PSI - Issue 79

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

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The convergence between the empirical histogram of the likelihood-ratio statistic and the reference χ 2 distribution is illustrated in Fig. 3. While the agreement is generally reasonable, some discrepancies remain. The bins fall below the χ 2 curve at low test-statistic values (up to about 0 . 5) and are above at higher values. In addition, the confidence levels based on the χ 2 distribution are shown.

Fig. 3: Empirical histogram vs. χ 2 PDF and nominal confidence levels (CL) for the load amplitude at the knee point S a , k .

4.2. Coverage results for the knee point N k

The results for the knee point N k (Fig. 4) are qualitatively similar to those for the load amplitude at the knee point S a , k . The presence of runouts tends to shift the knee point N k toward lower values. As with the load amplitude at the knee point S a , k , empirical coverage underestimates the nominal level across all investigated confidence levels. However, runouts appear to slightly improve empirical coverage for the knee point N k .

Fig. 4: Profile likelihoods and confidence intervals (CI) for the knee point N k .