In this paper a new compounding distribution with increasing, decreasing, bathtub shaped and unimodal hazard rate function is proposed. The new four-parameters distribution is a generalization of the complementary exponential power distribution. The raw-moments, density function of the order statistics, survival function, hazard rate function, quantiles, mean residual lifetime and reliability function are presented. The estimation of the new distribution in a special case Poisson complementary exponential power distribution is studied by the method of maximum likelihood and EM algorithm. Expression for asymptotic distribution for the maximum likelihood estimation of the parameters of the PCEP distribution are obtained and for determining the precision of the variance and covariance of the estimations, a simulation is used, Then experimental results are illustrated based on the real data set.

In this paper, we introduce a new five-parameters distribution with increasing, decreasing, bathtub-shaped failure rate, called as the Beta Weibull-Geometric (BWG) distribution. Using the Sterling Polynomials, the probability density function and several properties of the new distribution such as its reliability and failure rate functions, quantiles and moments, Renyi and Shannon entropies, moments of order statistics, mean residual life, reversed mean residual life are obtained. The maximum likelihood estimation procedure is presented in this paper. Also, we compare the results of fitting this distribution to some of their sub-models, using to a real data set. It is also shown that the BWG distribution fits better to this data set.

In this article, the total time on test  (TTT) transformation and its major properties are investigated. Then, the relationship between the TTT transformation and some subjects in reliability theory is expressed. The TTT diagram is also drawn for some well-known lifetime distributions, and a real-data analysis is performed based on this diagram. A new distorted family of distributions is introduced using the distortion function. The statistical interpretation of the new life distribution from the perspective of reliability is provided, and its survival function is derived. Finally, a generalization of the Weibull distribution is introduced using a new distortion function. A real data analysis shows its superiority in fitting in comparison to the traditional Weibull model.