PSI - Issue 79

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

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Regarding convergence to the χ 2 distribution, the behavior for the knee point N k is similar to that of the load amplitude at the knee point S a , k (Fig. 5), showing slightly better agreement in the presence of runouts. The better agreement is especially noticeable in the first histogram bins corresponding to confidence levels below 0.3. This improvement is also noticeable at higher confidence levels greater than 0.8.

Fig. 5: Empirical histogram vs. χ 2 PDF and nominal confidence levels (CL) for the knee point N k .

4.3. Coverage results for the slope k 1

For the slope k 1 (HCF regime, N i ≤ N k ), empirical coverage is consistently below the nominal levels. Unlike the knee-point parameters, the presence of runouts shows little to no e ff ect on the coverage of the slope k 1 . The corresponding likelihood-ratio histograms (Fig. 7) resemble those of the load amplitude at the knee point S a , k and the knee point N k and show the same general under- / overestimation pattern relative to the χ 2 reference.

Fig. 6: Profile likelihoods and confidence intervals (CI) for the slope k 1 .