A Comparative Study of Confidence Intervals for the Parameter of the XShanker Distribution with Applications

Abstract

Statistical modeling of lifetime and reliability data has gained significant interest in the last few decades because of its potential use in various disciplines, including engineering, biomedical sciences, agriculture, and medicine. For portraying complex data behaviors, especially when the hazard rate demonstrates non-constant patterns, traditional distributions such as the exponential and gamma distributions are frequently insufficient. The X-Shanker distribution offers a simple mixture distribution that retains a single parameter while providing greater flexibility in modeling positively skewed and heterogeneous data than some conventional lifetime distributions. Despite the increased interest in the X-Shanker distribution, no thorough comparative evaluation of confidence interval (CI) estimation approaches for its parameter has been conducted. Consequently, this work aimed to recommend four CIs for the X-Shanker distribution parameter. Using simulation studies and applications to two actual datasets, the likelihood, Wald, bootstrap-t, and bias-corrected and accelerated (BCa) bootstrap CIs were assessed. Monte Carlo simulations were used to evaluate their performance, with parameter values ranging from 0.2 to 2.5, sample sizes from 10 to 500, 2,000 simulation replications, and 2,000 bootstrap resamples. The empirical coverage rate (ECR) and average width were used as evaluation criteria. Likelihood and Wald intervals typically maintained coverage rates close to the nominal level of 0.95 across various parameter settings, including small sample sizes. Conversely, the bootstrap-t and BCa bootstrap CIs showed significant undercoverage, with empirical coverage rates decreasing to approximately 0.87–0.92 in some scenarios, although typically yielding narrower intervals. For the ECRs of all four methods, the values increasingly became similar and approached 0.95. The proposed methods were applied to two real datasets on bank customer waiting times and vinyl chloride concentrations in groundwater. The maximum likelihood estimate for the waiting-time data was 0.1945, and the four interval methods produced broadly similar 95% confidence intervals, with the bootstrap-based intervals being slightly narrower. For the vinyl chloride data, the likelihood and Wald CIs were narrower than the bootstrap-t and BCa bootstrap CIs. The results of model comparison based on information criteria and graphical analysis confirmed that the X-Shanker distribution provided a satisfactory fit for the two datasets. These empirical findings were consistent with the simulation results. In conclusion, the likelihood and Wald CIs are recommended when reliable coverage is the primary concern, particularly for small samples, whereas the bootstrap-based CIs may be preferred for larger samples when shorter interval widths are desired.