A New Version of the Exponentiated Exponential Distribution: Copula, Properties and Application to Relief and Survival Times

In this paper, 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 is performed. The new density can be right skewed and symmetric with “unimodal” and “bimodal” shapes. The new hazard function can be “constant”, “decreasing”, “increasing”, “increasing-constant”, “upsidedown-constant”, “decreasingconstant”. 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 usefulness and flexibility of the new distribution is illustrated by means of two real data sets.


Introduction and motivation
A random variable (RV) W is said to have the Exponentiated Exponential (EE) distribution if its probability density function (PDF) is given by π a,b (w)| (w>0,a>0 and b>0) = ab e −bw ( 1 − e −bw ) a−1 .
The corresponding cumulative distribution function (CDF) can be written as W a,b (w)| (w>0,a>0 and b>0) = Clearly, for a = 1, the EE reduces to the standard exponential (E) model. If 1 > a, the function π a,b (w) monotonically decreases with w. If a > 1, the function π a,b (w) attains a mode at w = 1 b log (a) . The statistical properties of the EE model have been studied by many authors. Many authors have derived and studied the EE model, see Zheng [66], Zheng and Park [67], Kundu and Pradhan [48], Aslam et al. [15], Aryal et al. [13], Khalil et al. [42], Abouelmagd et al. ( [1], [2]), Ibrahim et al. [35] and Bhatti et al. [16] among others. Recently, Alizadeh et al. [6] defined a new family based on the exponential model called the generalized odd generalized exponential family of distributions. Analogously, Hamedani et al. ( [33] and [34]) defined the the type I and type I general exponential class of distributions. Other works can be cited such as Korkmaz et al. [46] (exponential Lindley odd log-logistic family) and Yadav et al. [62] (Burr-Hatke exponential model). In the work, we introduce a new version of the EE model using the Odd-Burr generalized (OB-G) family called the OBEE (OBEE). On the other hand, some new bivariate type OBEE are derived. Due to Alizadeh et al. [6], the CDF of the the OB-G family is given by (3) where W Φ (w) = 1 − W Φ (w) and Φ refers to the parameter vector of the base line model. The PDF corresponding to (3) is given by where π Φ (w) = dW Φ (w) /dx . For β = 1, we get the Odd G (O-G) family. For α = 1, we have the proportional reversed hazard rate family (PRHR). The OBEE CDF is given by where Λ refers to the parameter vector of the new OBEE model. For β = 1, the OBEE reduces to the OEE. For α = 1, the OBEE reduces to the PRHREE. The PDF corresponding to (5) is given by whereȧ = aα − 1. The hazard function (HRF) can be derived from f Λ (w) /S Λ (w). For simulation of this new model, we obtain the quantile function (QF) of w (by inverting (5)), say w u = F −1 (u), as Equation (7) is used for simulating the OBEE model. Figure 1 gives some PDF plots for some selected parameters value. Figure 2 gives some HRF plots for some selected parameters value. Based on Figure 1 the OBEE density can be right skewed and symmetric with unimodal and bimodal PDFs. Based on Figure 2 the OBEE HRF can be"constant", "decreasing", "increasing", "increasing-constant", "upside-downconstant" and "decreasingconstant".

Useful representations
Due to Alizadeh et al. (2016), the PDF in (6) can be expressed as where a * = a (1 + κ) and and π a * ,b (w) refers to the density of the exponentiated exponential (EE) model with power parameter a * . By integrating (8), the CDF of W becomes where Π a * ,b (w) refers to the EE distribution with power parameter a * .

Asymptotics
The asymptotics of CDF, PDF and HRF as w → ∞ are given by and

Moments and incomplete moments
The ς th ordinary moment of W is given by then we obtain where where E(w) = µ ′ 1 is the mean of w. The ς th incomplete moment, say φ ς (t), of w can be expressed, from (9), as where γ (ζ, ϑ) is the incomplete gamma function.

Moment generating function (MGF)
The MGF M W (t) = E (exp (t W )) of W can be derived from equation (8) as where M a * ,b (T ) is the MGF of the EW model with power parameter a * .

Residual life and reversed residual life functions
The ρ th moment of the residual life The ρ th moment of the residual life of W is given by Therefore, The ρ th moment of the reversed residual life, say uniquely determines F Λ (w). We obtain Then, the ρ th moment of the reversed residual life of W becomes Table 1 gives Numericals results for the variance (V(Z)), mean (E (Z)), kurtosis (K(Z)), skewness (S(Z)) and dispersion index (DisIx(Z)). Based on Table 1, we note that: 1-The skewness of the OBEE distribution can range in the interval (−2.7792, 8.2978). 2-The spread for the OBEE kurtosis is much larger ranging from −46.275 to 35.526. 3-DisIx(Z) can be "between 0 and 1" or "equal 1" or more than 1.

BivariateOBEE-FGM (Type II) model Consider
The bivariate OBEE-FGM (Type-II) copula can be derived from

The bivariate OBEE via Renyi's entropy
Following Pougaza and Djafari [58], the joint CDF of the bivariate OBEE via Renyi's entropy can be written as then, the associated bivariate OBEE will be Then, we get the BOBEE type distribution via Renyi's entropy.

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

Maximum likelihood method
For getting the maximum likelihood estimates (MLE) of the vector Λ, we have the log-likelihood (ℓ(Λ)) function The components of the score vector ∂ℓ/∂Λ = U (Λ) = (∂ℓ/∂α, ∂ℓ/∂β, ∂ℓ/∂a, ∂ℓ/∂b) 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 OBEE distribution.

Simulations
In statistics, simulation is usually used for assessing the performance of a method, typically when there is a lack of theoretical background. 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: 1. Using the inversion method, we we generate N=1000 samples of size n from the OBEE distribution using (7). The StErs were computed by inverting the observed information matrix. β, a, b). 5. Repeated these steps for n = 50, 100, . . . , 300 with α = 1, 2, .., 100, β = 1, 2, .., 100, a = 1, 2, .., 100 and b = 1, 2, .., 100, so computing biases (  These figures (lefts) shows how the four biases vary with respect to n and also shows how the four MSEs vary with respect to n. From Figures 3, 4, 5 and 6, the biases for each parameter are generally negative and getting close to zero as n → ∞, the MSEs for each parameter decrease to zero as n → ∞.

Real data applications
We shall compare thefits of the OBEE distribution with those of other competitive models, namely: Exponential    [43] and Al-Babtain et al. [3]. For comparing models, we consider the Cramér-Von Mises (C · ) and the Anderson-Darling (A · ) and the Kolmogorov-Smirnov (KS) statistic. Moreover and for more accuracy, we consider another five goodness-of-fit measures: the Akaike Information Criterion (AIC) (C 1 ), Bayesian IC (C 2 ), Consistent AIC (C 3 ), Hannan-Quinn IC (C 4 ).

conclusions
In this article, we introduced and studied a new flexible version of the exponentiated exponential model called the odd Burr exponentiated exponential (OBEE) model. The new density can be right skewed and symmetric with unimodal and bimodal shapes. The new HRF can be "constant", "decreasing", "increasing", "increasingconstant", "upside-down-constant", "decreasing-constant". Some of its mathematical properties including the ordinary moments, incomplete moment, moment generating function are derived. Numerical calculations for the expected value, skewness, variance, kurtosis and the index of dispersion is presented.  Figure 10. The box plot, Q-Q plot and TTT plot for the survival times data. Figure 11. E-PDF, E-CDF and E-HRF for survival times data.
1" or more than 1. Some bivariate and multivariate OBEE type model have been also derived. Estimation of OBEE parameters is performed by maximum likelihood estimation method. We assessed the performance of the maximum likelihood method. The assessment can be performed graphically 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 other competitive 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 work, we can apply the Bagdonavičius-Nikulin goodness-of-fit test, Figure 12. Kaplan-Meier survival plot and P-P plot for survival times data.