Exponentiated Lomax Geometric Distribution : Properties and Applications

In this paper, a new four-parameter lifetime distribution, called the exponentiated Lomax geometric (ELG) is introduced. The new lifetime distribution contains the Lomax geometric and exponentiated Pareto geometric as new sub-models. Explicit algebraic formulas of probability density function, survival and hazard functions are derived. Various structural properties of the new model are derived including; quantile function, Re'nyi entropy, moments, probability weighted moments, order statistic, Lorenz and Bonferroni curves. The estimation of the model parameters is performed by maximum likelihood method and inference for a large sample is discussed. The flexibility and potentiality of the new model in comparison with some other distributions are shown via an application to a real data set. We hope that the new model will be an adequate model for applications in various studies.


Introduction
introduced an important and widely used lifetime model, the so-called Lomax distribution, and it used for stochastic modeling of decreasing failure rate.It has been applied in studies of income, size of cities, wealth inequality, engineering, queuing theory and biological analysis.
Studies about Lomax distribution have been discussed by several authors.Some properties and moments for the Lomax distribution have been discussed by Balakrishnan and Ahsanullah (1994).The discrete Poisson-Lomax distribution has been provided by Al-Awadhi and Ghitany (2001).The Bayesian and non-Bayesian estimation of the reliability has been studied by Abd-Elfattah et al. (2007) The cumulative distribution function (cdf) of Lomax distribution with shape parameter  and scale parameter  is given by ( ; , ) 1 ( 1) , , , 0. (1) The probability density function (pdf) of Lomax distribution is as follows ( 1) ( ; , ) , , , 0. (2) The exponentiated Lomax has been introduced by Abdul-Moniem and Abdel-Hameed (2012) by adding shape parameter to the cdf (1).The cdf of the EL takes the following form ( ; , , ) [1 ( 1) ] , , , , 0. (3) The corresponding pdf is as follows: ( ( ; , , ) ) ] , , , , 0. (4) Also, a discrete random variable, N is a member of a zero-truncated geometric random variable independent of ', Xswith probability mass function (pmf) given by: n P N n p p p n p

Construction of the Distribution
In this section, following the same approach of Adamidis and Loukas (1998), we introduce and study the exponentiated Lomax geometric distribution.The probability density function, distribution function, reliability and hazard rate function are obtained.
Let 12 , ,..., N X X X be a random sample of size N from the Exponentiated Lomax distribution with cdf(3) and N be a zero-truncated geometric random variable independent of ', Xswith pmf (5).
Based on cdf (7), some special distributions arisefrom the ELG distribution as follows: 1.As 0, p  the exponentiated Lomax (EL) is a limiting case of the ELG distribution.
Figure (1) gives some possible shapes of the density (6) for some selected parameter values.

Figure 1: Plots of the ELG densities function for some parameter values
In addition, the reliability and hazard rate functions of ELG distribution are as follows:   2 shows that the shapes of the hazard rate are increasing, decreasing and constant at some selected values of parameters.

Some Statistical Properties
In this section, some of statistical properties of the ELG distribution including, expansion for pdf (6) and cdf (7), quantile function, rth moment, and probability weighted moments are derived.Furthermore, Re'nyi entropy, distribution of order statistics, Bonferroni and Lorenz curves are provided.

Usefuel expansions
In this subsection, two useful expansions for the pdf (6) and cdf (7) are derived.We show that the pdf of ELG can be expressed as linear combinations of EL distributions.Also the two expansions are used to determine some mathematical properties of the ELG distribution.
Firstly, the pdf (6) of ELG can be expressed as linear combinations of EL distributions.
Using the following series expansions Then, the pdf (6) can be written as follows ( 1) 1 Then, by using the binomial expansion for the previous pdf, then it can be written as where, pdf (9) leads to the following infinite linear combination , (1 ), 1  Secondly; an expansion for [ ( ; )] s Fx is derived from cdf (7) through the expansions defined in (8) as follows

Quantile measures
The quantile function of ELG distribution, denoted by, ) where  is a uniform random variable on the unit interval (0,1).In particular the median of the ELG distribution, denoted by , m is obtained by substituting 0.5 u  in (12) as follows The Bowleyskewness (see Kenney and Keeping (1962)) based on quantiles can be calculated by Further, the Moors kurtosis (see Moors (1988)) is defined as   where Q(.) denotes the quantile function.Plots of the skewness and kurtosis for some choices of the parameter  as function of p, and for some choices of the parameter p as function of  are shown in Figures 3 and 4. We can detect from these figures that the skewness and kurtosis for p decreases as  increases from 0.5 to 2.5.Also, the skewness and kurtosis for  decreases as p increases from 0.1 to 0.7

