The Generalized Additive Weibull-G Family of Distributions

In this paper, we present a new family, depending on additive Weibull random variable as a generator, called the generalized additive Weibull generated-family (GAW-G) of distributions with two extra parameters. The proposed family involves several of the most famous classical distributions as well as the new generalized Weibull-G family which already accomplished by Cordeiro et al. (2015). Four special models are displayed. The expressions for the incomplete and ordinary moments, quantile, order statistics, mean deviations, Lorenz and Benferroni curves are derived. Maximum likelihood method of estimation is employed to obtain the parameter estimates of the family. The simulation study of the new models is conducted. The efficiency and importance of the new generated family is examined through real data sets.


Introduction
In recent years, the generated families of probability distributions have been broadly utilized for modeling real-data in many applied areas.These generated families are defined by adding one or more parameters to the baseline model.The generated families generalized and extended most of the formal distributions.Some of the generators are the beta-G by Eugene et al. (2002), gamma-G by Zografos and Balakrishanan (2009), Kumaraswamy-G by Cordeiro and de Castro (2011), generalized beta-G by Alexander et al. (2012), transformed-transformer (T-X) by Alzaatreh et al. (2013),Weibull-G by Bourguignon (2014), type I half-logistic-G by Cordeiro et al. (2016), additive Weibull-G by Hassan and Hemeda (2016) among others.
A more general family called the generalized Weibull-G family (GW-G) of distributions was introduced by Cordeiro et al. (2015).They considered a baseline cumulative distribution function (cdf) G(x;ξ), the probability density function (pdf) g(x;ξ) with parameter vector ξ and the Weibull distribution as a generator.They defined the cdf and pdf of the GW-G family as follows (1) where, Ḡ(x; ξ) = 1 − G(x; ξ) This article aims to introduce a new family of distribution called the GAW-G, which includes GW-G family as a special case, besides it contains several of the existing probability distributions.The current article can be arranged as follows.
The GAW generated family of distributions is formulated in Section 2. Four special models of GAW-G family are displayed in Section 3. Some structural properties of the GAW-G family are provided in Section 4. In Section 5, maximum likelihood estimators of the model parameters are derived.In Section 6, Simulation study results of four special models have been reported.In Section 7, an illustrative application based on a real data is investigated.A conclusion is provided in Section 8.

The Generalized Additive Weibull-G Family
In this section, we define a generalized additive Weibull generated family of continuous distributions by using the additive Weibull random variable as a generator.The reliability and hazard rate functions are defined and discussed analytically.Furthermore, the asymptotic of pdf, cdf and hazard function is explained.According to Lemonte et al. (2014), the cdf and pdf of AW distribution with shape parameters b, d and scale parameters a, c are given, respectively, by (4) To obtain the cdf of GAWCG, replacing the Weibull generator defined in (1) by the additive Weibull generator defined in (4) as the following where Φ = (a, b, c, d, ξ) and Ḡ(x; ξ) = 1 − G(x; ξ).The corresponding GAW-G pdf takes the following form A random variable having GAW-G density function (6) will be denoted by X ∼ GAW − G(x; Φ).Note that, for a=0 or c=0 the pdf (6) reduces to the GAW-G family defined by Cordeiro et al.(2015).Also, we obtain the same result for b = d, with scale parameter a + c.
Furthermore, the reliability and the hazard rate functions of GAW-G family are given, respectively, by The hazard rate function of GAW-G family is Here is a description regarding analytical behaviour of GAW-G family.The critical points of this family are the roots of the equation There is more than one root to this equation.When b = 1, it becomes to some extent a simpler model and then x = x 0 is a root of the equation } .

Some Special Models for GAW-G Family
In this section, some new special distributions, namely, GAW-uniform, GAW-Burr XII, and GAW-log logistic are introduced.

GAW-uniform Distribution
Consider the baseline distribution is uniform on the interval (0, θ), θ > 0 with the pdf and cdf, respectively The cdf of GAW-uniform (GAWU) distribution is obtained by substituting the pdf and cdf of uniform in (5) as follows

GAW-Gumbel Distribution
Consider the Gumbel distribution with location parameterλ ∈ R and scale parameter ν > 0 where the pdf and cdf for (λ ∈ R) are Inserting these equations into (5) and ( 6), the pdf and cdf of the GAW-Gumbel distribution will be obtained as follows The survival and hazard rate functions are given respectively as follows

GAW log-logistic Distribution
Assuming that the baseline distribution is log-logistic (see (Bennett (1983)) with the following pdf and cdf, As previously mentioned, the cdf and pdf of the generalized additive Weibull log -logistic(GAWLL) distribution are obtained by substituting the previous pdf and cdf in ( 5) and ( 6) as follows The survival and hazard rate functions take, respectively, the following forms

