A new three parameter Fréchet model with mathematical properties and applications

Abstract

A new three-parameter extension of the Fréchet model is proposed and studied. Some of its statistical properties are derived. A simple type Copula-based construction via Morgenstern family and via Clayton copula is used to derive many bivariate and multivariate extensions of the new model. We assessed the performance of the maximum likelihood estimators using a simulation study. The importance of the new model is shown via two applications to real data sets.

New Extension of Weibull Distribution: Copula, Mathematical Properties and Data Modeling

“…This data are given in Appendix(d).These data have also been analyzed by Smith and Naylor [51].For this data set,we shall compare the fits of the new distribution with some competitive models like OLEW,E-W,T-W.Based on the Table 9 we conclude that the proposed MOL-W model is the best model with CV M statistic = 0.10565 and AD statistic = 0.59106. Many other useful version can be used in more comparisons see Al-Babtain et al [1],Al-Babtain et al [2],Ibrahim et al [24],Ibrahim and Yousof [25], Ibrahim and Yousof [26]Ibrahim et al [27], Alshkaki [6]and Esmaeili et al [16]. The HRF of the MOL-W distribution exhibits "constant hazard rate (α = 1, β=1, a 1 = 1, a 2 = 1)", "upside down-constant (α = 0.5, β=0.5, a 1 = 1.01, a 2 = 1)", "decreasing hazard rate (α = 0.5, β=5, a 1 = 1, a 2 = 0.2)", "increasing-constant hazard rate (α = 0.5, β=0.15, a 1 = 1.25, a 2 = 1)", "increasing hazard rate (α = 2, β=1, a 1 = 1.5, a 2 = 1)", "J-hazard rate (α = 0.5, β=1, a 1 = 20, a 2 = 1)" and "decreasing hazard rate (α = 0.2, β=1, a 1 = 0.1, a 2 = 1)".…”

“…By means of two applications, it is noted that the MOL-W model provides better fits than other models each having the same number of parameters. By inserting (1) in 3, we obtain the cumulative distribution function (CDF) of the MOL-G class | (y≥0 and a1,a2,α,β>0) ,…”

The Three-parameters Marshall-Olkin Generalized Weibull Model with Properties and Different Applications to Real Data Sets

“…( 1 , 2 , … , ) represents the time to the first failure with CDF (5).Form another view, consider now a parallel system with independent components and suppose that a random variable has geometric distribution with the probability mass function ( = ) = −1̇−1 , = 1,2, . .…”

“…, represents the lifetime of the system. Therefore, the random variable follows (5). The reliability function (rf), hazard rate function (HRF) and cumulative hazard rate function (cHRF) of are, respectively, given by ( ) =…”

A new exponentiated Weibull distribution's extension: copula, mathematical properties and applications

“…It is the conditional probability distribution of a Poisson-distributed random variable (RV), given that the value of the RV is not zero. The probability mass function (PMF) of M is given by P (λ=1) [1] exp (−1)| (m=1,2,...) ,…”