PSI - Issue 79

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

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with one degree of freedom under standard regularity conditions. Hence, a (1 − α ) confidence set for λ is givenby R ( λ ) ≥ exp − 1 2 χ 2 1 , 1 − α , (6) i.e., the set of λ values where the profile likelihood ratio remains above the cuto ff . Here, α denotes the significance level, representing the probability that the true parameter value lies outside the constructed confidence interval. Graph ically, the confidence limits are defined by the intersections between R ( λ ) and the horizontal cuto ff line, see Fig. 1, where the horizontal cuto ff line is placed at (1 − α ) = 0 . 9.

Fig. 1: Schematic profile likelihood and confidence interval construction.

3. Methodology

3.1. Base model for simulation

The base model for sampling is given by Eq. (1) and the standard deviation σ S , log . The parameters of the base model are given in Table 1. Sampling is performed at evenly spaced logarithmic fatigue-life values. To account for physical plausibility, samples are truncated three standard deviations below S a , log = g ( N log = 4 | θ ). Any truncated sample is resampled uniformly in N between1 · 10 4 and1 · 10 7 until it falls outside the truncation region. To incorporate runouts, the right limit for sampling is extended and samples with fatigue life N > 1 · 10 7 are considered runouts. Load amplitude at the knee point Knee point Slope ( N i ≤ N k ) Slope ( N i > N k ) Standard deviation S a , k N k k 1 k 2 σ S , log 125 1 · 10 6 5 22 0.03 Table 1: Base model parameters used for data generation.

3.2. Design of experiments (DOE)

The aim of this study is to evaluate the actual coverage of profile-likelihood confidence intervals for the given S-N model with and without runouts for a sample size of n = 15. The DOE is summarized in Table 2. For each coverage setting, 1000 simulation runs are conducted, and each profile likelihood is constructed from 100 constrained likelihood evaluations. In this study, profile likelihoods are estimated for the load amplitude at the knee point S a , k , the knee point N k , and the slope k 1 .