Modified likelihood approach for Wald-typed interval of the shape parameter in Weibull distribution

Patchanok Srisuradetchai; Jatuporn Somsamai; Wikanda Phaphan; Patchanok Srisuradetchai; Jatuporn Somsamai; Wikanda Phaphan

doi:10.3934/math.2025001

AIMS Mathematics

2025, Volume 10, Issue 1: 1-20. doi: 10.3934/math.2025001

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Modified likelihood approach for Wald-typed interval of the shape parameter in Weibull distribution

1.
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12120, Thailand
2.
Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
3.
Research Group in Statistical Learning and Inference, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand

Received: 27 August 2024 Revised: 11 November 2024 Accepted: 19 November 2024 Published: 02 January 2025
MSC : 62F10, 62E10, 60E05

The Weibull distribution, widely used in lifetime analysis, is characterized by its shape parameter. We analytically derived Wald-type confidence intervals using standard and modified profile likelihood methods. Performance was assessed through a simulation study examining coverage probability (CP) and average length (AL) across twelve scenarios, varying the shape parameter from 0.5 to 10, the scale parameter from 0.5 to 5, and a range of sample sizes from 5 to 200. The proposed intervals were compared with traditional Wald, profile likelihood, and modified profile likelihood intervals. Our results indicated that the proposed intervals, especially those based on modified profile likelihood, consistently outperformed traditional methods, particularly with small sample sizes. Reductions in either the shape or scale parameter led to shorter AL, as the shape parameter was inversely related to CP. For larger sample sizes (over 30), all interval methods performed similarly, confirming the robustness of the derived intervals across sample sizes. Additionally, the methods were applied to real data on hospital-acquired urinary tract infections, demonstrating their practical utility in healthcare settings.
- confidence interval,
- lifetime distribution,
- profile likelihood,
- Wald-typed interval
Citation: Patchanok Srisuradetchai, Jatuporn Somsamai, Wikanda Phaphan. Modified likelihood approach for Wald-typed interval of the shape parameter in Weibull distribution[J]. AIMS Mathematics, 2025, 10(1): 1-20. doi: 10.3934/math.2025001

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Abstract

The Weibull distribution, widely used in lifetime analysis, is characterized by its shape parameter. We analytically derived Wald-type confidence intervals using standard and modified profile likelihood methods. Performance was assessed through a simulation study examining coverage probability (CP) and average length (AL) across twelve scenarios, varying the shape parameter from 0.5 to 10, the scale parameter from 0.5 to 5, and a range of sample sizes from 5 to 200. The proposed intervals were compared with traditional Wald, profile likelihood, and modified profile likelihood intervals. Our results indicated that the proposed intervals, especially those based on modified profile likelihood, consistently outperformed traditional methods, particularly with small sample sizes. Reductions in either the shape or scale parameter led to shorter AL, as the shape parameter was inversely related to CP. For larger sample sizes (over 30), all interval methods performed similarly, confirming the robustness of the derived intervals across sample sizes. Additionally, the methods were applied to real data on hospital-acquired urinary tract infections, demonstrating their practical utility in healthcare settings.

References

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