A New Flexible Distribution Based on the Zero Truncated Poisson Distribution: Mathematical Properties and Applications to Lifetime Data

The so called zero truncated Poisson (ZTP) distribution is a discrete probability model whose support is the set of only the positive integers ( ) + I . The ZTP is also known as the positive Poisson distribution or the conditional Poisson distribution. It is the conditional probability distribution of a Poisson-distributed random variable (RV), given that the value of the RV is 0. ≠ Thus, it is impossible for a ZTP RV to be zero, in this paper we will introduce a new flexible model based on the ZTP distribution for modeling lifetime data.


Introduction and Motivation
The so called zero truncated Poisson (ZTP) distribution is a discrete probability model whose support is the set of only the positive integers ( ) + I . The ZTP is also known as the positive Poisson distribution or the conditional Poisson distribution. It is the conditional probability distribution of a Poisson-distributed random variable (RV), given that the value of the RV is 0. ≠ Thus, it is impossible for a ZTP RV to be zero, in this paper we will introduce a new flexible model based on the ZTP distribution for modeling lifetime data. Suppose that a system (machine) has N subsystems functioning independently at a given time where N has ZTP distribution with parameter .
λ It is the conditional probability distribution of a Poisson-distributed random variable, given that the value of the random variable is not zero. The probability mass function (PMF) of N is given by And respectively. Suppose that the failure time of each subsystem has the TLW model defined by CDF and PDF in (2) Therefore, the marginal CDF of X is can be expressed as Equation (5) is called the CDF of the zero truncated Poisson Topp Leone Weibull (ZTPTLW) model. The corresponding PDF of (5) reduces to using the power series in (8) and after some algebra the PDF of the ZTPTLW in (7) can be expressed as where ( ) Equation (9) reveals that the density of X can be expressed as a linear representation of exponentiated-W (Exp-W) density. By integrating (9), we have is the CDF of the Exp-W odel with power parameter . γ Some generalization of the Weibull distribution studied in the literature includes, but are not limited to Yousof et al. [2], Afify et al. [3,4], Alizadeh et al. [5], Yousof et al. [6][7][8][9] Figure 1(a) displays some plots of the ZTPTLW density for different values of , λ α and .
b These plots reveal that the ZTPTLW density can be right-skewed model. The HRF plots of the ZTPTLW distribution given in Figure 1(b) can be unimodal, bathtub, increasing and decreasing shapes.
The justification for the wide practicality of the ZTPTLW lifetime model is based on the wider use of the W model, as well as we are motivated to introduce the ZTPTLW lifetime model because it exhibits the unimodal, the bathtub, the increasing and the decreasing hazard rates as illustrated in Figure 1(b). It is shown in above that the ZTPTLW lifetime model can be viewed as a mixture of the Exp-W distributions. It can be viewed as a suitable model for fitting the unimodal and right skewed data.
The proposed ZTPTLW lifetime model is much better than the Marshall Olkin extended W, gamma W, W Fréchet, Kumaraswamy W , beta W, transmuted modified W, Kumaraswamy transmuted W, modified beta W, the Mcdonald W and transmuted exponentiated generalized W models so the ZTPTLW lifetime model is a good alternative to these models in modeling aircraft windshield data as well as the new lifetime ZTPTLW model is much better than the WW, odd WW, W Log W, the gamma exponentiated-exponential and exponential-exponential geometric models so the ZTPTLW lifetime model is a good alternative to these models in modeling the survival times Guinea pigs.

Probability weighted moments (PWMs)
The ( ) ( , ) th s r PWMs of X following the ZTPTLW is defined by Using equations (5) and (6), we can write

Moments, incomplete moments and generating function
The ( ) th r ordinary moment of X is given by . , where ( , )

⋅ ⋅ B
is the beta function. Substituting (5) and (6) in equation (13) and using a power series expansion, we get The components of the score vector

Failure times
The data consist of 84 observations. Here, we will compare the fits of the ZTPTLW distribution with those of other competitive ones, namely: the gamma W (GaW) [ [26], and the transmuted exponentiated generalized W (TExGW) models, whose PDFs are given by: