Improving the XRani Distribution's Inference: Applying Confidence Intervals to PM2.5 Concentration Data Collected in Bangkok, Thailand

Abstract

This paper developed and assessed four methods for constructing confidence intervals (CIs) for the parameter of the XRani distribution, which is frequently applied in the analysis of lifetime data. The CIs examined include the likelihood-based CI, Wald-type CI, bootstrap-t CI, and the bias-corrected and accelerated (BCa) bootstrap CI. To evaluate their performance, both a simulation study and a real-world application were conducted. The evaluation criteria focused on empirical coverage probability (ECP) and average width (AW) across a range of scenarios. To enhance computational efficiency, an explicit analytical expression for the Wald-type CI was derived. Simulation results demonstrated that the likelihood-based and Wald-type CIs consistently achieved ECPs close to the nominal 0.95 confidence level across most scenarios. In contrast, for smaller sample sizes, the bootstrap-t and BCa bootstrap methods yielded lower ECPs. However, as sample sizes increased, the ECPs of both bootstrap methods gradually approached the nominal level. The parameter values were also found to influence performance: at lower parameter values, all CIs performed well, with the likelihood based and Wald-type methods maintaining an ECP close to 0.95. At higher parameter values and smaller sample sizes, however, the bootstrap-t and BCa methods exhibited diminished coverage probabilities. The practical utility of these CI methods was further demonstrated through the applications to PM2.5 concentration data collected in Bangkok, Thailand. The results from this empirical analysis corroborated the findings of the simulation study, thereby affirming the robustness and applicability of the proposed CI methods.