Abstract: This article introduces and discusses a new three-parameter lifespan distribution called Zero-Truncated Poisson Pareto

distribution ZTPP. that is built on compounding Pareto distribution as a continuous distribution and Zero-Truncated Poisson

distribution as a discrete distribution. Various statistical properties and reliability characteristics of the proposed distribution have

been investigated including explicit expressions for the moments, moment generating function, quantile function, and median.

With three parameters, the suggested distribution has an advantage over other distributions in that it makes estimating the model

parameters simpler. To estimate the unknown parameters of the ZTPP distribution, the maximum likelihood method, and L.

Received: October 22, 2022. Revised: December 19, 2022. Accepted: January 17, 2023. Published: February 16, 2023.

1. Introduction

he processes of installing distributions results in new

distributions. One such complex probability distribution

that is important in practical applications is the Poisson

composite distribution [1]. Since the nature of some data or

occurrences necessitates the use of composite distributions,

which are more flexible than standard distributions,

distribution is one of the composite distributions. Under its

name, this distribution involves many distributions. Recently,

new distributions have been proposed by integrating

continuous distribution with another discrete distribution. For

example, we can cite some of them such as the exponential

geometric by Adamidis.[2]; Silva proposed the generalized

exponential geometric [3]; Barreto-Souza et al. [4] proposed

the Weibull geometric; the Poisson exponential by Cancho et

al.[10]; and the Binomial-exponential 2 by Bakouch et al.

[11]. The Pareto Geometric by Nassar et al.[12] can also be

cited with some distributions related to the Pareto distribution

beta modified Weibull by Silva et al.[13]; the Pareto-type

distribution by Bourguignon et al.[14]; the bivariate Pareto by

Sankaran et al.[15]; and the beta generalized Pareto by

Mahmoudi et al.[16].

The shape and scale characteristics of the Generalized

Pareto Distribution GPD can be estimated using a variety of

methods. Moments-based approaches, maximum likelihood,

extension of the GPD and used parametric estimation.

Classical approaches, however, might not be appropriate in all

circumstances, as stated in [18]. That is why Zero-Truncated

Poisson inference could be advisable.

There aren't many approaches for combining the Pareto and

Poisson distributions. We can cite [20], who suggested using

conjugate prior distributions; thus in this paper, we derive

some structural properties of the (ZTP) and (P) distributions

based on a double integrating mechanism to it’s with a three-

parameter lifetime distribution, installing new distribution

The researchers in this research were interested in installing

the Zero-Truncated Poisson distribution with the Pareto

distribution, which resulted in a new distribution called the

Zero-Truncated Poisson Pareto distribution (ZTPP). Thus the

primary goal of this paper is propose a new life distribution

consisting of three parameters, which is a direct extension of

the Pareto distribution with two parameters. It is obtained by

integrating the Pareto distribution as a continuous distribution

distribution and the new distribution is called the Zero-

Truncated Poisson Pareto distribution (ZTPP). The probability

mass function of the Zero-Truncated Poisson distribution is:

Let  be the series of independent symmetric

distributions: 󰇛󰇜then the

distribution function for U and its dynasty function are:

respectively. The RV 󰇛󰇜 has a

cumulative distribution function (cdf) and its probability

density function (pdf), given by:

This paper is structured as follows: in Section II is derived

the cumulative distribution function, density function, and

failure function. We also present in Section III the statistical

characteristics of the distribution and in Section IV we present

two different methods of estimation: the maximum likelihood

method and the L. Moments method. In Section V we present

the application on real data and compare the results we

function, respectively: 󰇛󰇜 ,󰇛󰇜 , 󰇛󰇜where N

has ZTP distribution with parameter λ. is given by (1) then a

random variable 󰇛󰇜 is the

cumulative distribution function and its probability density

function, respectively:

with parameter λ. is given by (1) then a random variable 

󰇛󰇜 is the cumulative distribution function

and its probability density function, respectively: by equation

(2) and (3) we get cumulative distribution function and its

and L. Moments in (b).

The Maximum Likelihood method is one of the traditional

and widely used methods for estimating the parameters of the

model ZTPP that makes the logarithm of the Likelihood

function at its end and is easy to use analytically or

numerically with parameter estimators for large samples.

By deriving the function L for the parameters󰇛󰇜 is given



















Is a square and symmetric matrix that contains

approximate covariances of the Maximum Likelihood

estimators for the parameters of the ZTPP model of that

matrix, representing the covariance between each pair of

estimators and its main diagonal contains variances. It can be

used to estimate model parameters with confidence intervals.



