A GENERALIZED CLASS OF EXPONENTIATED MODI ED WEIBULL DISTRIBUTION WITH APPLICATIONS

In this paper, a new class of five parameter gamma-exponentiated or generalized modified Weibull (GEMW) distribution which includes exponential, Rayleigh, Weibull, modified Weibull, exponentiated Weibull, exponentiated exponential, exponentiated modified Weibull, exponentiated modified exponential, gamma-exponentiated exponential, gammaexponentiated Rayleigh, gamma-modified Weibull, gamma-modified exponential, gamma-Weibull, gamma-Rayleigh and gamma-exponential distributions as special cases is proposed and studied. Mathematical properties of this new class of distributions including moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics and Renyi entropy are presented. Maximum likelihood estimation technique is used to estimate the model parameters and applications to real data sets presented in order to illustrate the usefulness of this new class of distributions and its sub-models.


Introduction
Weibull distribution (Weibull, 1951) has exponential and Rayleigh as special sub-models and it is one of the most popular distributions for modeling lifetime data with monotone failure rates.However, as for non-monotone failure rates, Weibull distribution does not t very well.Recently, several modified Weibull distributions with additional parameters have been proposed and studied as lifetime distributions in reliability and lifetime data analysis.
Recently, several ways of generating new probability distributions from classic ones have been developed and discussed in the literature on distribution theory and its applications.Nelson (1982) stated that distributions with bathtub-shaped failure rate are complex and, therefore, di cult to model.The distribution pro-posed by Hjorth (1980) is an example.Rajarshi and Rajarshi (1988) presented a revision of these distributions, and Haupt and Schabe (1992) introduced a new lifetime model with bathtub-shaped failure rates.However, these models are not sufficient to address the various complex practical situations, so new classes of distributions were presented based on the modifications of the Weibull distribution to satisfy non-monotonic failure rate.For a review of these models, the reader can refer to Pham and Lai (2007), where the authors summarized some generalizations of Weibull distribution.Jones (2004) studied a family of distributions derived from the distribution of order statistics, the beta-generated family proposed by Eugene et al. (2002).Other generalizations include: the exponentiated Weibull (EW) (Gupta and Kundu, 1999), the modified Weibull (MW) (Lai et al., 2003), the beta exponential (BE) (Nadarajah and Kotz, WITH APPLICATIONS 2006).Some re-cent extensions are the generalized modified Weibull (GMW) (Carrasco et al., 2008), the beta modi ed Weibull (BMW) (Silva et al., 2010), the Weibull-G fam-ily (Burguignon et al., 2014), the Gamma-exponentiated Weibull distributions (GEW) (Pinho et al., 2012) and the McDonald exponentiated modified Weibull (McEMW) (Merovci and Elbatal, 2015).
Ristic and Balakrishnan (2011), provided a new family of distributions whose cumulative distribution function (cdf) were generated by equation (1).As a natural extension, in this paper we introduce a new distribution with ve pa-rameters, referred to as the gammaexponentiated modi ed Weibull (GEMW) distribution with the aim of attracting wider application in reliability, biology and other areas of research.This generalization contains as special sub-models several distributions such as the EW (Gupta and Kundu, 1999), MW, generalized Rayleigh (GR) (Kundu and Rekab, 2005) and a new sub-model, namely Gamma modi ed Weibull (GMW) distributions, along with several others.Due to its exibility in accommodating all the forms of the hazard function, the proposed GEMW distribution seems to be an important distribution that can be used in various problems in modeling survival data.The GEMW distribution is not only useful for modeling bathtub-shaped failure rate data but also suitable for testing goodness-of-t of some special sub-models such as the EW (Gupta and Kundu, 1999), MW, GMW (new) and GEW distributions.
The rest of the paper is organized as follows.In Section 2; we de ne the GEMW distribution and provide its hazard rate, reverse hazard and quantile functions.Expansions for its probability density function (pdf) and some special sub-models are presented as well.The moments, moment generating and characteristic functions are given in section 3: Section 4 is devoted to mean deviations about the mean and the median, Bonferroni and Lorenz curves.Section 5 contains results on the distribution of order statistics and Renyi entropy.In section 6; estimation of the parameters of the GEMW distribution via the method of maximum likelihood is presented.Applications are given in section 7, followed by concluding remarks in section 8:

