Control and Optimization in Applied Mathematics

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New Two-Parameter Weibull–Lindley Distribution‎: ‎Mathematical Properties‎, ‎Simulation‎, ‎and Applications

Articles in Press

Document Type : Research Article

Authors

1 LaPS Laboratory‎, ‎Department of Mathematics‎, ‎Badji Mokhtar–Annaba University‎, ‎Box 12‎, ‎Annaba 23000‎, ‎Algeria.

2 LAM2SIN Laboratory‎, ‎Department of Mathematics‎, ‎Badji Mokhtar–Annaba University‎, ‎Box 12‎, ‎Annaba 23000‎, ‎Algeria‎.

10.30473/coam.2025.75181.1321

Abstract

This study proposes the New Two-Parameter Weibull–Lindley Distribution (NTPWLD), a flexible lifetime model generated through a transformation of a one-parameter baseline survival function. Owing to its general structure, the NTPWLD accommodates diverse hazard rate shapes, including increasing, decreasing, and bathtub forms, and captures both light- and heavy-tailed behaviors relevant to survival analysis, engineering reliability, and biomedical applications. The work provides a full mathematical treatment of the distribution, deriving closed-form expressions for its density, distribution, survival, hazard, and quantile functions, along with ordinary and incomplete moments, the moment generating function, mean deviations, and Rényi entropy. Several reliability measures, such as mean residual life and stress–strength reliability, are also obtained. Parameter estimation is examined under various inferential approaches, with particular focus on maximum likelihood estimation. A Monte Carlo simulation study shows that the maximum likelihood estimator performs well across settings, displaying low bias, stability, and consistency. To incorporate uncertainty in lifetime data, fuzzy reliability measures are constructed using Zadeh’s extension principle and α-cut techniques. Applications to two real datasets demonstrate that the NTPWLD provides superior goodness-of-fit compared with several competing models based on AIC, BIC, AICC, and −2 log L, highlighting its practical value in both precise and fuzzy reliability environments.‎

Highlights

  • A new two-parameter probability model, the New One-Parameter Weibull–Lindley Distribution (NOPWLD), is proposed to flexibly model lifetime and reliability data.
  • The paper derives key mathematical characteristics of the NOPWLD, including its PDF, CDF, moments, entropy, survival and hazard functions, and mean deviations.
  • The NOPWLD exhibits unimodal density and supports increasing, decreasing, and bathtub-shaped hazard rates, making it suitable for various real-world reliability applications.
  • Multiple estimation techniques are examined—MLE, LSE, Anderson–Darling, Cramér–von Mises, and MPS—with simulation results identifying MLE as the most accurate and consistent method.
  • Reliability measures under parameter uncertainty are explored using fuzzy set theory and Zadeh’s extension principle, enhancing the model’s applicability to imprecise environments.
  • Applied to two real datasets, the NOPWLD demonstrates superior goodness-of-fit compared to established distributions, validated using AIC, BIC, AICC, and -2LogL criteria.

Keywords

Main Subjects

References
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Control and Optimization in Applied Mathematics

Articles in Press, Accepted Manuscript
Available Online from 26 December 2025
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