*Proof of Proposition 17.* Let $(\mathcal{R}_\alpha)_{0<\alpha<1}$ be a family of risk measures defined on $V$ and increasing with respect to $\alpha$. We denote by $\mathcal{P}$ its dual risk measure. For $Y \in V$, $s \in \mathbb{R}$ and $\alpha \in (0,1)$ we note $E_s(Y) = \{0 < \beta < 1 : \mathcal{R}_\beta(Y) \le s\}$ and $G_\alpha(Y) = \{t \in \mathbb{R} : 1-\alpha \ge \mathcal{P}_t(Y)\}$.

1. (Dual of monotony). Let $(Y_1, Y_2) \in V^2$ such that $Y_1 \le Y_2$. Let assume that for all $0 < \alpha < 1$, we have that $\mathcal{R}_\alpha$ is monotonic. For any $s \in \mathbb{R}$, since $\mathcal{R}$ is monotonic, we have $E_s(Y_2) \subset E_s(Y_1)$. Therefore $\mathcal{P}_s(Y_1) \le \mathcal{P}_s(Y_2)$. Hence, $\mathcal{P}_s(\cdot)$ is monotonic. Reciprocally, let assume that for all $s \in \mathbb{R}$, we have that $\mathcal{P}_s(\cdot)$ is monotonic. For any $0 < \alpha < 1$, since $\mathcal{P}$ is monotonic, we have $G_\alpha(Y_2) \subset G_\alpha(Y_1)$. Therefore, using Proposition 14, we have $\mathcal{R}_\alpha(Y_1) \le \mathcal{R}_\alpha(Y_2)$. Hence, $\mathcal{R}_\alpha$ is monotonic.

2. (Dual of positive homogeneity). Let $Y \in V$ and $\lambda > 0$. Let assume that for all $0 < \alpha < 1$, we have that $\mathcal{R}_\alpha$ is positively homogeneous. For any $s \in \mathbb{R}$, we have $E_{\lambda s}(\lambda Y) = E_s(Y)$. Therefore, $\mathcal{P}_{\lambda s}(\lambda Y) = \mathcal{P}_s(Y)$. Reciprocally, let assume that $\mathcal{P}$ respects the previous assertion. Then for any $0 < \alpha < 1$, we have $G_\alpha(\lambda Y) = \lambda G_\alpha(Y)$ and we conclude with Proposition 14.

3. (Dual of translation invariance). Let $Y \in V$ and $c \in \mathbb{R}$. Let assume that for all $0 < \alpha < 1$, we have that $\mathcal{R}_\alpha$ is invariant by translation. For any $s \in \mathbb{R}$, we have $E_{s+c}(Y+c) = E_s(Y)$. Therefore, $\mathcal{P}_{s+c}(Y+c) = \mathcal{P}_s(Y)$. Reciprocally, let assume that $\mathcal{P}$ respects the previous assertion. Then for any $0 < \alpha < 1$, we have $G_\alpha(Y+c) = G_\alpha(Y)+c$ and we conclude with Proposition 14.

4. (Dual of constant equivalence). Let $Y \in V$ and $c \in \mathbb{R}$ such that $Y = c$ a.s.. Let assume that for all $0 < \alpha < 1$, we have that $\mathcal{R}_\alpha$ is constant equivalent. For any $s \in \mathbb{R}$, if $s < c$ then $E_s(Y) = \emptyset$ and else $E_s(Y) = (0,1)$. Therefore $\mathcal{P}_s(Y) = \mathbb{1}_{\{s<c\}}$. Reciprocally, let assume that $\mathcal{P}$ respects the previous assertion. Then for any $0 < \alpha < 1$, we have $G_\alpha(Y) = [c, +\infty)$ and we conclude with Proposition 14.

5. (Dual of risk aversion). Let $Y \in V$ such that $Y$ is not constant a.s. Let assume that for all $0 < \alpha < 1$, we have that $\mathcal{R}_\alpha$ is risk-averse. For any $s \le E(Y)$, we have $E_s(Y) = \emptyset$. Therefore $\mathcal{P}_s(Y) = 1$. Reciprocally, let assume that $\mathcal{P}$ respects the previous assertion. Then for any $0 < \alpha < 1$, we have $G_\alpha(Y) \subset (E(Y), +\infty)$. Therefore, using Proposition 14, we have $\mathcal{R}_\alpha(Y) \ge E(Y)$. Moreover, according to Proposition 22, the dual risk measure is right-continuous. Hence, there exists $\eta > 0$ such that for all $s \in [E(Y), E(Y)+\eta)$, we have $|\mathcal{P}_s(Y) - \mathcal{P}_{E(Y)}(Y) < \alpha$. Since $\mathcal{P}_{E(Y)}(Y) = 1$, for all $s \in [E(Y), E(Y)+\eta)$, we have $\mathcal{P}_s(Y) < 1 - \alpha$. Hence, $G_\alpha(Y) \subset [E(Y) + \eta, +\infty)$. Therefore, using Proposition 14, we have $\mathcal{R}_\alpha(Y) \ge E(Y) + \eta > E(Y)$. $\square$