GAW-Burr XII Distribution
Considering the baseline distribution is Burr XII (see Burr (1942)) with the following pdf and cdf The cdf of GAW-Burr XII (GAWBXII) distribution is obtained by substituting the pdf and cdf of Burr-XII in ( 5) and (6) as follows The corresponding pdf is The survival and hazard rate functions are obtained, respectively, as follows Plots of pdf and hazard rate function for some parameter values for the selected distributions are represented through Figure 1.
From Figure 1, it appears that the shape of the distribution depend heavily on parameter values.In fact, the shape could be left skewed, symmetric and right skewed, which will depend on the values of the parameter.Thus this distribution could be suitable to model many kind of data.

Some Mathematical Properties
In this section, some general results of the GAW-G family are derived.
] d(i− j)+b j+m in power series, F(x; Φ) can be expressed as where The corresponding pdf can be expressed as Another expression of the cdf of GAW-G family can be written as where H(x; ξ) denotes the cdf of the mixture exponential The corresponding pdf can be expressed as where h(x; ξ) is the pdf of the exp-H(x; ξ) distribution

Quantile Function
The quantile function, say Q(u) = F −1 (u), of the GAW-G family is derived by inverting (5) as follows After some simplifications, the previous equation is reduced to where, x G = Q(u), and u has the uniform distribution on interval (0, 1).Hence the nonlinear equation ( 20) is solved numerically to obtain the generated number of the random variable X.

Moments
The rth moment of random variable X can be obtained from pdf (17) as follows where, I i, j,r = ∫ ∞ 0 kx r h i, j (x; ξ)dx and I i, j,m,r = ∫ ∞ 0 (k + m + 1)x r h i, j,m (x; ξ)dx.In particular, the mean and variance of GAW-G family are obtained as follows: The variance is Additionally, measures of skewness and kurtosis of family can be obtained, based on (21), according to the following relations Furthermore, the moment generating function of GAW-G family is as follows where, µ ′ r is the r th moment about origin, then the moment generating function of GAW-G family is obtained by using (21) as follows

Distribution of Order Statistics
Let X 1 , X 2 , X 3 , ......, X n be a simple random sample from GAW-G family with cdf (5) and pdf (6) and X 1:n , X 2:n , X 3:n , ......, X n:n denote the corresponding order statistics.The pdf of X r:n is obtained through the following Using the cdf (5) and pdf (6), the pdf of rth order statistic from GAW-G family takes the following form Substituting in ( 22), therefore Using the exponential expansion form for the following term: exp Substituting in (23) Therefore, where, In particular, the pdf of the smallest order statistic X 1:n is obtained from (24), by substituting r=1 where, Also, the pdf of the largest order statistic X n:n is obtained from (24), by substituting r=n where,

Incomplete Moments, Mean Deviations and Lorenz and Benferroni Curves
The r-th incomplete moment, say,m I r (t), of the GAW-G distribution is given by We can write from equation ( 17), Example 4.5.1 Consider the GAW-uniform distribution discussed in subsection 3.1.
The scatterings present in a population is, to some extent, to be measured by the totality of the deviations from a measure of central tendency like the mean or the median.The mean deviations about the mean of X may be used as measures of spread (or dispersion) in a population besides range and standard deviation.They are given by δ ) is the first incomplete function obtained from (25) with r = 1 and M is the median of X obtained by solving (20) for u = 0.5.

The Lorenz and Benferroni curves are defined by
, respectively, where x p = F −1 (p) can be computed numerically by ( 20) with u = p.These curves have significant roles in demography, economics, insurance, medicine and reliability.For details in this aspect, the readers are referred to Pundir et al.(2005) and references cited therein.

Moments of the Residual Life
The hazard rate, mean residual life, left truncated mean function are some functions related to the residual lifetime of a unit.These functions uniquely determine the cumulative distribution function, F(x).See, for instance, Gupta(1975) and Zoroa et al.(1990).
Definition 4.6.1 Let X, be a random variable denoting the lifetime of a unit is at age t.Then X t = X − t | X > t denotes the remaining lifetime beyond that age t.
The cdf F(x) is uniquely determined by the r-th moment of the residual life of X (for r = 1, 2, ...) [Navarro et al.(1998)], and it is given by In particular, if r = 1, then m 1 (t) represents an interesting function called the mean residual life (MRL) function that indicates the expected life length for a unit which is alive at age t.The MRL function has wide spectrum of applications in reliability/survival analysis, social studies, biomedical sciences, economics, population study, insurance industry, maintenance and product quality control and product technology.
Example 4.6.1 Consider again the GAW-uniform distribution discussed in subsection 3.1.
For the MRL function,

