A Generalization of Lomax Distribution with Properties, Copula and Real Data Applications

A new generalization of Lomax distribution is derived and studied. Some of its useful properties are derived. A simple clayton copula is used to generate many bivariate and multivariate type models. We performed graphical simulations to assess the finite sample behavior of the estimations. The new model is employed in modelling three real data sets.


Ordinary moment
The th ordinary moment of is given by Then, (7) Setting = 1 in (7), we have the mean of . The effects of the parameters , , on the mean ( 1 ′ ), variance (V ( )), skewness (S ( ) ) and kurtosis (K ( ) ) for given values are listed in Table 1
The first Ic-M of , referred to 1 ( ), is just determined from the above equation by setting = 1 . The first Ic-M has main applications related to the Bonferroni and Lorenz curves and the mean residual life and the mean waiting time. Moreover, the amount of scattering in a population is clearly measured, to some extent, by the totality of deviations from the mean and median.

Residual and reversed residual life functions
The . Then, we can write The th moment reversed residual life, say . The th moment of the reversed residual life of [ℓ 4 ] * ( ; * − , 1 + ).

BGOLLLx type via FGM Copula
A Copula is continuous in and ; actually, it satisfies the "stronger Lipschitz condition", where .

BGOLLLx and MvGOLLLx type via Clayton Copula
The "Clayton Copula" can be considered as

BGOLLLx type via modified FGM Copula
The modified version of the bivariate FGM copula defined as (Rodriguez-Lallena and Ubeda-Flores (2004)

5.Graphical simulations
To assess of the finite sample behavior of the MLEs, consider the following algorithm:  . The broken lines in Figure 2 corresponds to the biases being 0 . From Figure 2, the biases for each parameter decrease to zero as → ∞ , the MSEs for each parameter decrease to zero as → ∞ .

Comparing models
To illustrate the flexibility of the GOLLLx model, we provide three applications. The 1 st data set called breaking stress data. This data set consists of 100 observations of breaking stress of carbon fibrrs (in Gba) given by Nichols and Padgett (2006). The 2 nd data set called survival times. In this application, we work with the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, originally observed and reported by Bjerkedal, T.  Figure 3 gives the TTT plots. Based on Figure 3, the HRF of the three real data sets are increasing. Figure  4 gives the estimated PDFs. Figure 5 gives the estimated CDFs. Figure 6 gives the estimated HRFs. Figure 7 gives the P-P plots. Figure 8 gives

Conclusions
A new generalization of Lomax distribution is derived and studied. The new extension has only three parameters. Some of its useful mathematical properties are derived. We performed graphical simulations to assess the finite sample behavior of the estimations. The effects of all parameters on the mean, variance, skewness and kurtosis for given values are studied. we note that the new additional shape parameters and has an effect on the mean, variance, skewness and kurtosis. For the new Lx model, skewness can range in the interval (−183.1, 7514.7). However, for the standard Lx model, skewness can range in the interval (− 0.4104, 4.6476). For the GOLLLx model, kurtosis can range in the interval (− 1531.11, 56479275). However, for the standard Lx model, kurtosis can range in the interval (0.93244, 73.8). The new model is employed in modelling three real data sets. For all data sets, we compared the new Lx distribution with the standard Lx, the exponentiated Lx, the Burr XII, beta Burr XII, the Marshall-Olkin Burr XII, the Topp-Leone Burr XII, the Zografos-Balakrishnan Burr XII, beta exponentiated Burr XII, the five-parameters beta Burr XII, the five-parameters Kumaraswamy Burr XII and Kumaraswamy Burr XII distributions. The new Lx distribution is a useful alternative for the above-mentioned models in modeling breaking stress data, survival times of guinea pig's data and the Egyptian taxes revenue data.