called the type I quasi Lambert family is defined and applied for modeling real lifetime data. Some new bivariate type G families using "Farlie-Gumbel-Morgenstern copula", "modified Farlie-Gumbel-Morgenstern copula", "Clayton copula" and "Renyi's entropy copula" are derived. Three characterizations of the new family are presented. Some of its statistical properties are derived and studied. The maximum likelihood estimation, maximum product spacing estimation, least squares estimation, Anderson-Darling estimation and Cramer-von Mises estimation methods are used for estimating the unknown parameters. Graphical assessments under the five different estimation methods are introduced. Based on these assessments, all estimation methods perform well. Finally, an application to illustrate the importance and flexibility of the new family is proposed.
Highlights
In mathematics, the "Lambert function", called the "omega function" or "product logarithm", is a multivalued function, namely the branches of the inverse relation of the function f(W) = W exp(W) where W is any complex number and exp(W) is the exponential function
The cumulative distribution function (CDF) of the type I quasi Lambert (TIQL) family can be expressed as FΦ(x) = WΦ(x) exp[WΦ(x)] |x∈R, (1)
A copula is a multivariate CDF for which the marginal probability distribution of each variable is uniform on the interval [0,1] . copulas are used to describe the dependence between random variables
Summary
Et al (2018a)), exponential Lindley odd log-logistic G family (Korkmaz et al (2018b)), Marshall-Olkin generalized-. (2018a)), Marshall-Olkin generalized-G family (Yousof et al (2018b)), Burr-Hatke G family (Yousof et al (2018c)), Type I general exponential class of distributions (Hamedani et al (2017)), new extended G family (2018)), Type II general exponential class of distributions (Hamedani et al (2019)), exponential Lindley odd loglogistic-G family (Korkmaz et al (2018b)), dd power Lindley generator of probability distributions The PDF of the TIQL family can be expressed as a mixture of exp-G PDFs as fΦ(x) = ∑ cj,κ πκ∗(x; Ψ), j,κ=0 where πκ∗(x; Ψ) = dΠκ∗(x; Ψ)/dx s the PDF of the exp-G family with power parameter κ∗ > 0.
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