Definition
Based on a continuous cdf F(x) with survival function  ̅ (x) and pdf f(x) and the method proposed by Zografos and Balakrishnan (2009), Ristic and Balakrishnan (2011) proposed an alternative gamma-generator defined by the cdf and pdf: (1) and (2) respectively.For  =  ∈ N, equation ( 2) is the pdf of the n th lower record value of a sequence of i.i.d.variables from a population with density f(x).  is the incomplete gamma function.The pdf corresponding to (5) is given by for , , ,  > 0 and  ≥ 0. A random variable X having density ( 6) is denoted by X~GEM W(, , , , ).Several possible shapes of the GEMW pdf are shown in Figure 1, 2 and 3. Note that the parameters  controls the scale of the distribution, ,  and  controls its shape,  is a sort of accelerating factor in the imperfection time that works as a factor of fragility in the survival of the individual as time increases.

Hazard rate, reverse hazard and quantile functions
In this section, we provide the hazard rate, reverse hazard and quantile functions of the GEMW distribution.

Hazard rate and reverse hazard functions
Note that if X is a continuous random variable with cdf G(x); and pdf g(x); then the failure of hazard rate function (hrf), reverse hazard function (rhf) and mean residual life functions are given by ℎ  () = ()  ̅ () ,   () = () G(x) ; and   () = ∫  ̅ ()/ ̅ () ∞  , respectively.The functions ℎ  (),   () , and  ̅ () are equivalent.(Shaked and Shanthikumar, 1994).The hazard rate and reverse hazard rate functions of GEMW distribution are given by and respectively suitable for monotonic and non-monotonic empirical hazard behaviors which are more likely to be the case in real life situations.The density and hazard functions can exhibit different behavior depending on the values of the parameters when chosen to be positive, as shown in these plots.However, it is hard to analyze the shape of both the density and hazard function due to their complicated forms.

Quantile function
We can know more about this model by expanding the density function and analyzing the quantile function.The GEMW quantile function can be obtained by inverting  ̅ () = 1 − , where () =  and and the inverse incomplete gamma function can be implemented by using numerical methods.Consequently, random number can be generated based on (8). , +++ , for  ∈ .Therefore, the GEMW pdf can be written as Equation (13) shows that the GEMW density is indeed a linear combination of modified Weibull distribution.Hence, most of its mathematical properties can be immediately obtained from those of the modified Weibull distribution.For the convergence of equations ( 12) and ( 13), as well elsewhere in this paper, note that for > 0;

Some Sub-models
In this section, some sub-models of the GEMW distribution are presented.The GEMW distribution contains several special sub-models that are well known distributions.When β = 0, we obtain which is the gamma-exponentiated Weibull (GEW) distribution, given by Pinho et al.

Moments, moment generating and characteristic functions
In this section, the moments, moment generating function and characteristic function of the GEMW distribution are presented.Let Z be a random variable with a density function g(z), as in equation ( 6) and X be a random variable with modified Weibull density function   ().