**Proposition 23.** Let $(\mathcal{R}_\alpha)_{0<\alpha<1}$ be a family of risk measures defined on a subspace $V$ and increasing with respect to $\alpha$. We denote by $\mathcal{P}$ its dual risk measure. The dual property of $\forall Y \in V$, $\lim_{\alpha \to 1} = \sup(Y)$ is the following: for all $Y \in V$ and $s \in \mathbb{R}$, if $s > \sup(Y)$ then $\mathcal{P}_s(Y) = 0$, and if $s < \sup(Y)$ then $\mathcal{P}_s(Y) > 0$. We assume now that $V \subset L^1(\Omega)$. The dual property of $\forall Y \in V$, $\lim_{\alpha \to 0} = E(Y)$ is the following: for all $Y \in V$ and $s \in \mathbb{R}$, if $s < E(Y)$ then $\mathcal{P}_s(Y) = 1$, and if $s > E(Y)$ then $\mathcal{P}_s(Y) < 1$.

*Proof.* We only prove the first statement in the case $\sup(Y) < +\infty$. The proof of case $\sup(Y) = +\infty$ and of the second statement are similar. Let assume that for all $Y \in V$, we have $\lim_{\alpha \to 1} = \sup(Y)$. Then for $s > \sup(Y)$ we have $E_s(Y) = (0,1)$ and hence $\mathcal{P}_s = 0$. For $s < \sup(Y)$, using the limit, there exists $\eta > 0$ such that for all $\alpha \in (1-\eta, 1)$, we have $|\sup(Y) - \mathcal{R}_\alpha(Y)| < \sup(Y) - s$, i.e., $s < \mathcal{R}_\alpha(Y)$. Therefore, $E_s \subset (0, 1-\eta]$ and hence $\mathcal{P}_s \ge \eta > 0$. Reciprocally, let assume that for all $s \in \mathbb{R}$, if $s < E(Y)$ then $\mathcal{P}_s(Y) = 1$, and if $s > E(Y)$ then $\mathcal{P}_s(Y) < 1$. Let $\alpha \in (0,1)$. Then $(\sup(Y), +\infty) \subset G_\alpha(Y)$, hence using Proposition 14, we have $\mathcal{R}_\alpha(Y) \le \sup(Y)$. Let $\varepsilon > 0$. Then $\eta := \mathcal{P}_{\sup(Y)-\varepsilon}(Y) > 0$, hence $\sup(Y) - \varepsilon \in G_{1-\eta}$. Thus $\mathcal{R}_{1-\eta}(Y) \ge \sup(Y) - \varepsilon$. Therefore, for $\alpha \in (1-\eta, 1)$, we have $\sup(Y) - \varepsilon \le \mathcal{R}_{1-\eta}(Y) \le \mathcal{R}_\alpha(Y) \le \sup(Y)$. $\square$

# E Proofs of practical decision-making results

This part aims at proving the results from Section 4.

## E.1 General case

We focus here on the proofs of the results from Subsection 4.3. For the first results associated with Theorem 1, we begin by introducing two lemmas and their own proofs.

**Lemma 10.** For any $Z \in V_{\text{unsafe}}$ and any $\eta > \eta_*(\alpha, s, \beta, n)$, we have $\mathbb{P}(\hat{\mathcal{R}}_{\alpha,n}(Y) + \eta \le s) < \beta$.

**Lemma 11.** For any $Z \in V_{\text{unsafe}}$ and any $\delta \in (\delta_*(\alpha, s, \beta, n), 1-\alpha)$ we have $\mathbb{P}(\hat{\mathcal{P}}_{s,n}(Y) + \delta \le 1-\alpha) < \beta$.
**Proof of Lemma 10.** Let $E = \{\eta > 0 : \sup_{Z \in V_{\text{unsafe}}} \mathbb{P}(\hat{\mathcal{R}}_{\alpha,s}(Z) + \eta \le 1 - \alpha) < \beta\}$. If $E = \emptyset$ then $\eta_*(\alpha, s, \beta, n) = +\infty$. Hence there is no $\eta$ satisfying the condition $\eta > \eta_*(\alpha, s, \beta, n)$ and the statement of the lemma is empty. Else, the set $E$ is an interval of the type $[\inf E, +\infty)$ or $(\inf E, +\infty)$. Since $\eta > \eta_*(\alpha, s, \beta, n) = \inf E$, we hence have $\eta \in E$. This concludes the proof. $\square$

