A new generalized Lindley distribution, based on weighted mixture of two gamma distributions, is proposed. This model includes the Lindley, gamma and exponential distributions as and other forms of Lindley distributions as special cases. Lindley distribution based on two gamma with two consecutive shape parameter is investigated in some details. Statistical and reliability properties of this model are derived. The size-biased, the length-biased and Lorenze curve are established. Estimation of the underlying parameters via the moment method and maximum likelihood has been investigated and their values are simulated. Finally, fitting this model to a set of real-life data is discussed.

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The purpose of this paper is to provide further study of the Marshall–Olkin log-logistic model that was first described by Gui (Appl Math Sci 7:3947–3961, 2013). This model is both useful and practical in areas such as reliability and life testing. Some statistical and reliability properties of this model are presented including moments, reversed hazard rate and mean residual life functions, among others. Maximum likelihood estimation of the parameters of the model is discussed. Finally, a real data set is analyzed and it is observed that the presented model provides a better fit than the log-logistic model.Electronic supplementary materialThe online version of this article (doi:10.1186/s40064-016-2007-x) contains supplementary material, which is available to authorized users.

A new scale-invariant extension of the Lindley distribution and its power generalization has been introduced. The moments and the moment-generating functions of the proposed models have closed forms. The failure rate, the mean residual life, and the α -quantile residual life functions have been explored. The failure rate function of these models accommodates increasing, bathtub-shaped, and increasing then bathtub-shaped forms. The parameters of the models have been estimated by the maximum likelihood method for the complete and right-censored data. In a simulation study, the efficiency and consistency of the maximum likelihood estimator have been investigated. Then, the proposed models were fitted to four data sets to show their flexibility and applicability.