Moments
The moments of any probability distribution are necessary and important in any statistical analysis especially in applied work.Some of the most important features and characteristics of a distribution can be studied through moments such as; dispersion, skewness and kurtosis.
An explicit expression for the rth moment of ELG distribution about the origin is obtained by using pdf (10) as follows: ,0 0 ( ; , ) , ) ] .


, and using binomial expansion then the above integral is reduced to ) .

   
Furthermore, it is easy to show that, the moment generating () Mt function can be written as follows: where, ' r  is the rth moment.Then by using ( 13), the moment generating function of ELG distribution can be written as follows: ,, , , )

The probability weighted moments
Probability weighted moments (PWMs) were devised by Greenwood et al. (1979) primarily as an aid to estimate the parameters of distributions that are analytically expressible only in inverse form.PWMs are the expectations of certain functions of a random variable defined when the ordinary moments of the random variable exist.The PWMs of a random variable X are formally defined by , [ F( ) ] f( )(F( )) .( 14) The PWMs of ELG distribution is obtained by subsituting pdf (10) and cdf (11) in (14) as follows: where 1 , , ,  , , , 0 0  and using binomial expansion then , rs  takes the following form:


Hence, the PWMs of ELG distribution can be expressed as: (1 ).11

Re'nyi entropy
The entropy of a random variable Xis a measure of uncertainty variation.If X is a random variable which distributed as ELG, then the Re'nyi entropy, for 0   and 1   is defined by: Then by using pdf (6), the Re'nyi entropy of ELG can be written as follows: ( 1) 1 (1 By using the series expansion (8), then the above integration IP can be written as follows   ( 1) ,0 0 Therefore, the Re'nyi entropy of ELG distribution is as follows

Order statistics
In this subsection, a closed form expression for the pdf of the rth order statistics of the ELG distribution will be derived.Let  1 ,  2 , … ,   be a simple random sample from ELG distribution and let X1:n, X2:n,..., Xn:n denote the order statistics obtained from this sample.
According to David (1981), the pdf of rth order statistics is as follows


Substituting cdf (7), then the probability density function of rth order statistics Xr:n from ELG distribution is derived as follows Using the power series (8) and binomial expansion, then (15) takes the following form Then, by substituting pdf (6), using the series expansion (8) and binomial expansion in the previous equation,we have ( 1) Therefore, the pdf of rth order statistics of ELG can be expressed as follows : , , , , ( ) 0 , , , 0 ( ; ) ( ; , ), 0 , ( 16) ( 1) (1 ) , 2 ( , 1)( ) denotes the pdf of EL with parameters , ( ), m k r i     and . In particular, the pdf of the smallest order statistic X1:n is obtained by substituting r=1 in ( 16) as follows ( 1) ) denotes the pdf of EL with parameters , ( 1 ), m k i     and .

Bonferroni and Lorenz curves
Bonferroni and Lorenz curves are income inequality measures that are also useful and applicable to other areas including reliability, demography, medicine and insurance.The Bonferroni curve is calculated by the following form: Then by using pdf (10), the Bonferroni curve can be expressed as follows: ( 1) ( 1) 1 , , 0 0
Using the series expansion, then the previous integeral is as follows , then the Bonferroni curve can be written as follows ) Also, the Lorenz curve is calculated by the following form The Lorenz curve of ELG distribution takes the following form )

Parameter Estimation
In this section, estimation of the ELG model parameters; is obtained by using maximum likelihood method of estimation.
Let 12 , ,... n X X X be a simple random sample from the ELG distribution with set of parameters ( , , , ).

p     
The log-likelihood function, denoted by , lnl based on the observed random sample of size n from density ( 6) is given by: The partial derivatives of the log-likelihood function with respect to the unknown parameters are given by: 11 ln( ln 2 , ) The maximum likelihood estimators of the model parameters are determined by solving numerically the non-linear equations For interval estimation and hypothesis tests on the model parameters, the observed Fisher's information matrix must be obtained.The 4×4 unit observed information matrix () The approximate 100(1 )%   two sided confidence intervals for , , , p    are respectively, given by: 2 2 2 ˆˆˆv ar( ), var( ), var( ) Z is the upper 2  th percentile of the standard normal distribution and var (.)'s denote the diagonal elements of 1 ()  corresponding to the model parameters.