**Proof of Lemma 11.** Let $E := \{\delta > 0 : \sup_{Z \in V_{\text{unsafe}}} \mathbb{P}(\hat{\mathcal{P}}_{s,n}(Z) + \delta \le 1 - \alpha) < \beta\}$. The set $E$ is an interval of the type $[\inf E, 1 - \alpha)$ or $(\inf E, 1 - \alpha)$. Since $\delta \in (\delta_*(\alpha, s, \beta, n), 1 - \alpha) = (\inf E, 1 - \alpha)$, we hence have $\delta \in E$. This concludes the proof. $\square$

**Proof of Theorem 1.** We have $\mathbb{P}(\hat{\mathcal{R}}_{\alpha,n}(Y) + \eta_*(\alpha, s, \beta, n) < s) = \mathbb{P}(\exists k \in \mathbb{N}^* : \hat{\mathcal{R}}_{\alpha,n} + \eta_*(\alpha, s, \beta, n) + \frac{1}{k} \le s) = \mathbb{P}(\bigcup_{k \in \mathbb{N}^*} A_k)$, where $A_k := \{\hat{\mathcal{R}}_{\alpha,n}(Y) + \eta_*(\alpha, s, \beta, n) + \frac{1}{k} \le s\}$ for $k \in \mathbb{N}^*$. Since $(A_k)_{k \in \mathbb{N}^*}$ is an increasing sequence of events, the monotonous continuity of the probability gives $\mathbb{P}(\bigcup_{k \in \mathbb{N}^*} A_k) = \lim_{k \to +\infty} \mathbb{P}(A_k)$. Moreover, for any $k \in \mathbb{N}^*$, since $\eta_*(\alpha, s, \beta, n) + \frac{1}{k} > \eta_*(\alpha, s, \beta, n)$, according to Lemma 10, we have $\mathbb{P}(A_k) < \beta$. Hence, $\lim_{k \to +\infty} \mathbb{P}(A_k) \le \beta$. This proves the first inequality.

For the second inequality, we have similarly $\mathbb{P}(\hat{\mathcal{P}}_{s,n}(Y) + \delta_*(\alpha, s, \beta, n) < s) = \mathbb{P}(\bigcup_{k \in \mathbb{N}} B_k)$, where $B_k := \{\hat{\mathcal{P}}_{s,n}(Y) + \delta_*(\alpha, s, \beta, n) + \frac{1}{k} < 1 - \alpha\}$ for $k \in \mathbb{N}^*$. We conclude the proof with the same arguments. $\square$

**Proposition 24.** Criteria 6 and 8 are equivalent. Criteria 7 and 9 are equivalent.

*Proof.* We have almost surely:

$$ \begin{align*} \hat{\mathcal{R}}_{\alpha,n}(Y) + \eta_*(\alpha, s, \beta, n) < s &\iff \exists \varepsilon > 0 : \hat{\mathcal{R}}_{\alpha,n}(Y) + \eta_*(\alpha, s, \beta, n) + \varepsilon \le s, \\ &\iff \exists \eta > \eta_*(\alpha, s, \beta, n) : \hat{\mathcal{R}}_{\alpha,n}(Y) + \eta \le s, \\ &\iff \exists \eta > \eta_*(\alpha, s, \beta, n) : \hat{\mathcal{R}}_{\alpha,n}(Y) \le s - \eta, \\ &\iff \exists \eta > \eta_*(\alpha, s, \beta, n) : \hat{\mathcal{P}}_{s-\eta,n}(Y) \le s \quad \text{using Equation (12).} \end{align*} $$

This proves the first equivalence. For the second equivalence, if $\delta_*(\alpha, s, \beta, n) = 1 - \alpha$, then Criteria 7 and 9 are both never verified. Else, we have:

$$ \begin{align*} \hat{\mathcal{P}}_{s,n}(Y) + \delta_*(\alpha, s, \beta, n) < 1 - \alpha &\iff \exists \varepsilon > 0 : \hat{\mathcal{P}}_{s,n}(Y) + \delta_*(\alpha, s, \beta, n) + \varepsilon \le 1 - \alpha, \\ &\iff \exists \delta > \delta_*(\alpha, s, \beta, n) : \hat{\mathcal{P}}_{s,n}(Y) + \delta \le 1 - \alpha, \\ &\iff \exists \delta \in (\delta_*(\alpha, s, \beta, n), 1 - \alpha) : \hat{\mathcal{P}}_{s,n}(Y) + \delta \le 1 - \alpha, \\ &\iff \exists \delta \in (\delta_*(\alpha, s, \beta, n), 1 - \alpha) : \hat{\mathcal{P}}_{s,n}(Y) \le 1 - (\alpha + \delta), \\ &\iff \exists \delta \in (\delta_*(\alpha, s, \beta, n), 1 - \alpha) : \hat{\mathcal{R}}_{\alpha+\delta,n}(Y) \le s \quad \text{using Equation (12).} \end{align*} $$