 



 





ordinal statistics in a similar way. Therefore, the mean vector

and the variance and variance matrix include various elements

of the covariance and its properties, which are usually used to

summarize the features of multivariate distributions. To

overcome this drawback, Serfling and Xiao [22] proposed the

multivariate L-moments method, and its components are the

central moments, but this method does not assume the central

moments of the second and higher than the second must be

specified. Suppose x is a continuous random variable, the

cumulative distribution function󰇛󰇜 and the quantitative

function 󰇛󰇜, and that:



For any distribution, the first four moments can easily be



󰇛



󰇛

Step (1): Obtain the inverse distribution 󰇛󰇜 of the

distribution, which is given by: 󰇛󰇜 is the inverse function

of the function we obtained when U=F in the equation (7).

󰇛󰇜

Step (2): Obtain the theoretical probability-weighted moments

of the  ,where :

Letting 󰇛󰇛󰇜󰇜 , we have



Where:



  󰇟󰇛󰇜󰇠

󰇟󰇛󰇜󰇠

󰇟󰇛󰇜󰇠

Step(3): Replace the theoretical probability

weighted moment  and  by their

sample estimator, since the sample estimators are:

󰇡

  

distribution, the exponentiated pareto EP distribution, the

Pareto distribution P, and the BEP beta exponentiated pareto

distribution using an actual data sample. The models' fit is

compared with the outcomes. We take into consideration an

unedited data set that represents to remission times in months

includes 128 bladder cancer patients chosen at random. Lee et.

al [23] and Lemonte et.al [24] already examined these data. In

the case of bladder cancer, abnormal cells in the bladder

become out of control. Transitional cell carcinoma, the most

prevalent form of bladder cancer, mimics the typical urothelial

4.33, , 4.40, 4.50, 4.51, 4.34, 4.87, 4.98, 5.06, 5.17, 5.32, 5.32,

5.09, 5.34, 5.41, 5.49, 5.62, 5.71, 5.41, 5.85, 6.25, 6.54, 6.93,

6.94, 6.97, 6.76, 7.09, 7.26, 7.32, 7.39, 7.59, 7.28, 7.62, 7.63,

7.66, 7.87, 8.26, 8.37, 7.93, 8.53, 8.65, 9.02, 9.22, 9.47,

8.66,9.74, 10.06, 10.34, 10.66, 10.75, 11.25, 11.64, 11.79,

11.98, 12.02, 12.03, 12.63, 13.11, 12.07, 13.29, 13.80, 14.76,

20.28, 18.10, 21.73, 22.69, 25.74, 25.82, 23.63, 26.31, 32.15,

36.66, 43.01, 46.12, 34.26, 79.05.

minimum values rule of AIC, BIC, CAIC is taken into

consideration for selecting the best model to fit. These

parameters. Results show that our model satisfied the

minimum rule, hence it is the best one.

6. Discussion and Conclusion

In this study, we introduce a new distribution of life called

distribution as a discrete distribution, with three parameters,

thus in this paper, we derive some structural properties of the

(ZTP) and (P) distributions based on a double integrating

mechanism to it’s with three-parameter lifetime which resulted

in a new distribution called the Zero-Truncated Poisson Pareto

distribution (ZTPP) or in short ZTPP distribution. with three

parameters, the suggested distribution has an advantage over

other distributions in that it makes estimating the model

parameters simpler. To estimate the unknown parameters of

the ZTPP distribution, the maximum likelihood method and L,

method are employed. The distribution was applied to real

data to ensure the possibility of applying it to life data and

through the application, the distribution ZTPP was compared

with other distributions, as Pareto, BEP, BP, and EP, by the

comparison proving that the new distribution is better than the

distributions that had been compared with its in economics,

and other fields. Overall the result indicated the ZTPP is better

(CAIC), and (BIC) are minimum values by using the ZTPP

distribution as shown in table 1.

Anal., vol. 54, no. 4, pp. 935–944, 2010.

distribution with increasing failure rate: Exponential

truncated Poisson,” J. Basic Appl. Sci. Res., vol. 2, no.

2, pp. 1749–1762, 2012.

“Pareto Poisson–Lindley distribution with

applications,” J. Appl. Stat., vol. 40, no. 8, pp. 1717–

Cordeiro, “The Poisson Nadarajah–Haghighi

distribution: Properties and applications to lifetime

data,” Int. J. Reliab. Qual. Saf. Eng., vol. 27, no. 01, p.

“Binomial-exponential 2 Distribution: Different

Estimation Methods and Weather Applications,”

Pareto–geometric distribution,” J. Egypt. Math. Soc.,

140, no. 6. pp. 1353–1373, 2010, doi:

10.1016/j.jspi.2008.11.020.

multivariate L-moments: L-comoment matrices,” J.

Multivar. Anal., vol. 98, no. 9, pp. 1765–1781, Oct.

2007, doi: 10.1016/j.jmva.2007.01.008.

data analysis, vol. 476. John Wiley & Sons, 2003.

[24] A. J. Lemonte and G. M. Cordeiro, “An extended

Lomax distribution,” Statistics (Ber)., vol. 47, no. 4,

pp. 800–816, 2013.