Parameter Estimations for Some Modifications of the Weibull Distribution

Proposed by the Swedish engineer and mathematician Ernst Hjalmar Waloddi Weibull (18871979), the Weibull distribution is a probability distribution that is widely used to model lifetime data. Because of its flexibility, some modifications of the Weibull distribution have been made from several researches in order to best adjust the non-monotonic shapes. This paper gives a study on the performance of two specific modifications of the Weibull distribution which are the exponentiated Weibull distribution and the additive Weibull distribution.


Introduction
The Weibull distribution [1] is the most life-time probability distribution used in the reliability engineering discipline.Due to its wide applications [2], many researchers have developed various extensions and modified forms of the Weibull distribution with a number of parameters ranging from 2 to 5.These distributions have several desirable properties and nice physical interpretations.The literature that studies the various modifications of the Weibull distributions is extensive, for example: the two-parameter flexible Weibull extension of Bebbington et al. [3].Zhang and Xie [4] studied the characteristics and application of the truncated Weibull distribution which has a bathtub shaped hazard function.
A three-parameter model, called exponentiated Weibull distribution, was introduced by Mudholkar and Sri-vastave [5].The modified Weibull distribution of Sarhan and Zaindin [6] was studied by Gasmi and Berzig [7] in the case of type I censored data.Another three-parameter model was developed by Marshall and Olkin [8] and is called the extended Weibull distribution.Xie et al. [9] proposed a three-parameter modified Weibull extension with a bathtub shaped hazard function.Lai et al. [10] have described the modified Weibull (MW) distribution.A four-parameter additive Weibull distribution (AddW) was proposed by Xie and Lai [11].A second four-parameter beta Weibull distribution was proposed by Famoye et al. [12].Cordeiro et al. [13] introduced another four-parameter distribution called the Kumaraswamy Weibull distribution.A five-parameter modified Weibull distribution was introduced by Phani [14].The beta modified Weibull distribution was introduced by Silva et al. [15] and further studied by Cordeiro et al. [16].Recently, an extensive review of some discrete and continuous versions of the modifications of the Weibull distribution was introduced by Almalki and Nadarajah [17].The main objective of this article is in first step to estimate the three unknown parameters of the exponentiated Weibull distribution and the four unknown parameters of the additive Weibull distribution.Therefore, we use the maximum likelihood method to derive such estimates.In the second step, we study whether these distributions fit a set of real data of Aarset [18] better than other distributions.Two criteria are used for this purpose: the first one is the mean square distance MSD and the next one is the Kolmogorov-Smirnov test statistic.A real data set is analyzed and it is observed that the present distributions provide better fit than many existing well-known distributions.This paper will be organized as follows.In Section 2 we present the exponentiated Weibull distribution and the additive Weibull distribution.In Section 3, an application to real data is provided and different types of goodness-of-fit are applied to test the compatibility of the exponentiated Weibull distribution and the additive Weibull distribution in comparison to some other models.Mainly we use the mean square distance MSD and the Kolmogorov-Smirnov (K-S) test as a non-parametric test to illustrate how one can compare the exponentiated Weibull distribution and the additive Weibull distribution with some sub-models.Finally we conclude the paper in Section 4.

Exponentiated Weibull Distribution
The exponentiated Weibull (EW) distribution is proposed by Mudholkar and Srivastava [5] and studied first by Mudholkar et al. [19] and further by Mudholkar and Hutson [20].
The cumulative distribution function (CDF) and the survival function of the EW distribution, denoted by  distribution for different values of α , θ and λ .α θ λ distribution has the advantage that it possessed a closed form of cdf, therefore we can generate random values from it by using the explicit formula: ( ) , where 1, , i n =  , n is the sample size and U is a uniformly distributed random variable on the interval (0, 1).
Figure 3 illustrate the empirical cdf, the cdf and the 95% lower and upper confidence bounds for the cdf of 100 simulated data by setting 1 α = , 3 θ = and 1 λ = .

Parameter Estimation
To estimate the parameters of the After calculating the first partial derivatives of ( ) L t Θ and setting the results to zero, we get the following score functions: ( ) ) ( ) To get the MLE of the parameters α , θ and λ we have to solve the above system of three non-linear eq- uations with respect to α , θ and λ .The solution of this system of equations is not possible in closed form, so numerical technique such as the trust region method, which requires the second derivatives of the ( ) function is needed to get the MLE.We note that in order to accelerate the resolution of the system ( 7), ( 8), ( 9) by using the software MATLAB, we have introduced the following second partial derivatives of ( ) L t Θ : ( ) ( ) gives the estimated parameters of 10 N = simulations and the mean square error of each parameter, where:

Additive Weibull Distribution
The additive Weibull (AddW) distribution has four parameters α , β , θ and γ .This distribution is first in- troduced by Xie and Lai [11] and is denoted by AddW , , , α β θ γ .We remark, that this distribution has a bathtub shaped hazard function and it was obtained as the sum of two hazard functions of Weibull distributions.

