Systematic Undercoverage in Bootstrap Confidence Intervals for Xi Correlation Coefficient: A Simulation Study

Abstract

This study uses a comprehensive simulation to investigate the performance of bootstrap confidence intervals for the Pearson correlation coefficient and Xi correlation coefficient (XICOR). Different distributional settings (normal, lognormal, uniform, and t(3)), sample sizes (n = 10, 30, 100, 1000), and Gaussian copula dependence levels (rho 0 = 0.1, 0.3, 0.5, 0.8) were considered. For each scenario, the true population Pearson and XICOR values were separately estimated via a large-sample Monte Carlo reference. The coverage probability (CP), mean confidence interval width (MCIW) and absolute bias detection (BD) were used to evaluate the percentile, bias-corrected and accelerated (BCa) bootstrap methods. The results showed that bootstrap confidence intervals based on the Pearson correlation coefficient generally achieve coverage probabilities close to the nominal level across most scenarios, with some degradation observed under heavy-tailed distributions at large sample sizes. By contrast, confidence intervals based on XICOR showed a significant and systematic undercoverage problem across most sample sizes, especially for n >= 30, which worsened as sample size increased. Although XICOR tends to produce narrower intervals, these intervals were associated with low coverage and increased variability, indicating a false precision. The BCa method did not meaningfully improve XICOR's coverage performance. In conclusion, reducing bias alone is insufficient for reliable inference when dealing with non-smooth dependence measures. Classical bootstrap methods may be inappropriate for XICOR since they do not provide accurate quantification of uncertainty.