On Five-Parameter Lomax Distribution : Properties and Applications

A five-parameter continuous model, called the beta exponentiated Lomax distribution, is defined and studied. The model has as special sub-models some important lifetime distributions discussed in the literature, such as the logistic, Lomax, exponentiated Lomax, beta Lomax distributions, among several others. We derive the ordinary and incomplete moments, generating and quantile functions, mean deviations, Bonferroni, Lorenz and Zenga curves, mean residual life, mean waiting time and Rényi of entropy. The method of maximum likelihood is proposed for estimating the model parameters. We obtain the observed information matrix. Three real data sets demonstrate that the new distribution can provide a better fit than other classical lifetime models.


Introduction
The Lomax (or Pareto II) distribution has wide applications in many fields such as income and wealth inequality, medical and biological sciences, engineering, size of cities actuarial science, lifetime and reliability modeling.In the lifetime context, the Lomax model belongs to the family of decreasing failure rate (see Chahkandi and Ganjali, 2009) and arises as a limiting distribution of residual lifetimes at great age (see Balkema and de Hann, 1974).For more information about the Lomax distribution and Pareto family are given in Arnold (1983) and Johnson et al. (1994).Various generalizations of Lomax distribution have been studied, the exponentiated Lomax, discussed by Abdul-Moniem and Abdel-Hameed (2012), Marshall-Olkin extended Lomax defined by Ghitany et al. (2007), McDonald Lomax investigated by Lemonte and Cordeiro (2013), gamma Lomax introduced by Cordeiro et al. (2015), the Weibull Lomax distribution studied by Tahir et al. (2015) and recently the transmuted Weibull Lomax distribution given by Afify et al. (2015).
The random variable X with exponentiated Lomax (EL) distribution has the cumulative distribution function (cdf) given by ( ; , , ) 1 ( 1) , G x x  Let () Gx be the cdf of any random variable X .The cdf of a generalized class of distributions defined by Eugene et al. (2002) is given by  is the beta function and (.)  is the gamma function.The corresponding pdf for (3) is given by where ( ) ( )  is the baseline density function.Replacing (1) in (3), we obtain a new distribution, called beta exponentiated Lomax (BEL), with cdf given by   , ( , ) is the vector of the model parameters.Equation ( 5) can be expressed as follows where   is the well known hypergeometric function (Gradshteyn and Ryzhik, 2007).
The pdf corresponding to ( 5) is given by In Fig. and Plots of the HRF for different values of ( , , , , ) ab   are given in Fig. the density (6) yields the standard Lomax (SL) distribution.

Some Statistical Properties
We give a mathematical treatment of the new distribution including expansions for the density function, moments, incomplete moments, quantile function, mean deviations, Bonferroni, Lorenz and Zenga curves, mean residual life, mean waiting time and Rényi entropy.

Expansions for the Distribution and Density Functions
Equations ( 5) and ( 6) are straightforward to compute using any statistical software.However, we obtain expansions for () Fx and () fx in terms of an infinite (or finite) weighted sums of cdf 's and pdf 's of random variables having Lomax distributions, respectively.For any positive real number b and for | z | < 1, a generalized binomial expansion holds Therefore, the cdf of BEL can be expanded to obtain , ( , )     denotes the cdf of EL with parameters  ,  and () aj   .
Similarly, we can write the pdf (6) as where ( ; , , ( )) H x a j     denotes the EL density function with parameters  ,  and () aj Again, by using binomial expansion in equation ( 11), we obtain where

 and ( ; , ( 1)) h x i
  denotes the Lomax density with parameters  and ( 1) If b is an integer, then the summation in equations ( 10), ( 11) and ( 12) stops at 1 b  .Thus, the BEL density function can be expressed as an infinite linear combination of Lomax densities.Thus, some of its mathematical properties can be obtained directly from those properties of the Lomax distribution.

Moments and Moment Generating Function
As with any other distribution, many of the interesting characteristics and features of the BEL distribution can be studied through the moments.If we assume that Y is a Lomax distributed random variable, with parameters  and  , then the rth moment of Let X be a random variable having the BEL distribution (6).Using equation (12), it is easy to obtain the rth moment of The mean, variance, Skewness and Kurtosis can be obtained from (13).If 0 b  is integer and ( 1) ir  , the sum stops at 1 b  .

