Exponentiated Weibull-Lomax Distribution : Properties and Estimation

In this article, we introduce a new class of five-parameter model called the Exponentiated Weibull Lomax arising from the Exponentiated Weibull generated family. The new class contains some existing distributions as well as some new models. Explicit expressions for its moments, distribution and density functions, moments of residual life function are derived. Furthermore, Rényi and q–entropies, probability weighted moments, and order statistics are obtained. Three suggested procedures of estimation, namely, the maximum likelihood, least squares and weigthed least squares are used to obtain the point estimators of the model parameters. Simulation study is performed to compare the performance of different estimates in terms of their relative biases and standard errors. In addition, an application to two real data sets demonstrate the usefulness of the new model comparing with some new models.


Introduction
The Lomax or Pareto II distribution is originally used for modeling business failure data, and it has been widely applied in a variety of contexts studies.Atkinson and Harrison (1978) and Harris (1968) applied the Lomax distribution to income and wealth data.Bryson (1974) suggested Lomax distribution as an alternative to the exponential distribution for heavytailed data sets.Myhre and Saunders (1982) applied Lomax distribution in the right censored data.Different procedures of estimation for the Lomax distribution are suggested by Lingappaiah (1986).Moments of record values based on Lomax distribution were discussed by Ahsanullah (1991) and Balakrishnan and Ahsanullah (1994).Order statistics from non-identical right-truncated Lomax distribution and its applications were discussed by Childs et al. (2001).Abd-Elfattah et al. (2007) discussed the Bayesian and non-Bayesian estimation problem of the sample size in the case of type-I censored samples.Based on cumulative exposure model, the optimal times plans of changing stress level of simple stress for the Lomax distribution were determined by Hassan and Al-Ghamdi (2009).Abd-Elfattah and Alharbey (2010) discussed the estimation problem for the Lomax distribution based on generalized probability weighted moments.Nasiri and Hosseini (2012) obtained the Bayesian and non-Bayesian estimators for the Lomax parameters in presence of record values.The estimation problem of the unknown parameters for the Lomax distribution based on type-II progressively hybrid censored samples has been discussed by Ma and Shi (2013).Ahmad et al. (2015) obtained the Bayesian estimators of the shape parameter of the Lomax distribution under different loss functions.The optimal times of changing stress level for k-level step stress accelerated life tests based on adaptive type-II progressive hybrid censoring with product's life time following Lomax distribution have been investigated by Hassan et al. (2016).
The cumulative distribution function (cdf) and the probability density function (pdf) of Lomax distribution are given, respectively, by (；, ) = 1 − (1 + ) − , , , >0 g((；, ) =   (1 +   ) −(+1) , , , >0 where, 0   is the scale parameter and 0   is the shape parameter.In the literature, some of the extended and generalized forms of the Lomax distribution were derived and discussed by several authors.Ghitany  In recent years, new generated families of continuous distributions have attracted several statisticians to develop new models.These families are obtained by introducing one or more additional shape parameter(s) to the baseline distribution.A recent family of univariate distributions generated by Exponentiated Weibull random variables was suggested by Hassan and Elgarhy (2016) and then by Cordeiro et al. (2017).The cumulative distribution function of Exponentiated Weibull-generated (EW-G)family is defined by where ,0 a   are the two shape parameters and 0   is the scale parameter.The associated pdf is given by Note that; 1.For 1,   the cdf (3) reduces to the odd generalized exponential family (   the cdf (3) reduces to Burr X -G family (Yousof et al. 2017).In this study, we introduce a new five-parameter model, called the Exponentiated Weibull Lomax distribution based on the EW-G family.The rest of the paper is outlined as follows.In Section 2, we introduce the Exponentiated Weibull Lomax (EWL) distribution.In Section 3, we derive a very useful representation for the pdf and cdf of the proposed distribution besides some of its mathematical properties.In Section 4, three different methods of point estimation; namely maximum likelihood, least squares and weighted least squares are performed to obtain the point estimates of the model parameters.An extensive simulation study is performed to compare the performance of the different estimators in Section 5. Section 6 provides real data examples to illustrate the applicacbility of EWL distribution and finally we conclude the paper in Section 7.

For 1,
  the cdf (5) reduces to new model, called odd generalized exponential Lomax.

For
(1 Figure (1) provides plots of the pdf for some selected values of parameters.It is clear from Figure 1 that the EWL densities take various shapes such as symmetrical, right-skewed, reversed J-shaped and unimodal Also, Figure 2 shows that hazard rate shapes can take different shapes such as constant, increasing, decreasing, and reversed J shape.This fact implies that the EWL can be very useful for fitting data sets with various shapes.

Statistical Properties
This section provides some properties of the EWL distribution.

