PSI - Issue 79

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

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which corresponds to a log-normal distribution of the load amplitude S a , i This yields the complete S-N model S a , log , i = g ( N log , i | θ ) where the parameter vector is defined as θ = ( S a , k , log , N k , log , k 1 , k 2 ,σ S , log ) ⊺ .

2.2. Likelihood with failures and runouts

The model parameters θ are estimated by maximum likelihood estimation (MLE). The likelihood function L ( θ ) is constructed based on the probability density function (PDF, see Eq. 2) for failed samples and the survival function (SF, see Eq. 3) for runouts, i.e., right-censored observations. The PDF is

exp  −

2 σ 2  ,

( S a , log , i − µ i ) 2

1 √ 2 πσ 2

2 )

f ( S a , log , i | µ i , σ

(2)

=

where the mean µ i is given by the S-N model µ i = g ( N log , i | θ ) and the variance σ 2 is σ 2 S , log . The SF is

2 )

2 ) ,

SF ( S a , log , i | µ i ,σ

= 1 − F ( S a , log , i | µ i ,σ

(3)

where F ( · ) is the cumulative distribution function of the normal distribution of the logarithmic stress amplitude. With the PDF and SF, the (full) likelihood function for independent observations is

i =  

n 

i = 1 

2 ) 1 − δ i  , where δ

1 , if specimen i is a failure , 0 , if specimen i is a runout .

2 ) δ i · SF ( S

(4)

f ( S a , log , i | µ i ,σ

L ( θ ) =

a , log , i | µ i ,σ

2.3. Profile-likelihood confidence intervals

Beyond point estimation, the likelihood can be used to construct confidence intervals via the profile likelihood . Let the full parameter vector be θ =  λ ψ  ,

where λ is the scalar parameter of interest (e.g., S a , k , log ) and ψ denotes the vector of nuisance parameters. Let L ( ˆ θ ) denote the global maximum likelihood. The profile likelihood is then defined as

ψ 

L ( ˆ θ ) 

L ( λ, ψ )

R ( λ ) = max

(5)

,

i.e., the likelihood of the parameter of interest λ maximized over the nuisance parameters ψ and normalized by the global maximum likelihood L ( ˆ θ ). Therefore, R ( λ ) reaches its maximum value at the maximum likelihood estimate ˆ λ . As shown by (Mavrakakis, 2021, p. 139), the statistic − 2log R ( λ ) converges in distribution to a χ 2 random variable