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This work focuses on exploring the properties of past Tsallis entropy as it applies to order statistics. The relationship between the past Tsallis entropy of an ordered variable in the context of any continuous probability law and the past Tsallis entropy of the ordered variable resulting from a uniform continuous probability law is worked out. For order statistics, this method offers important insights into the characteristics and behavior of the dynamic Tsallis entropy, which is associated with past events. In addition, we investigate how to find a bound for the new dynamic information measure related to the lifetime unit under various conditions and whether it is monotonic with respect to the time when the device is idle. By exploring these properties and also investigating the monotonic behavior of the new dynamic information measure, we contribute to a broader understanding of order statistics and related entropy quantities.

Recently, there has been growing interest in alternative measures of uncertainty, including cumulative residual entropy. In this paper, we consider a mixed system consisting of n components, assuming that all components are operational at time t. By utilizing the system signature, we are able to compute the cumulative residual entropy of a mixed system’s remaining lifetime. This metric serves as a valuable tool for evaluating the predictability of a system’s lifetime. We study several results related to the cumulative residual entropy of mixed systems, including expressions, limits, and order properties. These results shed light on the behavior of the measure and provide insights into the predictability of mixed systems. In addition, we propose a criterion for selecting a preferred system based on the relative residual cumulative entropy. This criterion is closely related to the parallel system and provides a practical way to choose the best system configuration. Overall, the present study of cumulative residual entropy and the proposed selection criterion provide valuable insights into the predictability of mixed systems and can be applied in various fields.

For a given system observed at time t, the past entropy serves as an uncertainty measure about the past life-time of the distribution. We consider a coherent system in which there are n components that have all failed at time t. To assess the predictability of the life-time of such a system, we use the signature vector to determine the entropy of its past life-time. We explore various analytical results, including expressions, bounds, and order properties, for this measure. Our results provide valuable insight into the predictability of the coherent system’s life-time, which may be useful in a number of practical applications.

Measuring the uncertainty of the lifetime of technical systems has become increasingly important in recent years. This criterion is useful to measure the predictability of a system over its lifetime. In this paper, we assume a coherent system consisting of n components and having a property where at time t, all components of the system are alive. We then apply the system signature to determine and use the Tsallis entropy of the remaining lifetime of a coherent system. It is a useful criterion for measuring the predictability of the lifetime of a system. Various results, such as bounds and order properties for the said entropy, are investigated. The results of this work can be used to compare the predictability of the remaining lifetime between two coherent systems with known signatures.

The entropy of Tsallis is a different measure of uncertainty for the Shannon entropy. The present work aims to study some additional properties of this measure and then initiate its connection with the usual stochastic order. Some other properties of the dynamical version of this measure are also investigated. It is well known that systems having greater lifetimes and small uncertainty are preferred systems and that the reliability of a system usually decreases as its uncertainty increases. Since Tsallis entropy measures uncertainty, the above remark leads us to study the Tsallis entropy of the lifetime of coherent systems and also the lifetime of mixed systems where the components have lifetimes which are independent and further, identically distributed (the iid case). Finally, we give some bounds on the Tsallis entropy of the systems and clarify their applicability.

An alternate measure of uncertainty, termed the fractional generalized cumulative residual entropy, has been introduced in the literature. In this paper, we first investigate some variability properties this measure has and then establish its connection to other dispersion measures. Moreover, we prove under sufficient conditions that this measure preserves the location-independent riskier order. We then elaborate on the fractional survival functional entropy of coherent and mixed systems’ lifetime in the case that the component lifetimes are dependent and they have identical distributions. Finally, we give some bounds and illustrate the usefulness of the given bounds.

The fractional generalized cumulative residual entropy (FGCRE) has been introduced recently as a novel uncertainty measure which can be compared with the fractional Shannon entropy. Various properties of the FGCRE have been studied in the literature. In this paper, further results for this measure are obtained. The results include new representations of the FGCRE and a derivation of some bounds for it. We conduct a number of stochastic comparisons using this measure and detect the connections it has with some well-known stochastic orders and other reliability measures. We also show that the FGCRE is the Bayesian risk of a mean residual lifetime (MRL) under a suitable prior distribution function. A normalized version of the FGCRE is considered and its properties and connections with the Lorenz curve ordering are studied. The dynamic version of the measure is considered in the context of the residual lifetime and appropriate aging paths.

In this paper, the fractional cumulative entropy is considered to get its further properties and also its developments to dynamic cases. The measure is used to characterize a family of symmetric distributions and also another location family of distributions. The links between the fractional cumulative entropy and the classical differential entropy and some reliability quantities are also unveiled. In addition, the connection the measure has with the standard deviation is also found. We provide some examples to establish the variability property of this measure.

In contrast to many survival models such as proportional hazard rates and proportional mean residual lives, the proportional vitalities model has also been introduced in the literature. In this paper, further stochastic ordering properties of a dynamic version of the model with a random vitality growth parameter are investigated. Examples are presented to illustrate different established properties of the model. Potentials for inference about the parameters in proportional vitalities model with possibly time-varying effects are also argued and discussed.