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

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.