This concludes the proof. $\square$

## E.2 Application to quantile and failure probability

We deal here with the proof of the results from Subsection 4.4.

**Proposition 25.** Let $\delta_*(\alpha, s, \beta, n) := \inf \{\delta > 0 : \sup_{Z \in V_{\text{unsafe}}} \mathbb{P}(\hat{p}_{s,n}(Z) + \delta \le 1 - \alpha) < \beta\}$. Then we have $\delta_*(\alpha, s, \beta, n) = 1 - \alpha - \frac{1}{n}q_\beta(\mathcal{B}(n, 1 - \alpha))$.

**Lemma 12.** For any $\delta > 0$, we have $\sup_{Z \in V_{\text{unsafe}}} \mathbb{P}(\hat{p}_{s,n}(Z) + \delta \le 1 - \alpha) = F_{\mathcal{B}(n, 1-\alpha)}((1 - \alpha - \delta)n)$.
**Proof.** Let $Z \in V_{\text{unsafe}}$. We remind that $n\hat{p}_{s,n}(Z) \sim \mathcal{B}(n, p_s(Z))$. Hence, $\mathbb{P}(\hat{p}_{s,n}(Z) + \delta \le 1 - \alpha) = F_{\mathcal{B}(n,p_s(Z))}(1 - \alpha - \delta)n$. Moreover, since $p \mapsto F_{\mathcal{B}(n,p)}((1 - \alpha - \delta)n)$ is decreasing and $p_s(Z) > 1 - \alpha$, we have $\mathbb{P}(\hat{p}_{s,n}(Z) + \delta \le 1 - \alpha) \le F_{\mathcal{B}(n,p)}((1 - \alpha - \delta)n)$. Hence, $\sup_{Z \in V_{\text{unsafe}}} \mathbb{P}(\hat{p}_{s,n}(Z) + \delta \le 1 - \alpha) \le F_{\mathcal{B}(n,1-\alpha)}(1 - \alpha - \delta)$.

Moreover, since $p \mapsto F_{\mathcal{B}(n,p)}((1 - \alpha - \delta)n)$ is continuous, this upper-bound can be asymptotically attained. Thus, we obtain the desired equality. □

**Proof of Proposition 25.** Let $\delta > 0$. Using Lemma 12, we obtain:

$$ \sup_{Z \in V_{\text{unsafe}}} \mathbb{P}(\hat{p}_{s,n}(Z) + \delta \le 1 - \alpha) < \beta \iff F_{\mathcal{B}(n,1-\alpha)}(1 - \alpha - \delta) < \beta, \\ \iff 1 - \alpha - \delta < q_{\beta}(\mathcal{B}(n, 1 - \alpha)), \\ \iff 1 - \alpha - q_{\beta}(\mathcal{B}(n, 1 - \alpha)) < \delta. $$

The result follows immediately. □

**Proposition 26.** For any $Y \in E$, the following equivalence holds:

$$ \exists \delta \in (\delta_*(\alpha, s, \beta, n), 1 - \alpha) : \hat{q}_{\alpha+\delta,n}(Y) \le s \iff Y_{(k_*(\alpha,s,\beta,n),n)} \le s, $$

where $\delta_*(\alpha, s, \beta, n)$ and $k_*(\alpha, s, \beta, n)$ are respectively defined in Criteria 10 and 11 and with the convention that $Y_{(n+1,n)} = +\infty$.

**Proof.** We notice that $(\alpha + \delta_*(\alpha, s, \beta, n))n = k_*(\alpha, s, \beta, n) - 1$.

Assume that there exists $\delta \in (\delta_*(\alpha, s, \beta, n), 1 - \alpha)$ such that $\hat{q}_{\alpha+\delta,n}(Z) \le s$. Then $(\alpha + \delta)n > (\alpha + \delta_*(\alpha, s, \beta, n))n = k_*(\alpha, s, \beta, n) - 1$. Hence, $(\alpha + \delta)n \ge k_*(\alpha, s, \beta, n)$, and thus $s \ge Y_{((\alpha+\delta)n,n)} \ge Y_{(k_*(\alpha,s,\beta,n),n)}$.

