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Mathematics
Volume 14
Issue 1
10.3390/math14010098
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Closed-Form Solutions for the Weibull Distribution Parameters and Performance Lifetime Index with Interval-Censored Data

by
Zhengcheng Mou
1,
Yi Li
2,
Jyun-You Chiang
3 and
Tzong-Ru Tsai
4,*
1
Faculty of Science, Civil Aviation Flight University of China, Chengdu 641419, China
2
School of Big Data and Artificial Intelligence, Chengdu Technological University, Chengdu 611730, China
3
School of Statistics and Data Science, Southwestern University of Finance and Economics, Chengdu 611130, China
4
Department of Statistics and Data Science, Tamkang University, New Taipei City 251301, Taiwan
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(1), 98; https://doi.org/10.3390/math14010098 (registering DOI)
Submission received: 20 November 2025 / Revised: 19 December 2025 / Accepted: 23 December 2025 / Published: 26 December 2025
(This article belongs to the Section D1: Probability and Statistics)
Versions Notes

Abstract

In lifetime testing, reliably assessing the life performance index of the Weibull distribution under Type I interval-censored data is a critical task. Although maximum likelihood estimation (MLE) is a conventional approach for parameter estimation, closed-form solutions are unavailable for this data type. To address this limitation, four least-squares estimation methods based on data transformation are developed. The proposed estimations can provide closed-form solutions for the Weibull distribution and life performance index. The asymptotic unbiasedness and normality of the proposed estimators are rigorously established. Their effectiveness is further supported by simulation studies. Moreover, the practical relevance of the methods is illustrated with two real-data applications.
Keywords: Weibull distribution; least-squares estimation; life performance index; Type-I interval-censored data; data transformation Weibull distribution; least-squares estimation; life performance index; Type-I interval-censored data; data transformation

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MDPI and ACS Style

Mou, Z.; Li, Y.; Chiang, J.-Y.; Tsai, T.-R. Closed-Form Solutions for the Weibull Distribution Parameters and Performance Lifetime Index with Interval-Censored Data. Mathematics 2026, 14, 98. https://doi.org/10.3390/math14010098

AMA Style

Mou Z, Li Y, Chiang J-Y, Tsai T-R. Closed-Form Solutions for the Weibull Distribution Parameters and Performance Lifetime Index with Interval-Censored Data. Mathematics. 2026; 14(1):98. https://doi.org/10.3390/math14010098

Chicago/Turabian Style

Mou, Zhengcheng, Yi Li, Jyun-You Chiang, and Tzong-Ru Tsai. 2026. "Closed-Form Solutions for the Weibull Distribution Parameters and Performance Lifetime Index with Interval-Censored Data" Mathematics 14, no. 1: 98. https://doi.org/10.3390/math14010098

APA Style

Mou, Z., Li, Y., Chiang, J.-Y., & Tsai, T.-R. (2026). Closed-Form Solutions for the Weibull Distribution Parameters and Performance Lifetime Index with Interval-Censored Data. Mathematics, 14(1), 98. https://doi.org/10.3390/math14010098

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