Moments
As mention earlier in section 2. where  1 (, ),   and   given by and respectively.
There is also another relatively simpler form of the tth moment by using the Lambert W (.) function.We can get the following equation for  1 (, ), Substituting equation ( 18) in ( 14) gives a representation for moments of the MW (Lai et al., 2003) distribution in a relatively concise form with only a doubly infinite series.Thus, the moments of GEMW can be obtained from the moments above.Let ~(, , , , ) , then E(Z t ) can be expressed in terms of the t th moments of the same baseline MW distribution, which is where E(Xt) is defined by equation ( 14), and by substituting  1 (, ) from equation (18), one can also get another form of moments for GEMW distribution.Conditional expectations are very useful for lifetime models, so it is very important to know E(  | > ), which is given by: where Shusen Pu 1 ; Broderick O. Oluyede 2 , Yuqi Qiu 3 and Daniel Linder 4 595 and      are given by ( 16), (17) and Γ(, ) = ∫  −1  −  ∞  .Similarly, a relatively simpler representation for E(  | > ) can be obtained from equation (18), , where (  ) is the given by equation (14).Nothing that, () can be expressed as an infinite weighted sum, we have Where   () is the moment generating function of the MW distribution, which is, The corresponding characteristic function is with where   is given by equation ( 16).

Mean deviations, Bonferroni and Lorenz curves
In this section, mean deviations about the mean and the median, Bonferroni and Lorenz curves of the GEMW distribution are represented.

Mean deviations
Let ~(, , , , ), the mean deviation about the mean and the mean deviation about the median are defined by  1 () = ∫ | − |() , where and   and   are given by equation ( 16) and ( 17), respectively.It follows that

Bonferroni and Lorenz Curves
In this section, some inequality measures, namely Bonferroni and Lorenz curves are presented.These quantities have been applied to a wide variety of fields, such as studying of income and property in economics, reliability, demography, insurance and medicine.For ~(, , , , ), they are defined by respectively, where  = () and q =  −1 () is obtained from equation ( 8).Using similar methods in deriving the moments, we can show that where 597 and   and   are given by equation ( 16) and ( 17), respectively.We can reduce the curves in equation ( 25) to respectively.

Order Statistics and Renyi Entropy
In this section, the distribution of the ith order statistic and Renyi entropy for the GEMW distribution are presented.

Order Statistics
Consider  1 , … . .,   i.i.d random variables distributed according to (2).The pdf of the ith order statistic, say : , is given by Using the binomial theorem, the pdf of i th order statistic can be written as where  1 (, ) is given by equation ( 15).These moments are used in areas such as quality control, reliability and insurance, for prediction of future failure times from a set of past failures.

Estimation of Parameters
Let ~(, , , , ) and Δ = (, , , , )  be the parameter vector.The loglikelihood for a single observation  of X is given by , where ℓ  (Δ),  = 1,2, … … , n is given by equation (30).The equations obtained by setting the above partial derivatives to zero are not in closed form and the values of the parameters α, β, θ, λ, δ must be found by using iterative methods.The maximum likelihood estimates of the parameters, denoted by Δ ̂ is obtained by solving the nonlinear equation ( ) T = 0 , using a numerical method such as Newton-Raphson procedure.
We maximize the likelihood function using NLmixed in SAS as well as the function nlm in R (2011).These functions were applied and executed for wide range of initial values.This process often results or lead to more than one maximum, however, in these cases, we take the MLEs corresponding to the largest value of the maxima.In a few cases, no maximum was identified for the selected initial values.In these cases, a new initial value was tried in order to obtain a maximum.
The issues dealing with the existence and uniqueness of the MLEs are theo-retical interest and has been studied by several authors for di erent distributions including Seregin (2010), Santos Silva and Tenreyro (2010), Zhou (2009), and Xia et al. (2009).We hope to investigate this problem or issue for the GEMW distribution in the future.
Let Δ ̂= (α ̂, β ̂, k ̂, λ ̂, δ ̂) be the maximum likelihood estimate of Δ = (α, β, k, λ, δ) Under the usual regularity conditions and that the parameters are in the inte-rior of the parameter space, but not on the boundary, (Ferguson, 1996) we have: , where (∆) is the expected Fisher information matrix.The asymptotic behavior is still valid if (∆) is replaced by the observed information matrix evaluated at ∆ ̂ , that is J(∆ ̂).Elements of the observed information matrix are given in the Appendix.The multivariate normal distribution  5 (0, (∆ ̂)−1 ), where the mean vector 0 = (0, 0, 0, 0, 0)T , can be used to construct confidence intervals and confidence regions for the individual model parameters and for the survival and hazard rate functions.That is, the approximate 100(1 − η)% two-sided confidence intervals for α, β, k λ, and δ are given by: and  ̂±   2 and 3.
In order to compare the models, we use the criteria stated above.Note that for the value of the log-likelihood function at its maximum (L(∆ ̂ )), larger value is good and preferred, and for the Kolmogorov-Smirnov test statistic (K-S), smaller value is preferred.GEMW distribution is fitted to the data sets and these fits are compared to the fits using the Exponential, Weibull, EE, GE, GEE, GW, EW, MW, GEW and GMW distributions.