Quantile Function
Let , () ab Qu be the beta quantile function with parameters a and b .The quantile function of the BEL distribution, say ( ), x Q u  can be easily obtained as .
This scheme is useful to generate BEL random variates because of the existence of fast generators for beta random variables in most statistical packages, i.e. if V is a beta random variable with parameters a and b , then follows the BEL distribution.From (14) we conclude that the median m of X is (1 2).mQ  The Bowley skewness (SK) measure and Moors kurtosis (KR) (based on octiles) of the BEL distribution can be calculated using the formulae given below ) ( )

If
Y is a random variable with BXII distribution with parameters  and  , the rth incomplete moment ofY , for r   , is given by From this equation, we note that () Let X be a random variable having the BEL distribution (6).The rth incomplete moment of . The mean deviations about the mean and the median can be used as measures of spread in a population.Let and  be the mean and the median of the BEBXII distribution respectively.The mean deviations about the mean and about the median of X can be calculated as  respectively, where 1 () m  denotes the first incomplete moment and () F  follows from (5).

Mean Residual Life and Mean Waiting Time
The mean residual life function (MRL) at a given time t measures the expected remaining lifetime of an individual of age t.It is denoted by () mt .The MRL or life expectancy of BEL is defined as ( ( , ) .

Lorenz, Bonferroni and Zenga Curves
Lorenz and Bonferroni curves have been applied in many fields such as economics, reliability, demography, insurance and medicine, (see Kleiber and Kotz, (2003) for additional details).Zenga curve was presented by Zenga (2007) are the lower and upper means respectively.For the BEL distribution, these quantities are derived below 1.

Rényi Entropy
The entropy of a random variable X is a measure of uncertainty variation.The Rényi entropy is defined as where ( ) ( ) , 0 Based on the binomial expansion to the last factor in the above integrand yields Again, using the binomial expansion to the last factor, we obtain ( 1) ( ) ( 1) (1 ) .( , )

 
Using integration in above expression and simplifying,


Hence, the Rényi entropy reduces to

Estimation of Parameters
The maximum likelihood estimation (MLE) is one of the most widely used estimation method for finding the unknown parameters.Let 12 , ,...., n X X X be an independent random sample from BEBXII.The total log-likelihood is given by (1 ), where ( 1) .
The score vector ( , , , , ) ), where () p  is the digamma function which is the derivative of log (.)  .The maximum likelihood estimates (MLEs) of the unknown five parameters can be obtained by solving the system of nonlinear equations 0  , iteratively.
For interval estimation of ( , , , , ) ab   ) and hypothesis tests on these parameters, we obtain the observed information matrix since its expectation requires numerical integration.The 55  observed information matrix () whose elements are given in Appendix.
As a second application, we analyze a real data set on the active repair times (hours) for an airborne communication transceiver.This data set was analyzed by Jorgensen (1982).
Tables 1, 2 and 3 list the maximum likelihood estimates MLEs (the corresponding standard errors in parentheses) of the parameters of all the models for the three data sets respectively.The statistics: 2  (where denotes the log-likelihood function evaluated at the maximum likelihood estimates), the Anderson-Darling ( * A ) and Cramér-von Mises ( * W ) are reported in Tables 1, 2 and 3.In general, the distribution which has the smaller values of these statistics is the better the fit to the data.The results show that the BEL distribution provides a significantly better fit than the other six models.All the computations were done using the MATH-CAD PROGRAM.

Concluding remarks
In this paper, we proposed a new distribution, named the beta exponentiated Lomax distribution which extends the Lomax distribution.Several properties of the new distribution were investigated, including ordinary and incomplete moments, mean deviations, Rényi entropy, and reliability.The model parameters are estimated by maximum likelihood and the information matrix is derived.Three applications of the beta exponentiated Lomax distribution to real data show that the new distribution can be used quite effectively to provide better fits than the exponentiated Lomax (Abdul-Moniem and Abdel-Hameed, 2012), beta Lomax and McDonald Lomax (Lemonte and Cordeiro, 2013), Weibull Lomax (Tahir et al., 2015), gamma Lomax (Cordeiro et al., 2015) and recently, the transmuted Weibull Lomax (Afify et al., 2015).We hope that the proposed model may attract wider applications in many areas such as engineering, survival analysis, hydrology, economics, and so on.(1 ) , ( 1) (

1. The pdf of the BEL for different values of the parameters
On Five-Parameter Lomax Distribution: Properties and Applications Pak.j.stat.oper.res.Vol.XII No.1 2016 pp185-199  Fig.

Table 1 : MLEs (standard errors in parentheses) for BEL, BL, EL, McL, TWL, WL and GL models and the statistics 2,
* W * A

Table 2 : MLEs (standard errors in parentheses) for BEL, BL, EL, McL, TWL, WL and GL models and the statistics 2, 
*W and *A ;

second data set.
* W * A

Table 3 : MLEs (standard errors in parentheses) for BEL, BL, EL, McL, TWL, WL and GL models and the statistics 2
,  * W and * A ;

third data set
* W * A ,