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In this study, we focused on investigating the properties of residual Tsallis entropy for order statistics. The reliability of engineering systems is highly influenced by order statistics, for example, when modeling the lifetime of a series system and the lifetime of a parallel system. The residual Tsallis entropy of the ith order statistic from a continuous distribution function and its deviation from the residual Tsallis entropy of the ith order statistics from a uniform distribution were investigated. In the mathematical framework, a method was provided to represent the residual Tsallis entropy of the ith order statistic in the continuous case with respect to the case where the distribution was uniform. This approach can provide insight into the behavior and properties of the residual Tsallis entropy for order statistics. We also investigated the monotonicity of the new uncertainty measure under different conditions. An investigation of these properties leads to a deeper understanding of the relationship between the position of the order statistics and the resulting Tsallis entropy. Finally, we presented the computational results and proposed estimators for estimating the residual Tsallis entropy of an exponential distribution. For this purpose, we derived a maximum likelihood estimator. [ABSTRACT FROM AUTHOR]

Recently, extropy has emerged as an alternative measure of uncertainty instead of entropy. When it comes to quantifying uncertainty regarding the remaining lifetime of a component, entropy has proven to be less effective. Therefore, the concept of residual entropy was introduced to address this limitation. Similar to the residual entropy, the residual extropy was formulated and used to investigate the uncertainty in the residual lifetime of a unit. Systems in the real world exhibit a pervasive property of uncertainty that affects future events and past events. For this reason, the concept of past extropy was introduced to specifically capture and analyze the uncertainty associated with past events. This paper focuses on stochastic aspects, including stochastic orderings, which provide useful inequalities related to past extropy when applied to order statistics and lower record values. It is worth noting that the past extropy of the i th-order statistics and record values in the continuous case is related to the past extropy of the i th-order statistics and record values evaluated from the uniform distribution. The monotonicity of the past extropy of order statistics is examined and some insights into the past extropy of lower data set values are also given. Finally, some computational results are presented. In fact, an estimator for the extropy of the exponential distribution is proposed. For this purpose, the maximum likelihood estimator is derived. The proposed method is easy to implement and apply from a computational point of view. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper explores the concept of residual extropy as an uncertainty measure for order statistics. We specifically derive the residual extropy for the ith-order statistic and establish its relationship with the residual extropy of the ith-order statistic from a random sample generated from a uniform distribution. By employing this approach, we obtain a formula for the residual extropy of order statistics applicable to general continuous distributions. In addition, we offer two lower bounds that can be applied in situations where obtaining closed-form expressions for the residual extropy of order statistics in diverse distributions proves to be challenging. Additionally, we investigate the monotonicity properties of the residual extropy of order statistics. Furthermore, we present other aspects of the residual extropy of order statistics, including its dependence on the position of order statistics and various features of the underlying distribution. [ABSTRACT FROM AUTHOR]

A fundamental factor in relevant applications is the predictability of the life cycle of a coherent system consisting of more than one component. In this context, we examine how entropy can be applied to evaluate the degree of predictability. In particular, in order to calculate the Tsallis entropy of the past life, we consider a scenario in which all components of the system fail at a given time t and use the system signature to calculate the Tsallis entropy of the past life. We examine a number of analytical results, e.g., expressions, thresholds and orders for the measure at issue in our study. The results may provide insights into the predictability of a coherent system's life cycle. [ABSTRACT FROM AUTHOR]

In parallel with the concept of Rényi entropy for residual lifetime distributions, the Rényi entropy of inactivity time of lifetime distributions belonging to asymmetric distributions is a useful measure of independent interest. For a system that turns out to be inactive in time t, the past entropy is considered as an uncertainty measure for the past lifetime distribution. In this study, we consider a coherent system that includes n components and has the property that all the components of the system have failed at time t. To assess the predictability of the coherent system's lifetime, we use the system's signature to determine the Rényi entropy of its past lifetime. We study several analytical results, including expressions, bounds, and order properties for this measure. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a mixed system with n components, where at time t , all system components are functioning. We then use the system signature to evaluate the extropy of the excess lifetime of the mixed system, which is a useful criterion for predicting the lifetime of the system. We give several results including expressions, bounds, and order conditions for the above measure. Finally, based on the relative extropy, we establish a criterion for selecting a preferred system that is closely related to the parallel system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The measurement of uncertainty across the lifetimes of engineering systems has drawn more attention in recent years. It is a helpful metric for assessing how predictable a system's lifetime is. In these circumstances, Renyi entropy, a Shannon entropy extension, is particularly appealing. In this paper, we develop the system signature to give an explicit formula for the Renyi entropy of the residual lifetime of a coherent system when all system components have lived to a time t. In addition, several findings are studied for the aforementioned entropy, including the bounds and order characteristics. It is possible to compare the residual lifespan predictability of two coherent systems with known signatures using the findings of this study. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]