Useful Expansions
Here, expansions for the pdf and cdf of Exponentiated Weibull-Lomax distribution are derived.The pdf (6) can be rewritten as follows: )  )] −1 (7) Since the generalized binomial theorem, for 0   is real non integer and 1, z  is given by: ( −1  )  (8) Then, by applying the binomial theorem (8),where a is real non integer and the power series for the exponential function in (7), then the pdf of EWL distribution becomes: Hence, the pdf of EWL distribution takes the following form , Applying the power series for the exponential function and the binomial expansion ( 10) in ( 12), then we obtain where () is the cdf of the Exponentiated Lomax distribution with parameters , and q   .

Quantile Function
The quantile function, say 1 ( ) ( ) of X can be obtained by inverting (5) as follows where u is a uniform random variable on the unit interval   0,1 .In particular, the median is obtained by subsituting u  0.5 in (14).Furthermore, the variability of the skewness and kurtosis on the shape parameters ,  and a can be investigated based on quantile measures.Bowley skewness based on quantile has been introduced by Kenney and Keeping (1962) and given by: The Moors kurtosis (see Moors (1988)) based on quantiles is given by: where Q(.) denotes the quantile function.Plots of the skewness and kurtosis for some choices of the parameters  and  and a are shown in Figures 3 and 4.   While the skewness of EWL and WL are equal at 0.5,1.5   as seen in Figure 3(ii).The skewness of EWL is less than the skewness of WL at 0.8   as a function of  , while skewness of EWL is greater than the skewness of WL at 2.5   as a function of  (see Figure 3(iii)).
We can detect from Figures 4(iv) that the kurtosis of EWL is less than the kurtosis of WL at 0.5, 2   as a function of a. Also, the kurtosis of EWL is less than the kurtosis of WL at 0.8, 2.5

 
as a function of  (see Figure 4(vi)).While the kurtosis of EWL is greater than the kurtosis of WL at 0.5,1.5

 
( see Figure 4(v)).Generally, these plots show that all the values of skewness and kurtosis decrease when the values of the parameters increase.

Moments
The rth moment of EWL distribution can be obtained by using pdf (11) as follows x y and using the binomial expansion, hence the rth moment of EWL distribution takes the following form:  15), we can obtain the first four moments about zero.Furthermore, the moment generating functionof EWL distribution is obtained as follows: + )

The Probability Weighted Moments
The probability weighted moments (PWM) of a random variable X following the EWL distribution, say , , rs  is formally defined by: Inserting ( 11) and ( 13) into ( 16), hence, the PWM of EWL distribution takes the following form + )

Moments of Residual Life Function
The residual life plays an important role in life testing situations and reliability theory.The nth moment of the residual life is given by: The nth moment of the residual life of a random variable has EWL distribution is obtained by inserting pdf (11)

Order Statistics
Order statistics play a vital role in many areas of statistical theory and practice.We derive an explicit expression for the density function of the rth order statistic : rn X in a random sample 1: 2: : ... Therefore, the pdf of rth order statistics of Exponentiated Weibull Lomax can be expressed as a mixture of Exponentiated Lomax densities with parameters ,  and Further, the pdf of the smallest order statistics is obtained by subsituting 1 r  in (20) as follows

Different Estimation Methods
This section concerns with the point estimates of the model parameter for EWL distribution using three different methods.The maximum likelihood estimators, least squares estimators and weighted least squares estimators are derived in the following subsections.

Maximum Likelihood Estimators
The maximum likelihood (ML) estimators of the unknown parameters for the expoentiated Weibull Lomax distribution are obtained.Let 1 ,..., n XX be observed values from the EWL distribution with set of parameters ( , , , , ) .

T a     
The log-likelihood function for the vector of parameters  can be written as The elements of the score function U() = (  ,   ,   ,   ,   are given by , ML estimators of the model parameters are determined by solving numerically the nonlinear equations 0, 0, 0, 0  and 0 U   simultaneously by using mathematical package.... ) 2 , with respect to , , , a    and  .It is very hard to obtain a closed form solution, so mathematical software will be applied.

Weighted Least Squares Estimators
Here with respect to the unknown parameters , ,  and .Therefore, the WLS estimators will be obtained by minimizing the following quantity with respect , , , a    and  .

Simulation Study
In this section, an extensive simulation study is conducted to compare the performance of the different estimators in the sense of their relative biases (RBs) and standard errors (SEs) for different sample sizes and for different parameter values.1000samples of small, moderate and large sample sizes are generated from EWL distribution with different set of parameters.Without loss of generality, we take the scale parameter  to be known and equal one throughout the experiment and six sets of parameters are considered.The RBs and SEs of the ML, LS and WLS estimates of the models parameters are listed in Tables (1, 2 and 3).The simulation study is carried out as follows: Step 1: Generate 1000 random samples of size 10, 20, 30, 50 and 100 from the EWL distribution.
Step 4: The RBs and SEs of different estimates of unknown parameters are computed.
All the results of the simulation are listed in Tables (1, 2, and 3).Some conclusion can be deducted about the performance of different estimators: 1.For all different values of estimates and different methods of estimation we can realize that the SEs decrease as sample size increases.(see Tables (1, 2, and 3)).
2. The SEs of ML estimates, for all parameters values, are the largest among the other estimates in all cases (see Tables (1, 2, and 3)).
3. Based on Tables 1, 2 and 3, the SEs for the β estimate increase as the value of the parameter β increases, for all different methods of estimation.
4. The SEs of the estimate  are the smallest for set of parameters 2 and 4, for different methods of estimation and different sample sizes (see Tables 1 and 2).
5. Depending on Tables 1 and 2, both the RBs and SEs for  decrease when the value of  increases.

