Extended Poisson Lomax Distribution

The main goal of this article is to introduce a new extension of the continuous Lomax distribution with a strong physical motivation. Some of its statistical properties such as moments, incomplete moments, moment generating function, quantile function, random number generation, quantile spread ordering and moment of the reversed residual life are derived. Two applications are provided to illustrate the importance and flexibility of the new model.


1.Introduction and physical motivation
In this work we introduce a new extension of the Lomax (Lx) distribution by compounding the Burr X Lomax (BrXLx) model with the zero truncated Poisson (ZTP) distribution. A random variable (r.v.) is said to have the BrXLx distribution (see Yousof et al. (2017) and Abouelmagd (2018)) if its cumulative distribution function (CDF) is given ( with corresponding PDF as The plots of the ZTPBrXLx PDF and HRF are displayed in Figures 1 and 2    This rest of this work is organized as follows: In Section 2, we derive some properties of the new model. Maximum likelihood estimation for the unknown model parameters under uncensored case is addressed in Section 3. In Sections 4, the potentiality of the proposed model is illustrated by means of two real data sets. Finally, Section 5 provides some concluding remarks.

2.Mathematical properties 2.1 Useful expansions
Consider the power series using (6), the PDF in (5) can be expanded as Consider the following power series and >0is a realnon-integer) .
Applying (8) to the quantity in Equation (7) we have Applying the power series to the quantity , Equation Applying (11) to (10) for the quantity , Equation (10) becomes where and , is the PDF of the exp-Lx model with power parameter [2(1 + ) + ] . Equation (12) reveals that the density of can be expressed as a linear mixture of exp-Lx densities. So, several mathematical properties of the new model can be obtained from those of the exp-Lx distribution. Similarly, the CDF of the ZTPBrXLx model can also be expressed as a mixture of exp-Lx CDFs given by

Theorem Let be a continuous r.v. having the exp-Lx model with power parameter , then the th ordinary and incomplete moments ( ′ ( ) and ( ) ) of the exp-Lx r.v. (defined in this subsection) given by
and where is the complete beta function and is the incomplete beta function.

Moments and incomplete moments
The ℎ ordinary moment of , say ′ , follows from (12) as where , , The ℎ incomplete moment of is defined by We can write from (12) (14) gives the mean of . Two important applications of the first incomplete moment ( )| ( =1) are related to the mean deviations about the mean and median and to the Bonferroni and Lorenz curves.

Quantile spread ordering
The quantile spread (QS) of the r.v. ∼ ZTPBrXLx ( , , , ) having CDF (5)  . The QS of a distribution describes how the probability mass is placed symmetrically about its median and hence can be used to formalize concepts such as "peakedness" and "tail weight traditionally associated with kurtosis". So, it allows us to separate concepts of "kurtosis" and "peakedness" for asymmetric models. Let 1 and 2 be two r.v.s following the ZTPBrXLx ( , , , ) with quantile spreads [ ] 1 and [ ] 2 , respectively. Then 1 is called smaller than 2 in quantile spread order, denoted by 1 0.5,1)) .
The following properties of the QS order can be obtained:

3.Estimation
We will consider the method of the maximum likelihood to estimation the unknown parameters ( , , , ) of the new model from the complete samples. Let 1 , ⋯ , be a random sample (r.s.) from the ZTPBrXLx models with parameter vector = ( , , , ) , the log-likelihood function for is given by The above log-likelihood function can be maximized numerically by via R (optim) or SAS (PROC NLMIXED) or Ox program (sub-routine MaxBFGS), among others. For interval estimation of the parameters, the elements of the 4 × 4 observed information matrix ( ) can be derived and evaluated numerically (see section 4).

4.Modeling data
In this section, we provide two applications to show empirically the potentiality of the ZTPBrXLx model. In order to compare the fits of the ZTPBrXLx distribution with other competing distributions, we consider the Cramér-von Mises ( * ) and the Anderson-Darling ( * ) statistics. These two statistics are widely used to determine how closely a specific CDF fits the empirical distribution of a given data set. We provide two applications to two real data sets to prove the importance and flexibility of the new model. We compare the fit of the new model with competitve models namely: The Burr X Lomax (BrXLx):

Application 1: modeling failure times
This data set represents the data on failure times of 84 aircraft windshield given in Murthy et al. (2004) figure 3, the ZTPBrXLx lifetime model is much better than gamma Lomax, beta Lomax, exponentiated Lomax and Lomax models so the exponentiated Lomax, model is a good alternative to these models in modeling aircraft windshield data.   figure 4, the ZTPBrXLx lifetime model is much better than gamma Lomax, beta Lomax, exponentiated Lomax and Lomax models so the new model is a good alternative to these models in modeling the service times data.

5.Concluding remarks
The main goal of this article is to introduce a new extension of the continuous Lomax distribution with a strong physical motivation. Some of its statistical properties such as moments, incomplete moments, moment generating function, quantile function, random number generation, quantile spread ordering and moment of the reversed residual life are derived. Two applications are provided to illustrate the importance and flexibility of the new model. The new lifetime model is much better than other competitive models such as the gamma Lomax, the Burr X Lomax, the beta Lomax, the exponentiated Lomax and the original Lomax model so the new model is a useful alternative to these models in modeling the failure and service times data. Figure 4: Estimated PDF, P-P plot, Estimated CDF, HRF for data set II.