Applications
In this section, we present two examples to illustrate the flexibility of the GEMW distribution and its sub-models for data modeling.
Plots of the fitted densities, the histogram of these data and probability plots are presented in Figure 5  For the Aarset data, the LR test statistic of the hypotheses H0: GEW against Ha: GEMW and H0: GMW against Ha: GEMW are ω1 = 17.5 with p-value= 2.87 × 10 −5 and ω2 = 20.2 with p-value= 6.98 × 10 −6 .Therefore, we reject H0 in favor of Ha and conclude that the GEMW distribution is a significant better than the GEW and GMW distributions.Moreover, the values of the statistics AIC, AICC, BIC and KS show that model GEMW is a "better" fit for this data.Also, the value of SS for the GEMW distribution is the smallest.
Similarly, we can also conduct the LR test for the Meeker and Escobar data WITH APPLICATIONS , which implies that we should reject H0 in favor of Ha and conclude that the GEMW distribution is a significant better fit for the Meeker and Escobar data.In addition, the values of the statistics AIC, AICC, BIC and KS clearly show that model GEMW is a "better" fit for this data.Furthermore, the value of SS for the GEMW distribution is the smallest.

Concluding Remarks
A new class of distributions called the gamma-exponentiated or generalized modified Weibull (GEMW) distribution is proposed and studied.The GEMW distribution has several sub-models such as the GEW, EW, EE, GW, GE, GEE, GER, GR, MW, GEV, EV, GMW, Weibull, Rayleigh and exponential distributions as special cases.The density of this new class of distributions can be expressed as a linear combination of MW density functions.The GEMW distribution possesses hazard function with flexible behavior.We also obtain closed form expressions for the moments, distribution of order statistics and entropy.Maximum likelihood estimation technique is used to estimate the model parameters.Finally, the GEMW model is fitted to real data sets to illustrate the usefulness, flexibility and applicability of this class of distributions.

Figure 2 :
Figure 2: Graphs of GEMW pdf with varying values of δ and λ; respectively

Figure 3 :
Figure 3: Graph of GEMW pdf when only the parameter k changes
3, the GEMW distribution is a linear combination of modified Weibull distribution.Recall the general results for the beta-modi ed Weibull (BMW) distribution given by Nadarajah et al. (2011).Let ~(, , , , ) with pdf where , , α > 0 and γ, λ ≥ 0 .Note that   () is a sub-model of BMW distribution (Silva et al., 2010) with the parameters: Using the moments of BMW distribution, the moments of the MW distribution (Nadarajah et al., 2011) are as follow: A GENERALIZED CLASS OF EXPONENTIATED MODI ED WEIBULL DISTRIBUTION WITH APPLICATIONS

Figure 5 :
Figure 5: Fitted pdf and observed probabilities for Aarset data The estimated covariance matrix for the GEMW distribution (Meeker and Escobar Data) is given by

Figure 6 :
Figure 6: Fitted pdf and observed probabilities for Meeker and Escobar data

Table 1 :
GEMW Descriptive Statistics of Application Data Sets

Table 3 :
GEMW Estimation for Meeker and Escobar data