Applications
In this section, the flexibility of ELG model is examined by comparing it with some other distributions.Two real data sets are used to show that ELG distribution can be applied in practice and can be a better model than some others.
For the two sets of data; the ELG is compared to Lomax (L), exponentiated Lomax, kumaraswamy Lomax (KL), Weibull Lomax (WL) and exponentiated Pareto (EP) distributions.The density functions for kumaraswamy Lomax, Weibull Lomax and exponentiated Pareto distributions are as follows; ( ; , , , ) ( ; , ) where k is the number of models parameter, n is the sample size and L ln is the maximized value of the log-likelihood function under the fitted models.The better model is corresponding to the lower values of, AIC, CAIC, BIC, and k-s statistics.The results for the previous measures to the mentioned models are listed in Table 1.The values in Table1 indicate that the most fitted distribution to the data is ELG distribution compared to other distributions considered here (EL, EP, KL, WL, L).

Table1: Measurements for all models based on the first data set
Plots of the estimated cumulative and estimated densities of the fitted models for the first set of data are described below, Figure 5.Estimated densitiesof models for the first data set The second data set contains 100 observations on breaking stress of carbon fibers (in Gba) studied by Nichols and Padgett (2006).The second set of data are as follows: Furthermore, the graphical comparison corresponding to these fittedmodelsto conform our claim is illustrated in Figures 7 and 8.
)ln ( Ghitany et al. (2007) introduced Marshall-Olkin extended Lomax.Hassan and Al-Ghamdi (2009) determined the optimal times of changing stress level for simple stress plans under a cumulative exposure model for the Lomax distribution.Hassan et al. (2016) discussed 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.Some extensions of the Lomax distribution have been constructed by several authors.Abdul-Moniem and Abdel-Hameed (2012) introduced the exponentiated Lomax (EL) by adding a new shape parameter to the Lomax distribution.Lemonte and Cordeiro (2013) investigated beta Lomax, Kumaraswamy Lomax and McDonald Lomax distributions.The gamma-Lomax distribution has been suggested by Cordeiro et al. (2013).Five-parameter beta Lomax distribution has been investigated by Rajab et al. (2013).The Weibull Lomax and Gumbel-Lomax distributions have been introduced by Tahir et al. (2015a) and (2015 b).

Figure 3 :Figure 4 :
Figure 3: Skewness of the ELG distribution.(a) As function of p for some values of  with 0.5   and 1   (b) As function of for some values of p with 1.5   and 1  

7 
Kurtosiswhere, (.,.)  stands for beta function.Hence the rth moment for ELG distribution takes the following form: The entries of Fisher's information matrix for ELG are given in the Appendix.

Figure 6 .
Figure 6.Estimated cumulative densities for the first data set

Figure 7 .Figure 8 .
Figure 7.Estimated densities of models for the second data

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Again, Figures 7 and 8show that ELG model is the best fitted model for the second dataIn the present study, we propose a new distribution called exponentiated Lomax geometric distribution.The subject distribution is derived by compounding exponentiated Lomax and geometric distributions.The density function of ELG can be expressed as a mixture of EL density functions.The ELG distribution includes the Lomax geometric and exponentiated Pareto geometric as new distributions.Explicit expressions for moments, probability weighted moments, Bonferroni and Lorenz curves, order statistics and R´enyi's entropy are derived.The estimation of parameters along with the information matrix is derived.Applications of the exponentiated Lomax geometric distribution to realdata show that the new distribution can be used quite effectively to provide better fits as compared toLomax, exponentiated Lomax, Kumaraswamy Lomax, Weibull Lomax and exponentiated Pareto distributions.26.Tahir, M. H., Hussain, M. A., Cordeiro, G. M, Hamedani, G. G., Mansoor, M. and Zubair, M. (2015 b).The Gumbel-Lomax distribution: properties and applications, Journal of Statistical Theory and 15 (1), 61-79.27.Zakerzadeh, H. and Mahmoudi, E. (2012).A new two parameter lifetime distribution: model and properties, arXiv:1204.4248[stat.CO]