Procedure 2 Simulation and dual-criterion estimation procedure

Initialization: Fix the design parameters p, m, and K. In the following examples, we set $m=5$, the target level $p=0.25$, and $K\in \{15,30,50\}$. Select an initial threshold ${\tilde{s}}_1$ based on expert knowledge and perform K trials at this level to obtain a first estimate of $(c_T,a_T)$. Since ${\tilde{s}}_1$ is not calibrated to exactly match the target level p, the resulting probability $p_1$ may differ from p. Therefore, p is adjusted using ${\widehat{p}}_1$ so as to satisfy ${\widehat{p}}_1{p}^{m-1}\approx \alpha$. For each stage $j=2,\dots ,m$: 1. Generation of K trials: Perform K trials at the current threshold ${\tilde{s}}_j$, leading to binary observations $$Y_{j,i}={\mathbb{1}}_{{\tilde{R}}_{j,i}\ge {\tilde{s}}_j},i=1,\dots ,K,$$ where ${\tilde{R}}_{j,i}$ are simulated under $G_{{(c,a)}_{j-1}}$. Compute the empirical conditional probability $${\widehat{p}}_j={\displaystyle \frac{1}{K}}\sum _{i=1}^K{Y}_{j,i},$$ and construct a confidence interval $I_{1-\gamma }\left({\widehat{p}}_j\right)$ with confidence level $1-\gamma =0.8$ (see Section 5.2). 2. Parameter space restriction: Define the set of admissible parameters ${\mathcal{S}}_j$ as those for which the theoretical conditional probability under the Generalized Pareto model lies within the confidence interval $I_{1-\gamma }$ (see (11) and (12)). 3. Stability criterion: Among the candidates in ${\mathcal{S}}_j$, select the parameter pair ${(\widehat{c},\widehat{a})}_j$ by minimizing the discrepancy between successive conditional quantile estimates: $${(\widehat{c},\widehat{a})}_j=arg\underset{(c,a)\in {\mathcal{S}}_j}{min}\left|({\tilde{s}}_{j-1}-{\tilde{s}}_{j-2})-G_{(c,a+c{\tilde{s}}_{j-2})}^{-1}(1-{\widehat{p}}_{j-1})\right|.$$ From the estimated parameters, retrieve the $(1-p)$-quantile of the estimated conditional distribution $G_{{(\widehat{c},\widehat{a})}_j}$: $${\tilde{s}}_{j+1}=G_{{\left(\widehat{c},\widehat{a}\right)}_j}^{-1}(1-p)+{\tilde{s}}_j,$$ 4. Final estimator: The estimator of the target extreme quantile ${\tilde{q}}_{1-\alpha }$ is given by the final threshold ${\tilde{s}}_m$.