A New Lifetime Model: Copulas, Properties and Real Lifetime Data Applications

A new four parameter lifetime model called the Weibull generalized Lomax is proposed and studied. The new density function can be "right skewed", "symmetric" and "left skewed" and its corresponding failure rate function can be "monotonically decreasing", " monotonically increasing" and "constant". The skewness of the new distribution can negative and positive. The maximum likelihood method is employed and used for estimating the model parameters. Using the "biases" and "mean squared errors", we performed simulation experiments for assessing the finite sample behavior of the maximum likelihood estimators. The new model deserved to be chosen as the best model among many well-known Lomax extension such as exponentiated Lomax, gamma Lomax, Kumaraswamy Lomax, odd log-logistic Lomax, Macdonald Lomax, beta Lomax, reduced odd log-logistic Lomax, reduced Burr-Hatke Lomax, special generalized mixture Lomax and the standard Lomax distributions in modeling the "failure times" and the "service times" data sets.

The WG-Lx model could be useful in statistical modeling in following cases: 1-The real-life data sets with " monotonically increasing HRF (asymmetric increasing HRF)".

2-
The real-life data sets which have no extreme observations.

3-
The real-life data sets which their nonparametric Kernel density estimation is bimodal and symmetric with right and left simple tail.
The WG-Lx model proved its applicability against many well-known Lx extensions in following cases: In statistical modeling of the failure times of the aircraft windshield observations, the WG-Lx model is better  than many well-known Lx extension such as the McDonald Lx extension, the special generalized mixture Lx  extension, the odd log-logistic Lx extension, the Gamma Lx extension, the Burr-Hatke Lx extension, the  transmuted Topp-Leone Lx extension, the exponentiated Lx extension, the proportional reversed hazard rate Lx extension and the Kumaraswamy Lx extension under the Akaike-Information-Criteria, Consistent-Information-Criteria, Bayesian-Information-Criteria and Hannan-Quinn-Information-Criteria.

2-
In modeling the service times of the aircraft windshield, the WG-Lx model is better than many well-known Lx extension such as the McDonald Lx extension, the special generalized mixture Lx extension, the odd loglogistic Lx extension, the Gamma Lx extension, the Burr-Hatke Lx extension, the transmuted Topp-Leone Lx extension, the exponentiated Lx extension, the proportional reversed hazard rate Lx extension and the Kumaraswamy Lx extension under the Akaike-Information-Criteria, Consistent-Information-Criteria, Bayesian-Information-Criteria and Hannan-Quinn-Information-Criteria.

BvWG-Lx-FGM (Type I) model
Here, we consider the following functional form for both ( ) and ( ) as

BvWG-Lx-FGM (Type III) model
Consider the following functional form for both ( ) and ( ) which satisfy all the conditions stated earlier where The corresponding bivariate copula (henceforth, BvWG-Lx-FGM (Type III) copula) can be derived from and are two absolutely continuous functions on (0,1). Then, the associated BvWG-Lx will be ( 1 , 2 ) = − 1 2

BvWG-Lx type via Clayton Copula
The Clayton Copula can be considered as . Let us assume that ∼ WG-Lx ( 1 ) and ∼ WG-Lx ( 2 ) . Then, setting Then, the BvWG-Lx type distribution can be derived as

MvWG-Lx extension via Clayton Copula
A straightforward -dimensional extension from the above will be Then, the MvWG-Lx extension can be expressed as

3.Mathematical properties Useful representations
Due to , the PDF in (6) can be expressed as and (1+ ), , ( ) is the PDF of the Lx model with power parameter 1 + . By integrating Equation (8), the CDF of becomes where (1+ ), , ( ) is the CDF of the Lx distribution with power parameter 1 + .

Moments and incomplete moments
The ℎ ordinary moment of is given by , (1 + , − 1 + 1) | ( >1) , and is the mean of . The ℎ incomplete moment, say , ( ) , of can be expressed, from (9) as where The first incomplete moment given by (11) with = 1 as The index of dispersion IxDis or the variance ( 2 ) to mean ratio can derived as IxDis ( ) = 2 / 1 ′ . It is a measure used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard statistical model.

Numerical analysis
By analyzing the 1 ′ , 2 , skewness ( 1 ) , kurtosis ( 2 ) and IxDis ( ) numerically in Table 1, it is noted that, the 1 of the WG-Lx distribution can be negative and also positive. The spread for the 2 of the WG-Lx model is ranging from − 1129.85 to 311.698. The IxDis ( ) for the WG-Lx model can be in (0,1) and also > 1 so it may be used as an "under-dispersed" and "over-dispersed" model.