Moments of the Reversed Residual Life
In some life testing aspects, instead of relating uncertainty to the future, it may relate to the past.When the state of a system is observed only at a preassigned inspection time t and if it is found to be at "'down"' state, then failure lies on the past i.e. the instant in (0, t) at which it has failed.Therefore, study of a notion that is complementary to the residual life, in the sense that it deals with the past time instead of future seems worthwhile [see Di Crescenzo and Longobardi (2002)].
Definition 4.7.1 Let X be a random variable denoting the lifetime of a unit is down at age t.Then Xt = t − X | X < t denotes the idle time or inactivity time or reversed residual life of the unit at age t.
In case of forensic science, one may be interested in estimating Xt to have an idea about the exact time of death of a living creature.In Insurance study, it represents the unpaid period of a policy holder due to death.For details, see Block et al.(1998), Chandra and Roy(2001), Maiti andNanda(2009), andNanda et al.(2003).The r-th moment of Xt (for r = 1, 2, ...) is given by In particular, if r = 1, then m1 (t) represents a function called the mean idle time or inactivity time (MIT) or reversed residual life (MRRL) function that indicates the expected inactive life length for a unit which is first observed down at age t.The properties of MIT function have been explored by Ahmad et al. (2005) and Kayid and Ahmad (2004).
Example 4.7.1 Consider again the GAW-uniform distribution discussed in subsection 3.1.
Using ( 17), we have For the MIT (or MRRL) function,

Estimation of Model Parameters
In this section, the maximum likelihood estimators of the model parameters Φ = (a, b, c, d, ξ)of GAW-G family from complete samples are derived.LetX 1 , X 2 , ...., X n be a simple random sample from GAW-G family with observed values x 1 , x 2 , ...., x n .The log likelihood function of ( 6) is obtained as follows For simplicity, let and ln L(Φ) to be l, then Differentiating with respect to each parameter and setting the result equals to zero, the maximum likelihood estimators will be obtained.The partial derivatives of l with respect to each parameter are given by where and The maximum likelihood estimates (MLEs) of the model parametersare determined by solving the non-linear equations ∂l ∂a = 0, ∂l ∂b = 0, ∂l ∂c = 0, ∂l ∂d = 0, ∂l ∂ξ = 0 .These equations cannot be solved analytically but some software's can be used to solve them numerically.

Simulation Study
In this section, we have conducted simulation study for above mentioned four Generalized Additive Weibull model.We have generated samples of sizes n = 20, 40, 100 from each model and parameters have been estimated by the maximum likelihood method.1000 such repetitions are made to calculate the bias and mean square error (mse) of these estimates using the formula for estimates of any parameter η by Bias η , respectively.All the computations are made using R-Software.
From the Tables 1-4, it is observed that 1.As sample size n increases, bias decreases.That shows accuracy of the MLE of the parameters, 2. As sample size n increases, MSE decreases.That shows consistency (or preciseness) of the MLE of the parameters.

Concluding Remarks
We have introduced and studied a new generalized family of distributions, called the generalized additive Weibull-G (GAW-G) distribution.The GAW-G family generalizes the Weibull-G family [see, Cordeiro et al.(2015)] and includes several new distributions.Properties of the GAW-G family include: an expansion for the density function and expressions for the quantile function, moment generating function, ordinary moments, incomplete moments, mean deviations, Lorenz and Benferroni curves, reliability properties including mean residual life and mean inactivity time, and order statistics.Four new distributions, namely, GAW-Uniform, GAW-Gumble, GAW-Log logistic and GAW-Burr XII are defined and discussed in some details.The maximum likelihood method is employed to estimate the model parameters.Simulation study has been conducted to study the accuracy and consistency of the MLE of the parameters.A real data set is used to demonstrate the flexibility of distribution belonging to the introduced family.These special models give better fits than other models.We hope the findings of the paper will be quite useful for the practitioners in various fields of probability, statistics and applied sciences.

Figure 1 .
Figure 1.Graph of Generalized Additive Weibull distributions and their corresponding hazard rates.

Figure 2 .
Figure 2. Plots of the Histogram, estimated pdf and cdf for failure times data (pressure at 90 percentage)

Table 1 .
Bias and Mean Square Error (MSE) of the MLE of parameters of GAW-uniform distribution

Table 2 .
Bias and Mean Square Error (MSE) of the MLE of parameters of GAW-Gumbel distribution

Table 3 .
Bias and Mean Square Error (MSE) of the MLE of parameters of GAW log-logistic distribution

Table 6 .
Summarized results of fitting different distributions for Kevlar 49/epoxy strands failure times data