A new generalized version of the mixed poisson distribution, called the poisson-beta exponential (PBE) distribution, is obtained by mixing the poisson and the beta exponential (BE) distributions. Estimation of the parameters, using the method of moments and maximum likelihood estimators, is discussed. We show the consistency of the new model parameters using simulation study. Examples are given for fitting the PBE distribution to data, and the fit model is compared with that obtained using other distributions.

Gompertz-poisson distribution is a three-parameter lifetime distribution with increasing, decreasing, increasing-decreasing and unimodal shape failure rate function and a composition of Gompertz and poisson distributions cut at zero point that in this paper estimated the parameters of the distribution by maximum likelihood method and in order to confirm the calculated estimates, based on random sample with volumes of 100, 200, 300, 400 and 500 of Gompertz-poisson distribution simulation study was conducted. Also with the help of two real data sets and comparing Gompertz-poisson distribution with several other distributions of lifetime we show this distribution is a good model for data fitness related to lifetime.

In this study, based on Bayesian Generalized Linear Models, correlation between the parameters of two poisson distributions was computed. Due to lack of the closed form for posterior distribution, hierarchical Bayesian statistics using the Metropolis- Hastings algorithm to calculate the correlation of two poisson distributions is presented. In this regard, the highest posterior density for coefficient of variation in the model are calculated. Using Bayesian Deviance Information Criterion (DIC) has been shown that a poisson- lognormal model can assess the correlation between the parameters better than the poisson-gamma model. Finally, the proposed method is used to simulated data of BANK TEJARAT.

Background & Aim: In the survival data with Long-term survivors the event has not occurred for all the patients despite long-term follow-up, so the survival time for a certain percent is censored at the end of the study. Mixture cure model was introduced by Boag, 1949 for reaching a more efficient analysis of this set of data. Because of some disadvantages of this model non-mixture cure model was introduced by Chen, 1999, which became well-known promotion time cure model. This model was based on the latent variable distribution of N. Non mixture cure models has obtained much attention after the introduction of the latent activating Scheme of Cooner, 2007, in recent decades, and diverse distributions have been introduced for latent variable.Methods & Materials: In this article, generalized poisson-inverse Gaussian distribution (GPIG) will be presented for the latent variable of N, and the novel model which is obtained will be utilized in analyzing long-term survival data caused by skin cancer. To estimate the model parameters with Bayesian approach, numerical methods of Monte Carlo Markov chain will be applied. The comparison drawn between the models is on the basis of deviance information criteria (DIC). The model with the least DIC will be selected as the best model.Results: The introduced model with GPIG, with deviation criterion of 411.775, had best fitness than poisson and poisson-inverse Gaussian distribution with deviation criterion of 426.243 and 414.673, respectively.Conclusion: In the analyzing long-term survivors, to overcome high skewness and over dispersion using distributions that consist of parameters to estimate these statistics may improve the fitness of model. Using distributions which are converted to simpler distributions in special occasions, can be applied as a criterion for comparing other models.

Although the random sum distribution has been well-studied in probability theory, inference for the mean of such distribution is very limited in the literature. In this paper, two approaches are proposed to obtain inference for the mean of the poisson-Exponential distribution. Both proposed approaches require the log-likelihood function of the poisson-Exponential distribution, but the exact form of the log-likelihood function is not available. An approximate form of the log-likelihood function is then derived by the saddlepoint method. Inference for the mean of the poisson-Exponential distribution can either be obtained from the modified signed likelihood root statistic or from the Bartlett corrected likelihood ratio statistic. The explicit form of the modified signed likelihood root statistic is derived in this paper, and a systematic method to numerically approximate the Bartlett correction factor, hence the Bartlett corrected likelihood ratio statistic is proposed. Simulation studies show that both methods are extremely accurate even when the sample size is small.

Background & Objectives: Changing the pattern of mortality gives important perspective of health determinants. The aim of this study is to detect location and time of mortality pattern change in country using statistical change point method during 1971-2009 Years.Methods: We assume for years before and after k0, yt has a poisson distribution with means  l0 and l1, respectively. We used several methods for estimation change point in real data by assume poisson model.Results: Using two simulated and real data analysis showed that the change point has been occurred in year 1993 and this confirmed by all methods.Conclusion: Our findings have shown that the change pattern of mortality trend in Iran is related to improvement of health indicators and decreasing mortality rate in Iran.

