Time-Dependent Reliability Analysis Based on Quantile Function and Fourth-Order Moment Saddle-Point Approximation Under Small Sample Conditions

Li, Zhaozhan; Li, Zhaowang; Jia, Jufang; He, Xiangdong

doi:10.3390/machines14020142

Open AccessArticle

Time-Dependent Reliability Analysis Based on Quantile Function and Fourth-Order Moment Saddle-Point Approximation Under Small Sample Conditions

School of Mechanical Engineering and Automation, Dalian Polytechnic University, Dalian 116034, China

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^†

These authors contributed equally to this work.

Machines 2026, 14(2), 142; https://doi.org/10.3390/machines14020142

Submission received: 9 December 2025 / Revised: 7 January 2026 / Accepted: 12 January 2026 / Published: 26 January 2026

(This article belongs to the Section Machine Design and Theory)

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Abstract

Time-dependent reliability analysis (TRA) is essential for the life-cycle risk assessment of engineering systems. However, existing TRA methods often require extensive statistical data or computationally expensive surrogate models, which limit their applicability under sparse sampling conditions. This paper presents a novel TRA framework that integrates probability-weighted moments (PWMs), a maximum entropy-based quantile function, and a fourth-order moment saddle-point approximation (FMSPA). The PWM method provides asymptotically unbiased moment estimates for small samples, while the maximum entropy-based quantile function directly computes the first four central moments of random variables. Subsequently, the FMSPA method is extended for use in TRA to efficiently evaluate the cumulative failure probability at each time point. The effectiveness of the proposed method is demonstrated through three engineering examples. The results indicate that the method delivers accurate and efficient TRA under sparse sampling conditions.

Keywords: time-dependent reliability analysis; probability-weighted moments; maximum entropy quantile function; fourth-order moment saddle-point approximation

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MDPI and ACS Style

Li, Z.; Li, Z.; Jia, J.; He, X. Time-Dependent Reliability Analysis Based on Quantile Function and Fourth-Order Moment Saddle-Point Approximation Under Small Sample Conditions. Machines 2026, 14, 142. https://doi.org/10.3390/machines14020142

AMA Style

Li Z, Li Z, Jia J, He X. Time-Dependent Reliability Analysis Based on Quantile Function and Fourth-Order Moment Saddle-Point Approximation Under Small Sample Conditions. Machines. 2026; 14(2):142. https://doi.org/10.3390/machines14020142

Chicago/Turabian Style

Li, Zhaozhan, Zhaowang Li, Jufang Jia, and Xiangdong He. 2026. "Time-Dependent Reliability Analysis Based on Quantile Function and Fourth-Order Moment Saddle-Point Approximation Under Small Sample Conditions" Machines 14, no. 2: 142. https://doi.org/10.3390/machines14020142

APA Style

Li, Z., Li, Z., Jia, J., & He, X. (2026). Time-Dependent Reliability Analysis Based on Quantile Function and Fourth-Order Moment Saddle-Point Approximation Under Small Sample Conditions. Machines, 14(2), 142. https://doi.org/10.3390/machines14020142

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Time-Dependent Reliability Analysis Based on Quantile Function and Fourth-Order Moment Saddle-Point Approximation Under Small Sample Conditions

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