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<p>Bounded random variables arise naturally in physical, engineering, and reliability systems when measurements represent proportions, efficiencies, normalized intensities, or constrained state variables. In this paper, a flexible bounded stochastic framework generated through a beta transformation of the Kumaraswamy (Kw) baseline is introduced, yielding a four-parameter family capable of capturing diverse boundary behaviors and hazard rate (HR) structures. Rigorous theoretical properties of the proposed model are developed, including structural identifiability, limiting behavior at the boundaries, shape characteristics of the probability density function (PDF) and HR functions, and explicit stochastic representations. Closed-form expressions for moments, probability-weighted moments are derived under mild regularity conditions, together with comprehensive information-theoretic characterizations based on Shannon, Rényi, and Tsallis entropies, as well as Kullback–Leibler divergence relative to baseline models. Likelihood-based inference is studied in detail, with explicit score functions, Fisher information, and asymptotic properties of the maximum likelihood estimators (MLE) established. An illustrative application to bounded measurements from an engineered system demonstrates the practical relevance of the theoretical results. The proposed framework provides a mathematically rigorous and interpretable tool for uncertainty quantification and reliability analysis of bounded physical quantities.</p>

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A flexible bounded stochastic framework for uncertainty and reliability in physical systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月30日

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  • Tmader Alballa,
  • Abdus Saboor,
  • Muhammad Danish,
  • Hamiden Abd El-Wahed Khalifa

摘要

Bounded random variables arise naturally in physical, engineering, and reliability systems when measurements represent proportions, efficiencies, normalized intensities, or constrained state variables. In this paper, a flexible bounded stochastic framework generated through a beta transformation of the Kumaraswamy (Kw) baseline is introduced, yielding a four-parameter family capable of capturing diverse boundary behaviors and hazard rate (HR) structures. Rigorous theoretical properties of the proposed model are developed, including structural identifiability, limiting behavior at the boundaries, shape characteristics of the probability density function (PDF) and HR functions, and explicit stochastic representations. Closed-form expressions for moments, probability-weighted moments are derived under mild regularity conditions, together with comprehensive information-theoretic characterizations based on Shannon, Rényi, and Tsallis entropies, as well as Kullback–Leibler divergence relative to baseline models. Likelihood-based inference is studied in detail, with explicit score functions, Fisher information, and asymptotic properties of the maximum likelihood estimators (MLE) established. An illustrative application to bounded measurements from an engineered system demonstrates the practical relevance of the theoretical results. The proposed framework provides a mathematically rigorous and interpretable tool for uncertainty quantification and reliability analysis of bounded physical quantities.