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A new framework for generating lifetime distributions is proposed, which is called the Topp-Leone Exponentiated Power Lindley (TL-EPL) distribution. Submodels of the TL-EPL distribution, such as the Topp-Leone Power Lindley, Topp-Leone Generalized Lindley, and Topp-Leone Lindley, are introduced. Some statistical characteristics of the distributions are investigated (i.e., mean, variance, and functions of survival, hazard, and quantile). The maximum likelihood estimation is used to estimate the parameters of each distribution. Some real data sets are fitted in order to illustrate the usefulness of the proposed distribution.

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In this paper, a new mixture distribution for count data, namely the negative binomial-new generalized Lindley (NB-NGL) distribution is proposed. The NB-NGL distribution has four parameters, and is a flexible alternative for analyzing count data, especially when there is over-dispersion in the data. The proposed distribution has sub-models such as the negative binomial-Lindley (NB-L), negative binomial-gamma (NB-G), and negative binomial-exponential (NB-E) distributions as the special cases. Some properties of the proposed distribution are derived, i.e., the moments and order statistics density function. The unknown parameters of the NB-NGL distribution are estimated by using the maximum likelihood estimation. The results of the simulation study show that the maximum likelihood estimators give the parameter estimates close to the parameter when the sample is large. Application of NB-NGL distribution is carry out on three samples of medical data, industry data, and insurance data. Based on the results, it is shown that the proposed distribution provides a better fit compared to the Poisson, negative binomial, and its sub-model for count data.