Parameter Estimation
Now, we introduce the estimation of the model parameters by using the method of maximum likelihood.Let   ( ) , , , n t t t  be a random sample of the AddW distribution with unknown parameters α , β , θ and γ .By setting ( ) , , , , the likelihood function of this sample is given by: ( ) ( ) ( ) The log-likelihood function has the following form: ( ) ( ) ( ) After calculating the first partial derivatives of ( ) ln ; i L t Θ and setting the obtained expressions equal to zero, we get the following score functions: ( ) ( ) ( ) ( ) To get out the MLE of the unknown parameters, we have to solve the above system of four non-linear equations with respect to α , β , θ and γ .The solution of this system of equations is not possible in closed form, so numerical technique such as the trust region method is needed to get the MLE.We obtain the second partial derivatives of ( ) ln ; i L t Θ as follows: ( ) ( ) ) gives the estimated parameters of 10 simulations and the mean square error of each parameter.

Analysis of a Real Data Set
In this section, we analyze a real data set to demonstrate the performance of the EW and AddW distributions in practice.A sample of 50 components taken from Aarset [18] has been studied.For this data set, we compare at first the results of the fits of the EW distribution (EWD) against ED, GED, RD and WD which are sub-models of the EW distribution.In the second step the fits of the AddW distribution (AddWD) will be compared against WD, MWD, and LRFD which are sub-models of the AddW distribution.Table 3 gives the often used lifetimes of 50 devices introduced by Aarset.Table 4 and Table 5 show the MLE of the parameters, the log-likelihood function values and the MSD on the one hand for the ED, RD, GED, WD, EWD and on the other hand for the WD, MWD, LRFD, and the AddW models.Table 6 and Table 7 show the observed K-S test statistic values for each models EWD and AddWD and their correspondent sub-models and the p-value for each one.Figure 7 and Figure 8 show the plots of the empirical and fitted scaled TTT-Transforms, the empirical and parametric cumulative density functions, the empirical and fitted hazard and probability density functions for the models EWD, AddWD and their correspondent submodels.However in Figure 9 we have a comparison between the two models EW and AddW.We note that for comparison purpose, we use the mean square difference between the empirical cdf and the fitted cdf, denoted by MSD.The MSD is computed by the following relation: where ˆi F and i F are the estimated and the empirical cdf computed at the cumulative failure times i t and r is the size of the data set.
Based on the results shown in Table 4 and Table 5, we could deduce that: • compared with the MSD of the ED and the WD, the EWD is not the best fit of the Aarset data; • the MSD of the AddWD has the lowest value compared with each sub-models, so the AddWD is the best in fitting the Aarset data; • the MSD of the AddWD is smaller than the MSD of the EWD which indicates that the AddWD fits the given data better than the EWD.value (0.1230).We can immediately observe from Figure 7, Figure 8 and Figure 9 that: 1) the data set has a bathtub shaped hazard rate, 2) one can see the closeness of the fitted pdf using the AddWD model, 3) the AddWD fits the data set better than all other distributions used here, because its fitted curve is closer to the empirical curve.

Conclusion
In this paper, we show the performance of two models called the exponentiated Weibull distribution and the additive Weibull distribution by using an empirical comparison with the sub-models of each one such as the exponential distribution, the Rayleigh distribution, the generalized Weibull distribution, the linear failure rate distribution, the Weibull distribution and the modified Weibull distribution.The maximum likelihood estimations of the unknown parameters for these distributions are discussed.A real data set of Aarset is studied by using the EW and the AddW distributions.The results of the comparisons showed that the additive Weibull distribution provided a better fit for the Aarset data set than some of the often-used distributions.

Figure 1 .
Figure 1.Plots of cumulative distribution function and survival function of

Figure 2 .
Figure 2. Plots of probability density function and hazard rate function of the three parameters α , θ and λ as follows:

Figure 3 .
Figure 3. Cdf and empirical cdf of the γ distribution generalizes the following distributions: 1) linear failure rate distribution

Figure 4
Figure 4 shows respectively the cumulative distribution function and the survival function of the additive Weibull distribution for different values of α , β , θ and γ .The probability density function of the

Figure 5
Figure5shows the probability density function and the hazard rate function of the

Figure 6
Figure 6 illustrates the empirical cdf, the cdf and the 95% lower and upper confidence bounds for the cdf of the 100 simulated data by setting 1.5 α = , 0.5 β = , 3 θ = and 0.8 γ = .

Figure 4 .
Figure 4. Plots of cumulative distribution function and survival function of the

Figure 5 .
Figure 5. Plots of probability density function and hazard rate function of the

Figure 6 .
Figure 6.Cdf and empirical cdf of the

Figure 7 .
Figure 7. (a) The empirical and estimated scaled TTT-Transform plots of the ED, RD, GED, WD and EWD models; (b) The empirical and estimated cumulative density function of the ED, RD, GED, WD and EWD models; (c) Empirical and estimated hazard rate functions of the ED, RD, GED, WD and EWD models; (d) Empirical and estimated PDF of the ED, RD, GED, WD and EWD models, for Aarset data.

Figure 8 .
Figure 8.(a) The empirical and estimated scaled TTT-Transform plots of the WD, LRFD, MWD and AddWD models; (b) The empirical and estimated cumulative density function of the WD, LRFD, MWD and AddWD models; Empirical and estimated hazard rate functions of the WD, LRFD, MWD and AddWD models; (d) Empirical and estimated pdf of the WD, LRFD, MWD and AddWD models, for Aarset data.

Table 4 .
MLE of the parameter(s), log-likelihood function values and the MSD of sub-models of the EWD.

Table 5 .
MLE of the parameter(s), log-likelihood function values and the MSD of sub-models of the AddWD.

Table 6 .
The MLE of the parameter(s), K-S values and the associated p-values.

Table 7 .
The MLE of the parameter(s), K-S values and the associated p-values.