The Poisson Topp Leone Generator of Distributions for Lifetime The Poisson Topp Leone Generator of Distributions for Lifetime Data: Theory, Characterizations and Applications Data: Theory, Characterizations and Applications

: We study a new family of distributions defined by the minimum of the Poisson random number of independent identically distributed random variables having a Topp Leone-G distribution (see Rezaei et al., (2016)). Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics, reliability and entropies are derived. Maximum likelihood estimation of the model parameters is investigated. Some special models of the new family are discussed. An application is carried out on real data set applications sets to show the potentiality of the proposed family.

respectively. Suppose 1 , . . . , be independent identically random variable (iid) with common CDF Topp Leone-G and be random variable with Equation (3) can be expressed as Using equations (2) which holds for | | < 1 and > 0 real non-integer. Using the power series in and after some algebra the PDF of the PTL-G class in can be expressed as where Equation (8) reveals that the density of can be expressed as a linear representation of exp-G densities. So, several mathematical properties of the new family can be obtained by knowing those of the exp-G distribution. The CDF of the PTL-G family can also be expressed as a mixture of exp-G densities. By integrating, we obtain the same mixture representation where ( ) is the CDF of the exp-G family with power parameter ( ). The rest of the paper is outlined as follows. In Section 2, we define three special models and provide the plots of their PDF's and hazard rate functions (HRF's). In Section 3, we derive some of its mathematical properties including probability weighted moments (PWMs), residual life and reversed residual life functions, ordinary, incomplete moments and generating functions, finally order statistics and their moments are introduced at the end of the section. Some characterizations results are provided in Section 4. Maximum likelihood estimation of the model parameters is addressed in Section 5. In section 6, simulation results to assess the performance of the proposed maximum likelihood estimation procedure are discussed. In Section 7, we provide the applications to real data to illustrate the importance of the new family. Finally, some concluding remarks are presented in Section 8.

The PTL-generalized half normal (PTL-GHN) distribution
The parent generalized half normal distribution has CDF given by The CDF and PDF of PTL-GHN distribution are given by

Mathematical properties 3.1 Probability weighted moments
The PWMs are expectations of certain functions of a random variable and they can be defined for any random variable whose ordinary moments exist. The PWM method can generally be used for estimating parameters of a distribution whose inverse form cannot be expressed explicitly. The ( , )th PWM of following the PTL-G family, say , , is formally defined by Using equations (5) and (6), we can write Then, the ( , ) th PWM of can be expressed as Henceforth, denotes the exp-G distribution with power parameter ( ) .

Residual life and reversed residual life functions
The th moment of the residual life, say The th moment of the residual life of is given by Another interesting function is the mean residual life (MRL) function or the life expectation at age defined by The mean inactivity time (MIT) or mean waiting time (MWT) also called the mean reversed residual life function is given by 1 ( ) = [( − )| ≤ ] , and it represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, ) .The MIT of the PTL-G family of distributions can be obtained easily by setting = 1 in the above equation.

Moments, incomplete moments and generating function
The th ordinary moment of is given by Then we obtain Setting = 1 in (11), we have the mean of . The last integration can be computed numerically for most parent distributions. The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships. The th central moment of , say , follows as The main applications of the first incomplete moment refer to the mean deviations and the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The th incomplete moment, say ( ) , of can be expressed from (9) as The mean deviations about the mean [ 1 = (| − 1 ′ |)] and about the median [ 2 = (| − |)] of are given by 1 is easily calculated from (5) and 1 ( ) is the first incomplete moment given by (12) with = 1 . Ageneral equation for 1 ( ) can be derived from (12) as is the first incomplete moment of the exp-G distribution. The moment generating function (mgf) ( ) = ( ) of can be derived from equation (9) as where ( ) is the mgf of . Hence, ( ) can be determined from the exp-G generating function.

