A New Lifetime Parametric Model for the Survival and Relief Times with Copulas and Properties

In this article, we introduce a new generalization of the Exponentiated Exponential distribution. Various structural mathematical properties are derived. Numerical analysis for mean, variance, skewness and kurtosis and the dispersion index are performed. The new density can be right skewed and symmetric with "unimodal" and "bimodal" shapes. The new hazard function can be "constant", "monotonically decreasing", " monotonically increasing", "increasing-constant”, “upside-down-constant", "decreasing-constant". Many bivariate and multivariate type model have been also derived. We assess the performance of the maximum likelihood method graphically via the biases and mean squared errors. The applicability of the new life distribution is illustrated by means of two real data sets.


Incomplete moment, MGF and dispersion index
The ℎ incomplete moment, say , ( ) , of can be expressed from (7) as where ( 1 , 2 ) is the incomplete gamma function such that .
The MGF can be derived via (7) as Based on Gupta and Kundu (2001a) and Gupta and Kundu (2007), another formula for the MGF can be derived as is the beta function. So, the characteristic function (CF) of can be easily written as The dispersion index (DisIx) or the variance to the mean ratio of the GOLLEE model can derived as ]. ( ) can be "between 0 and 1 " or "equal 1 " or more than 1.

Moments of residual life (MRL)
The ℎ MRL is Therefore, ).

Moments of the reversed residual life (MRRL)
The ℎ MRRL is Then, the ℎ MRRL of becomes

Maximum likelihood method
Let 1 , … , be a random sample from the GOLLEE distribution with parameters , , and . Let be the 4 × 1 parameter vector. For getting the maximum likelihood estimates (MLE) of , we have the log-likelihood (ℓ) function where , = 1 − − . The score vector components are available if needed. We can compute the maximum values of the unrestricted and restricted log-likelihoods to obtain likelihood ratio (LR) statistics for testing some sub-models of the GOLLEE distribution.
The maximum likelihood estimators and the Bayesian estimators are equivalent asymptotically (Ibragimov (1962) and Chao (1970)), that can be expressed as A direct result of the above theorem is that all asymptotic properties of the MLEs also hold for the Bayesian estimators. Also, since the determination of the MLE is independent of the loss function and the prior measure, the asymptotic properties of Bayesian estimators hold for all priors and loss functions in a certain class. .

Bivariate GOLLEE-FGM (Type II) model:
Consider the following functional form for both ( ) and ( ) which satisfy all the conditions stated earlier where The corresponding bivariate copula can then be derived from Assume that ∼ GOLLEE ( 1 ) and ∼ GOLLEE ( 2 ). Then, setting = =

The Multivariate GOLLEE extension
A straightforward -dimensional extension from the above will be

Simulations
In this Section, we assess the performance of the maximum likelihood (ML) method. The assessment can be performed numerically or graphically. Graphically, we can perform the simulation experiments to assess of the finite sample behavior of the ML estimators (MLEs) via the biases and mean squared errors (MSEs). The following algorithm is considered for the assessment:     Hannan-Quinn Information Criteria ( 4 ), the Cramér-Von Mises ( * ) and the Anderson-Darling ( * ). Additionally, the Kolmogorov-Smirnov (K.S) test and its corresponding p-value is also performed.

Modeling failure (relief) times
The first data set called the failure or relief times (in minutes). This data represents the lifetime data relating to relief times of patients receiving an analgesic (see Gross and Clark (1975)). This data was recently analyzed by  and Al-Babtain et al. (2020). Table 2 lists the MLEs, SEs confidence intervals (C.I.s). Table 3 lists the 1 , 2 , 3 , 4 * , * , K.S. and p-value. Figure 7 gives the total time in test test (TTT) plot for the relief times data along with the corresponding box plot. Based on Figure 7, the HRF of the relief times is "increasing HRF" and this data has only one EV observation. Figure 8 gives the estimated PDF, estimated CDF, estimated HRF and P-P plot for relief times data. Figure 9 below gives Kaplan-Meier survival plot for relief times data. Based on Table 3, we conclude that the proposed lifetime GOLLEE model is much better than the "Exponential", "Odd Lindley-E", "Marshall-Olkin-E", "Moment-E", "Burr-Hatke-E", "generalized Marshall-Olkin-E", "Beta-E", "Marshall-Olkin-Kumaraswamy-E", "Kumaraswamy-E", "Burr type X-E" and "Kumaraswamy-Marshall-Olkin-E" models with 1 = 39.62, 2 = 43.60,     The proposed GOLLEE lifetime model is much better than the E, ME, MOE, GzMOE, KrE, BE, MOKE, KMOE models.

Modeling survival times
The second data set called the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal (1960). This data was recently analyzed by Ibrahim et al. (2020) and Al-Babtain et al.
(2020). Table 4 lists the MLEs, SEs confidence intervals (C.I.s). Table 5 lists the 1 , 2 , 3 , 4 , * , * , K.S. and p-value. Figure 10 gives the TTT plot along with the corresponding box plot for the survival times data. Based on Figure 10, the HRF of the survival times is "increasing HRF" and this data has only four EV observations. Figure 11 gives the estimated PDF, estimated CDF, estimated HRF and P-P plot survival times data. Figure 12 gives the Kaplan-Meier survival plot survival times data. Based on Table 5, we conclude that the GOLLEE model is much better than the Exponential, Odd Lindley Exponential, Marshall-Olkin Exponential, Moment Exponential, The Burr-Hatke Exponential, generalized Marshall-Olkin Exponential, Beta Exponential, Marshall-Olkin Kumaraswamy Exponential, Kumaraswamy Exponential, the Burr X Exponential and Kumaraswamy Marshall-Olkin Exponential models with 1 = 204.44, 2 = 213.54, 3 = 205.03, 4 = 208.06, * = 0.35, * = 0.058, K.S=0.0723 and p-value=0.846 so the new lifetime model is a good alternative to these models in modeling relief times data set. According to Figures 11 and 12, the GOLLEE distribution provides adequate fits to the empirical functions.    Figure 11: Estimated PDF, estimated CDF, estimated HRF and P-P plot survival times data.

Conclusions and future works
In this article, we introduced and studied a new generalization of the exponentiated exponential distribution. Various structural mathematical properties including explicit expressions for the moment generating function, the ordinary moments, incomplete moment are derived. Numerical analysis for mean, variance, skewness and kurtosis and the dispersion index is performed for illustrating the importance and flexibility of the new model. Many bivariate and multivariate type extensions have been also derived. The estimation of the model parameters is performed by maximum likelihood method. The new density can be right skewed and symmetric with unimodal and bimodal shapes.
The new hazard function can be "constant", "monotonically decreasing", " monotonically increasing", "increasingconstant", "upside-down-constant", "decreasing-constant". We assessed the performance of the maximum likelihood estimation method using a graphical simulation study via the biases and mean squared errors. The usefulness and flexibility of the new distribution is illustrated by means of two real data sets. The new model is much better than many useful models in modeling relief times and survival times data sets according to the Akaike Information Criterion, the Consistent Akaike Information Criterion, the Hannan-Quinn Information Criterion, the Bayesian Information Criterion, the Cramér-Von Mises, the Anderson-Darling statistics.
As a future related work, authors can apply many new useful goodness-of-fit tests for right censored validation such as the Nikulin-Rao-Robson goodness-of-fit test, modified Nikulin-Rao-Robson goodness-of-fit test, Bagdonavicius-Nikulin goodness-of-fit test, modified Bagdonavicius-Nikulin goodness-of-fit test, to the new BuXENH model as