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Maximum of the Renyi entropy and the Tsallis entropy are generalization of the maximum entropy for a larger class of Shannon entropy. In this paper we introduce the maximum Renyi entropy and some of the attributes of distributions which have maximum Renyi entropy investigated. The form of distributions with maximum Renyi entropy is power so we state some properties of these distributions and we have a new form of the Renyi entropy. After pointing the topics of minimum Renyi divergence, some other points in this relation have been discussed. An another form of Renyi divergence have also obtained. Therefore we discussed some of the economic applications of the maximum entropy. Meanwhile, the review of the Csiszar information measure, the general form of distributions with minimum Renyi divergence have obtained.

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Entropy plays a fundamental role in reliability and system lifetesting areas. In the recent studies, much attentions have been paid to use quantile functions properties and their applications as an alternate approac in distinguishing statistical models and analysis of data. In the present paper, quantile based residual Tsallis entropy is introduced and its properties in continuous models are investigated. Considering distributions of certain lifetime, explicit versions for quantile based residual Tsallis entropy are obtained and their properties monotonicity are studied and characterization based on this entropy is investigated. Also quantile based Tsallis divergence is introduced and quantile based residual Tsallis divergence is obtained. Finally, an estimator for the quantile based residual Tsallis entropy is introduced and its performance is investigate by study simulation.

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In this paper, the entropy characteristics of the lifetime of coherent systems are investigated using the concept of system signature. The results are based on the assumption that the lifetime distribution of system components is independent and identically distributed. In particular, a formula for calculating the Tsallis entropy of a coherent system's lifetime is presented, which is used to compare systems with the same characteristics. Also, bounds for the lifetime Tsallis entropy of coherent systems are presented. These bounds are especially useful when the system has many components or a complex structure. Finally, a criterion for selecting the preferred system among coherent systems based on the relative Tsallis entropy is presented.