Some generating functions (GF)
The moment generating function (MGF) can be derived using (8)  The 1 st cumulant is the mean ( 1 = 1 ′ ), the 2 nd cumulant is the variance, and the 3 rd cumulant is the same as the 3 rd central moment 3 = 3 . But 4 th and higher order cumulants are not equal to central moments. In some cases, theoretical treatments of problems in terms of cumulants are simpler than those using moments. When two or more RVs are statistically independent, the ℎ order cumulant of their sum is equal to the sum of their ℎ order cumulants. Moreover, the cumulants can be also obtained from

5.Applications
In this section, we provide two real life applications to two real data sets to illustrate the importance and flexibility of the WG-Lx model. We compare the fit of the WG-Lx with some well-known competitive models (see Table 2)   Figure 6). Based on Figure 6, we note that no extreme values were found in the two real life data sets. For checking the normality, the Quantile-Quantile (QQ) plot is sketched (see Figure 7). Based on Figures 7, we note that the normality is nearly exists. For exploring the HRF for real data, the total time test (TTT) plot is provided (see Figure 8). Based on Figure 8, we note that the HRF is "monotonically increasing" for the two real life data sets. For exploring the initial shape of real data nonparametrically, kernel density estimation (KDE) is provided (see Figure 9). Figure 9 show nonparametric KDE for exploring the data. Figures 10 and 11 give, Probability-Probability (P-P) plot (top left), estimated PDF (EPDF) (top right), estimated CDF (ECDF) (bottom left) and estimated HRF (EHRF) (bottom right) for data set I and II respectively.    Tables 3 and 4. Table 3 gives the MLEs and standard errors (SEs) for failure times data. Table 4 gives the −l and goodness-of-fits statistics for failure times data. For service times data: the analysis results of are listed in Tables 5 and 6. Table 5 gives the MLEs and SEs for service times data. Table 6 give the −l and goodness-of-fits statistics for the service times data. Based on Tables 4 and 6, we note that the WG-Lx model gives the lowest values for the AIC, CAIC, BIC, HQIC, * and * among all fitted models. Hence, it could be chosen as the best model under these criteria.   Figure 10: P-P plot, EPDF, ECDF and EHRF for data set I.   Figure 11: P-P plot, EPDF, ECDF and EHRF for data set II.

6.Conclusions
A new four parameter lifetime model called the Weibull generalized Lomax (WG-Lx) is proposed and studied. The WG-Lx density function can be "right skewed", "symmetric", "left skewed" and "uniformed density". The WG-Lx failure rate function can be "monotonically decreasing", " monotonically increasing" and "constant". The new WG-Lx density can be expressed as a mixture of the exponentiated Lomax model. The skewness of the WG-Lx distribution can negative and positive. The spread for the kurtosis of the WG-Lx model is ranging from − 1129.85 to 311.698. The index of dispersion for the WG-Lx model can be in (0,1) and also > 1 so it may be used as an "under-dispersed" and "over-dispersed" model. The maximum likelihood method is used to estimate the WG-Lx parameters. Using the "biases" and "mean squared errors", we performed simulation experiments for assessing the finite sample behavior of the maximum likelihood estimators. It is noted that, the biases for all parameters are generally negative and tends to 0 as → ∞ and the mean squared errors for all parameter decrease to 0 as → ∞ . The WG-Lx model deserved to be chosen as the best model among many well-known Lomax extension such as exponentiated Lomax, gamma Lomax, Kumaraswamy Lomax, odd log-logistic Lomax, Macdonald Lomax, beta Lomax, reduced odd log-logistic Lomax, reduced Burr-Hatke Lomax, reduced WG-Lx, special generalized mixture Lomax and the standard Lomax distributions in modeling the "failure times" and the "service times" data sets. As a future potential work, the WG-Lx model can be validated using many new useful goodness-of-fit statistic tests in case of the right censored data sets such as the goodness-of-fit test of Nikulin-Rao-Robson (NRR) for right censored data, the modified NRR goodnessof-fit test for right censored data, the goodness-of-fit test of Bagdonavičius-Nikulin (BgN) for right censored data, modified BgN goodness-of-fit test for right censored data as recently performed by Ibrahim