The problem of finding tolerance intervals receives very much attention of researchers and are widely used in various statistical fields, including biometry, economics, reliability analysis and quality control. Tolerance interval is a random interval that covers a specified proportion of the population with a specified confidence level. In this paper, we compare approximate tolerance intervals for the poisson random variable. Approximate tolerance intervals are constructed based on approximate confidence intervals for the parameter of the poisson distribution such as Wald interval, Wald interval with continuity correction, score interval, variance stabilizing interval, recentered variance stabilizing interval and Freeman and Tukey interval. Coverage probabilities and expected widths of the proposed tolerance intervals are evaluated by a simulation. Proposed tolerance intervals are used by using an application example.

The aim of this article is to obtain some necessary and sufficient conditions for  functions, whose coefficients are probabilities of the Miller-Ross-type poisson distribution series, to belong to certain subclasses of analytic and univalent functions. Furthermore, we consider an integral operator related to the Miller-Ross type poisson distribution series.

Introduction In lifetime studies consider that different components cause the failure of the unit/item under study but of the same type that is not entirely observed. In this case, the failure time of the unit/item is recorded and evaluated based on the information obtained from the observation and as the minimum value among other components affecting failure. The experimenter cannot identify the component that led to the unit’, s failure. In the study of series systems, the minimum component lifetime among the effective components leads to failure and is observed. In recent literature, Adamidis and Loukas (1998) used the geometric distribution function as the number of failure components and introduced a two-parameter exponential-geometric distribution with a descending failure rate. In applying compounding distributions of lifetime study, the experimenter may face the phenomenon of censoring. Because there are cases in which the units/items, although alive, are lost or removed. In this study, type-II of censoring has been investigated. Recently, the inverse Weibull distribution in censored data has been studied by Ateya (2017), Singh and Tripathi (2018). This paper presents the inverse Weibull-poisson distribution function in the series system of the type-II censored sample. Material and Methods This paper considers the classical and Bayesian estimation of parameters of inverse Weibull-poisson distribution function under the type-II censoring. Since the normal equations are not solved analytically, the EM algorithm, as the numerical method, is used in estimating the maximum likelihood methods. Little and Rubin (1983) showed that the EM algorithm is more reliable than the Newton-Raphson method in the case of incomplete data. Then, the Fisher information matrix for censored data is obtained with the principle of Louis (1982), and the approximate confidence intervals can be calculated. Parameters are estimated under the square error and LINEX loss functions while Gamma distribution is prior distribution. In Bayesian estimation, since the posterior distribution is not obtained in closed form, parameters are estimated with Markov chain Monte Carlo techniques and samples are generated by Gibbs sampling via the Metropolis-Hastings algorithm. Finally, the Bayesian confidence intervals are obtained using Kundu (2008), and the HPD intervals are constructed with Chen and Shao (1999) methods. Results and Discussion To evaluate the performance of estimators in terms of MSE’, s and their corresponding confidence intervals, it is generated 10000 samples for different sample sizes and three censoring schemes. Conclusion The simulation results show that with decreasing number of censors for a fixed sample size, the estimation of the parameters is closer to the actual values, and the MSE are reduced. Moreover, with an increasing sample size, the MSE of parameters is reduced for a fixed censoring scheme. For the 30% censoring scheme, Bayesian estimators of the parameters under the square error loss function have small MSE. The maximum likelihood estimators of parameters for the 10% censoring scheme have small MSE. The simulation results using confidence intervals show that the length of confidence intervals is reduced for a fixed sample size with decreasing number of censored. Moreover, the classical confidence intervals have the shortest interval length for all censoring schemes. The length of the HPD confidence intervals is shorter than the Bayesian confidence intervals.

Background & Aim: Mixed poisson and mixed negative binomial distributions have been considered as alternatives for fitting count data with over-dispersion. This study introduces a new discrete distribution which is a weighted version of poisson-Lindley distribution. Methods & Materials: The weighted distribution is obtained using the negative binomial weight function and can be fitted to count data with over-dispersion. The p. m. f., p. g. f. and simulation procedure of the new weighted distribution, namely weighted negative binomial-poisson-Lindley (WNBPL), are provided. The maximum likelihood method for parameters estimation is also presented. Results: The WNBPL distribution is fitted to several datasets, related to genetics and compared with the Poison distribution. The goodness of fit test shows that the WNBPL can be a useful tool for modeling genetics datasets. Conclusion: This paper introduces a new weighted poisson-Lindley distribution which is obtained using negative binomial weight function and can be used for fitting over-dispersed count data. The p. m. f., p. g. f. and simulation procedure are provided for the new weighted distribution, namely the weighted negative binomial-poisson Lindley (WNBPL) to better inform parents from possible time of occurrence reflux and treatment strategies.