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We propose a new class of continuous models called the Weibull Generalized G family with two extra positive shape parameters, which extends several well-known models. We obtain some of its mathematical properties including ordinary and incomplete moments, generating function, order statistics, probability weighted moments, entropies, residual, and reversed residual life functions. Characterizations based on a ratio of two truncated moments, in terms of hazard function and based on certain functions of the random variable are presented. We estimate the model parameters by the maximum likelihood method. We assess the performance of the maximum likelihood estimators in terms of biases and mean squared errors by means of two simulation studies. The usefulness of the proposed models is illustrated via three real data sets.

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In this paper, we introduce and study quantile version of the Shannon entropy function via doubly truncated (interval) lifetime, which includes the residual and past lifetimes as special case. We aim to study the use of proposed measure in characterization of distribution functions. Further, we describe a stochastic order and a weighted distribution based on this entropy and show their properties. Finally, some results have been obtained for some distributions such as Uniform, Exponential, Pareto I, Power function and Govindarajulu. Also by analysing a real data the subject has been illustrated. 

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‎This paper deals with the dynamic survival past extropy as a measure of uncertainty in the past lifetime‎ ‎distributions‎. ‎We introduce a quantile version of the extropy function in past lifetime‎. ‎Various properties of the proposed measure are obtained‎. ‎Additionally‎, ‎some stochastic comparisons‎ ‎and bounds are derived and the performance of the dynamic survival past extropy of parallel and series system is studied as well‎.