PSI - Issue 79

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

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Fig. 7: Empirical histogram vs. χ 2 PDF and nominal confidence levels (CL) for the slope k 1 .

4.4. Summary

Figure 8 summarizes the coverage behavior of the load amplitudes at the knee point S a , k , the knee points N k , and the slopes k 1 for confidence levels from 0 . 10 to 0 . 98. All empirical coverages tend to fall below nominal levels. The presence of runouts appears to improve coverage primarily for the knee point N k , while the e ff ect on the load amplitude at the knee point S a , k is minor for nominal confidence levels greater than 0.8, and the e ff ect on the slope k 1 is minor across all investigated confidence levels.

Fig. 8: Summary of empirical coverage vs. nominal confidence level and percent error.

5. Discussion

This study indicates a systematic undercoverage of profile-likelihood confidence intervals in the investigated test setup. Potential contributors include the small sample size and limited identifiability of certain parameters. According to Wilks’ theorem Wilks (1938), convergence is asymptotic, i.e., holds as n → ∞ . Nevertheless, the bias remains moderate for sample size n = 15, typically below 15%, and below 10% for higher confidence levels ( > 0 . 9). Thus, despite the bias, profile-likelihood confidence intervals provide meaningful uncertainty quantification for bilinear S-N models, particularly when censoring due to runouts is appropriately handled in the likelihood. In addition, the bias can likely be reduced by the use of small-sample corrections like Bartlett’s correction Bartlett (1937).