Order statistics
Order statistics make their appearance in many areas of statistical theory and practice. Let 1 , … , be a random sample from the PTL-G family of distributions and let (1) , … , ( ) be the corresponding order statistics. The PDF of th order statistic, say : , can be written as where (⋅,⋅) is the beta function. Substituting (5) and (6) in equation (12) and using a power series expansion,we get The L-moments are analogous to the ordinary moments but can be estimated by linear combinations of order statistics. They exist whenever the mean of the distribution exists, even though some higher moments may not exist, and are relatively robust to the effects of outliers. Based upon the moments in equation (14), we can derive explicit expressions for the L-moments of as infinite weighted linear combinations of the means of suitable PTL order statistics. They are linear functions of expected order statistics defined by

Characterizations
In this section we present certain characterizations of PTL-G distribution. The first characterization is based on hazard function and the second one is in terms of the ratio of two truncated moments.

Characterization based on hazard function
It is known that the hazard function, ℎ , of a twice differentiable distribution function, , satisfies the first order differential equation For many univariate continuous distributions, this is the only characterization available in terms of the hazard function.
The following characterizations establish a non-trivial characterization for (5) in terms of the hazard function which is not of the trivial form given in (14).  (5) if and only if its hazard function ℎ ( ) satisfies the differential equation Proof. If has of (5), then clearly which is the hazard function of (5). Remark 4.1.1. For = 1 , the differential equation (15) will have the following form

Characterizations in terms of two truncated moments
In this subsection we present characterizations of (5) in terms of a simple relationship between two truncated moments. Our first characterization result employs a theorem due to Glänzel (1987) , see Theorem 1 below. Note that the result holds also when the interval is not closed. Moreover, it could be also applied when the does not have a closed form. As shown in Glänzel (1990), this characterization is stable in the sense of weak convergence. is defined with some real function . Assume that 1 , 2 ∈ 1 ( ) , ∈ 2 ( ) and is twice continuously differentiable and strictly monotone function on the set . Finally, assume that the equation 1 = 2 has no real solution in the interior of . Then is uniquely determined by the functions 1 , 2 and , particularly where the function is a solution of the differential equation The general solution of the differential equation in Corollary 4.2.1 is where is a constant.

Estimation
Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The maximum likelihood estimators (MLEs) enjoy desirable properties and can be used for constructing confidence intervals and regions and also in test statistics. The normal approximation for these estimators in large samples can be easily handled either analytically or numerically. So, we consider the estimation of the unknown parameters of this family from complete samples only by maximum likelihood. Let 1 , … , be a random sample from the PTL-G distribution with parameters , and . Let be the × 1 parameter vector. To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ.

Simulation of PTL-E distribution
In this section, we study the performance of the PTL-E distribution by conducting various simulations for different sizes (n=50, 150, 300) by using R-Language. We simulate 500 samples for the true parameters values I: = 4, = 5.5, = 0.5 in order to obtain average estimates (AEs), biases and mean square errors (MSEs) of the parameters. They are listed in Table 1. The small values of the biases and MSEs indicate that the maximum likelihood method performs quite well in estimating the model parameters of the proposed distribution.

Application
In this section, we use a real data set to show that the PTL-Weibull distribution can be a better model than one based on Beta-exponential, Kumaraswamy-exponential, PTL-Weibull, Modified Weibull and Weibull distribution. The data set given below and represents the failure times of 50 components (per 1000h) The values in table Tab1 indicate that the PTL-W distribution leads to a better fit than the Weibull distribution, modified Weibull distribution, exponential distribution, beta exponential distribution, Kumaraswamy exponential distribution and PTL-E distribution. Also, from Table 2 PTL-G family of distribution is better than Beta-G and Kumaraswamy-G distribution.
Many other useful real data sets can be found and analyzed see Ibrahim

Concluding remarks
We have proposed and presented results on a new family of distributions called Poisson Topp Leone-G family of distributions. This family of distributions have applications in Reliability, Economics and Survival data analysis. Properties of this family are studied including reliability properties, quantile function, series expansion of CDF and PDF, moments, moment generating function, mean deviation. Expression for ℎ order statistics is given, and estimation of parameters are carried out by Maximum likelihood method. A special sub-model is discussed in detail for illustration propose. Finally, an application is carried out on real data set to check the performance of the proposed family which provides consistently better fit than other models.      Figure 5: Estimated densities of the models for data set. Figure 6: Fitted CDFs plots of the considered distribution for the real data set.