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

This paper introduces a new flexible four-parameter lifetime model. Various of its structural properties are derived. The new density is expressed as a linear mixture of well-known exponentiated Weibull density. The maximum likelihood method is used to estimate the model parameters. Graphical simulation results to assess the performance of the maximum likelihood estimation are performed. We proved empirically the importance and flexibility of the new model in modeling four various types of data.


Moments
First we have where , using (7) and (8) the CDF of the MOL-G family in (5) can be expressed as where τ 0 = ζo η0 and for k 1 ≥ 1 we have , the PDF of the MOL-W model can also be expressed as a mixture of exponentiated W (Exp-W) PDF. By differentiating F α,β,a1,a2 (y), we obtain the same mixture representation where π ω (y) is the Exp-W PDF with power parameter (ω). Equation (9) reveals that the MOL-W density function is a linear combination of Exp-W densities. Thus, some structural properties of the new family such as the ordinary and incomplete moments and generating function can be immediately obtained from well-established properties of the Exp-W distribution. The r [th] ordinary moment of Y is given by then we obtain where ϱ The last expressions can be computed numerically . The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships. The moment generating function Clearly, the first one can be derived from equation (9) as setting s = 1, 2, 3, 4 in (11) we get and

Entropies
The Rényi entropy of a random variable Y represents a measure of variation of the uncertainty. The Rényi entropy is defined by Using the PDF (6), we can write where The ζ-entropy, say H ζ (Y ), can be obtained as

Order statistics
Suppose Y 1 , . . . , Y m is any random sample from any MOL-W distribution. Let Y :m denote the i [th] order statistic. The PDF of Y :m can be expressed as Then where is given in Section 3 and the quantities ξ s+ −1,k1 can be determined with ξ s+ −1,0 = w s+ −1 0 and recursively for k 1 ≥ 1

Residual life and reversed residual life functions
The m [th] moment of the residual life, say the m [th] moment of the residual life of Y is given by The mean residual life (MRL) at age τ can be defined as , which represents the expected additional life length for a unit which is alive at age τ . The MRL of Y can be obtained by setting m = 1 in the last equation. The m [th] The mean inactivity time (MIT) or mean waiting time (MWT) also called the mean reversed residual life function is given by and it represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, τ ).The MIT of the MOL-W distribution of distributions can be obtained easily by setting m = 1 in the above equation.

Via FGM family
A Copula is continuous in t and ν; actually, it satisfies the stronger Lipschitz condition, where we can esaily get the get the joint CDF of the MOL-W using the FGM family a 1 ,a 2 (t)  B α 1 ,β 1 ,a 1 ,a 2 The joint PDF can then be derived from c ζ (t, ν) = 1 + ζt · ν · | (t · =1−2t and ν · =1−2ν) or from c ζ (t, ν) = f (x 1 ,

Type-II
Let W (t) and M (ν) be two functional form satisfying all the conditions stated earlier where

B-MOL-W type via Renyi's entropy
Using the theorem of Pougaza [48] where H(t, ν) = x 2 t + x 1 ν − x 1 x 2 , the associated B-MOL-W can be derived from

Estimation
Let Y 1 , . . . , Y m be a random sample from the MOL-W distribution with parameters α, β and a 1 . Let Ψ =(α, β, a 1 ) be the 3 × 1 parameter vector. For determining the MLE of Ψ, we have the log-likelihood function ℓ = ℓ(Ψ) = m log α + m log β + m log a 1 + ma 1 log a 2 The components of the score vector are where z = ∂ ∂α s , p = ∂ ∂β s , d = ∂ ∂a1 s and q = ∂ ∂a2 s . Setting the nonlinear system of equations U α = U β = and U a1 = 0 and solving them simultaneously yields the MLE Ψ = ( α, β, a 1 , a 2 ) . To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ.

Graphical assessment
Graphically, we can perform the simulation experiments to assess of the finite sample behavior of the MLEs. The assessment was based on the following algorithm: 4. Compute the biases and mean squared errors given for Ψ = α, β, a 1 , a 2 . We repeated these steps for m = 50, 100, . . . , 500 with α = β = a 1 = a 2 = 1, so computing biases

Applications
In this section, we provide four applications of the OLEW distribution to show empirically its potentiality. In order to compare the fits of the MOL-W distribution with other competing distributions, we consider the Cramér-von Mises ( CV M (statistic) ) and the Anderson-Darling ( AD (statistic) ) . These two statistics are widely used to determine how closely a specific CDF fits the empirical distribution of a given data set. These statistics are given by (1 + 1/2m) , and respectively, where z = F (y s ) and the y s 's values are the ordered observations. The smaller these statistics are, the better the fit. The required computations are carried out using the R software. The MLEs and the corresponding standard errors (in parentheses) of the model parameters are given in Tables 2, 4 Tables 3, 5, 7  and 9 and Figures 6, 7, 8 and 9, the MOL-W model is a potential model for modeling the "symmetric bimodal" real data, the "asymmetric bimodal heavy tailed right skewed" real data, "asymmetric bimodal right skewed" real data and "asymmetric bimodal heavy tailed left skewed" real data as illustrated in Section 6.

Modeling cancer data
This data set represents the remission times (in months) of a random sample of 128 bladder cancer patients as reported in Lee and Wang [38]. This data are given in Appendix (b). We compare the fits of the MOL-W distribution with other competitive models, namely: The TMW, MBW, transmuted additive W distribution (TA-W) (Elbatal and Aryal [15]), and the W ( [52]) distributions with corresponding densities (for y > 0). Based on the figures in Table 5 we conclude

Modeling survival times
The second real data set corresponds to the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli reported by Bjerkedal [9].This data are given in Appendix (c).We shall compare the fits of the MOL-W distribution with those of other competitive models, namely: Odd Lindley exponentiated W (OLEW), the Odd W-W (OW-W) (Bourguignon et al. [10]), the gamma exponentiated-exponential (GaE-E) (Ristic and Balakrishnan [50]) distributions, whose PDFs (for y > 0).Based on the figures in Table 7 we conclude that the proposed MOL-W model is much better than all other models with CV M (statistic) = 0.19885and AD (statistic) = 1.15606.   ECDF plot (f), P-P plot (g), EHRF plot (h) for data set III.

Application 4: Glass fibers data
This data consists of 63 observations of the strengths of 1.5 cm glass fibres, originally obtained by workers at the UK National Physical Laboratory. 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)". We proved the wide flexibility of the new model numerically and graphically. Simple type Copula-based construction is presented to derive many bivariate and multivariate type models. The maximum likelihood method is used to estimate the model parameters. Graphical simulation results to assess the performance of the maximum likelihood estimation are performed. We proved empirically the importance and flexibility of the new Lehmann Weibull model in modeling various types of data. The new distribution has a high ability to model different types of real data sets such as the "symmetric bimodal" real data, the "asymmetric bimodal heavy tailed right skewed" real data, "asymmetric bimodal right skewed" real data and "asymmetric bimodal heavy tailed left skewed" real data.