Applications to Real Data
In this section, two real data sets are provided to illustrate the importance and flexibility of EWL distribution comparing with main four models; Exponentiated generalized modified Weibull (EGMW) (Aryal and Elbatal (2015)), Beta modified Weibull (BMW) (Silva et al. (2010)),Kumaraswamy Lomax (KL); and Weibull Lomax (WL).The method of maximum likelihood is used to estimate the unknown parameters of the selected models.The following statistics: -2log-likelihood function ( 2 ln )  evaluated at the parameter estimates, Akaike information criterion (AIC), the corrected Akaike information criterion (CAIC), the Hannan-Quinn information criterion (HQIC), Anderson-Darling (A * ) criterion and Cramérvon Mises (W * ) criterion are used to compare all the models.However, the better distribution corresponds to the smaller values of AIC, BIC, CAIC, HQIC, A * and W * criteria.Furthermore, we plot the histogram for each data set and the estimated pdf of the models.Moreover, the plots of empirical cdf of the data sets and estimated cdf of the models are displayed in Figures 5 and 6.

Conclusion
In this paper, we present a new class of distributions, called the Exponentiated Weibull Lomax, based on Exponentiated Weibull-G family.The EWL distribution generalizes the Weibull Lomax distribution presented by Tahir et al. (2016 a) and at the same time, provides some new models.Some properties of the EWL distribution such as, moments, mean residual life, order statistics, quantile, Re'nyi and q-entropies are derived.The maximum likelihood, least squares, and weighted least squares estimators are obtained and simulation study is provided to compare the model performance of the estimates.An application of the EWL distribution to two real data sets show that the new distribution can be used quite effectively to provide better fits than Kumaraswamy-Lomax, Weibull-Lomax, beta modified Weibull and Exponentiated generalized modified Weibull models.
et al. (2007) suggested Marshall-Olkin extended Lomax distribution.Abdul-Moniem and Abdel-Hameed (2012) introduced the Exponentiated Lomax distribution by adding shape parameter to the distribution function of Lomax distribution.Elbatal and Kareem (2014) proposed the Kumaraswamy Exponentiated Lomax distribution.Lemonte and Cordeiro (2013) investigated beta Lomax, Kumaraswamy Lomax and McDonald Lomax.Cordeiro et al. (2015) introduced the gamma-Lomax based on gamma generated family.Ashour and Eltehiwy (2013) introduced the transmuted Exponentiated Lomax distribution.Shams (2013) introduced Kumaraswamy-generalized Lomax distribution.Tahir et al. (2016a) introduced the Weibull Lomax based on Weibull generated family.The Gumbel-Lomax has been introduced by Tahir et al. (2016b).Rady et al. (2016) introduced more flexible model through applying power transformation, named as the power Lomax distribution.
   is the set of parameters.The pdf of EWL distribution is obtained by inserting the pdf (1) and cdf (2) into (4) as the following (; )

Figure 1 :
Figure 1: Plots of the EWL pdf for some parameters

Figure 2 :Fx
Figure 2: Plots of the EWL hrf for some parameters

𝑓 1 :
() = ∑ ∑  ,,,,, ℎ ++++ (), ∞ ,,,,, −1 =0  ,,,, = (−1) ++ denotes the corresponding ordered sample.According toJohnson et al. (1995), the expectation and the variance of distribution are independent of the unknown parameter and are given by (( : )FX is cdf for any distribution and : in X is the ith order statistic.Hence, the least squares (LS) estimators can be obtained by minimizing the sum of squares errors ∑(( : ) the unknown parameters.So the LS estimators of the unknown parameters , , , a    and  are obtained by minimizing the following quantity

Figure 6 :
Figure 6: Densities and distributions of models for the second data set Another interesting function is the mean residual life (MRL),which represents the expected additional life length for a unit which is alive at age x .The MRL of the EWL Suppose 1 ,..., n XX is a random sample of size  from EWL distribution and suppose the weighted least squares (WLS) estimators of the unknown parameters for EWL are derived.Again, let 1 ,...,

Table 1 :
Results of simulation study of RBs and SEs of estimates for different values of parameters (, , , ) for the Exponentiated Weibull Lomax distribution

Table 2 :
Results of simulation study of RBs and SEs of estimates for different values of parameters (, , , ) for the Exponentiated Weibull Lomax distribution.

Table 3 :
Results of simulation study of RBs and SEs of estimates for different values of parameters (, , , ) for the Exponentiated Weibull Lomax distribution.

Table 6 :
ML estimates and their SEs (in parentheses) for the second data set Figure 5: Densities and distributions of models for the first data set

Table ( 7
) Model selection criteria for the second data set 