Uncertainty quantification based on residual Tsallis entropy of order statistics

Mansour Shrahili; Mohamed Kayid; Mansour Shrahili; Mohamed Kayid

doi:10.3934/math.2024910

AIMS Mathematics

2024, Volume 9, Issue 7: 18712-18731. doi: 10.3934/math.2024910

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Research article

Uncertainty quantification based on residual Tsallis entropy of order statistics

Mansour Shrahili ,
Mohamed Kayid ^,

Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 25 February 2024 Revised: 19 May 2024 Accepted: 30 May 2024 Published: 04 June 2024
MSC : 62N05, 94A17

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.
- order statistics,
- residual Tsallis entropy,
- Shannon entropy,
- residual lifetime
Citation: Mansour Shrahili, Mohamed Kayid. Uncertainty quantification based on residual Tsallis entropy of order statistics[J]. AIMS Mathematics, 2024, 9(7): 18712-18731. doi: 10.3934/math.2024910

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Abstract

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.

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