Assume now that $Y_{(k_*(\alpha,s,\beta,n),n)} \le s$. Let define $\delta = \delta_*(\alpha, s, \beta, n) + \frac{1}{2n} > \delta_*(\alpha, s, \beta, n)$. Since $Y_{(k_*(\alpha,s,\beta,n),n)} \le s$, we have $k_*(\alpha, s, \beta, n) \le n$, and hence $\delta < \delta_*(\alpha, s, \beta, n) + \frac{1}{n} \le 1 - \alpha$. Moreover, $\hat{q}_{\alpha+\delta,n}(Y) \le \hat{q}_{\alpha+\delta_*(\alpha,s,\beta,n)+\frac{1}{n},n}(Y) = Y_{(k_*(\alpha,s,\beta,n),n)} \le s$. This concludes the proof. □

**Proposition 27 (Wilks quantile).** Let $Y \in E$. Let us define $k_W := \min\{k \in \{1, ..., n+1\} : \mathbb{P}(Y_{(k,n)} \le q_{\alpha}(Y)) < \beta\}$, with the convention that $Y_{(n+1,n)} = +\infty$. Then $k_W = n - q_{\beta}(\mathcal{B}(n, p_{q_{\alpha}}(Y))) + 1$.

If $Y$ admits a PDF, then $k_W = k_*(\alpha, s, \beta, n)$, where $k_*(\alpha, s, \beta, n)$ is defined in Criterion 11.

**Proof.** Let $k \in \{1, ..., n+1\}$. We have:

$$ Y_{(k,n)} \le q_{\alpha}(Y) \iff \text{Card}\{Y_1, ..., Y_n : Y_i \le q_{\alpha}(Y)\} \ge k, \\ \iff \text{Card}\{Y_1, ..., Y_n : Y_i > q_{\alpha}(Y)\} \le n - k, \\ \iff n\hat{p}_{q_{\alpha}(Y),n}(Y) \le n - k. $$

Now, $n\hat{p}_{q_{\alpha}(Y),n}(Y) \sim \mathcal{B}(n, p_{q_{\alpha}(Y)}(Y))$. Hence, $\mathbb{P}(Y_{(k,n)} \le q_{\alpha}(Y)) = F_{\mathcal{B}(n,p_{q_{\alpha}(Y)}(Y))(n-k)}$. Therefore,

$$ \mathbb{P}(Y_{(k,n)} \le q_{\alpha}(Y)) < \beta \iff F_{\mathcal{B}(n,p_{q_{\alpha}(Y)}(Y))}(n-k) < \beta, \\ \iff n-k < q_{\beta}(\mathcal{B}(n, p_{q_{\alpha}(Y)}(Y))), \\ \iff n - q_{\beta}(\mathcal{B}(n, p_{q_{\alpha}(Y)}(Y))) + 1 \le k. $$

Thus, $k_W = n - q_{\beta}(\mathcal{B}(n, p_{q_{\alpha}(Y)}(Y))) + 1$. When $Y$ admits a PDF, then $F_Y(q_{\alpha}(Y)) = \alpha$, hence $p_{q_{\alpha}(Y)}(Y) = 1 - \alpha$ and the result follows. □

**Proposition 28.** For any $\eta > 0$, if the median $q_{\frac{1}{2}}(\mathcal{B}(n, 1 - \alpha))$ is lower or equal to the expectation $(1 - \alpha)n$, then we have $\sup_{Z \in V_{\text{unsafe}}} \mathbb{P}(\hat{q}_{\alpha,n}(Z) + \eta \le s) \ge \frac{1}{2}$.

**Corollary 4.** Let us define $\eta_*(\alpha, s, \beta, n) := \inf\{\eta > 0 : \sup_{Z \in V_{\text{unsafe}}} \mathbb{P}(\hat{q}_{\alpha,n}(Z) + \eta \le s) < \beta\}$. If the median $q_{\frac{1}{2}}(\mathcal{B}(n, 1 - \alpha))$ is lower or equal to the expectation $(1 - \alpha)n$ and $0 < \beta < \frac{1}{2}$, then $\eta_*(\alpha, s, \beta, n) = +\infty$.
**Proof.** Using Proposition 28, we know that the set $\{\eta > 0 : \sup_{Z \in V_{\text{unsafe}}}{P(\hat{q}_{\alpha,n}(Z) + \eta \le s) < \beta}\}$ is empty. Thus, $\eta_*(\alpha, s, \beta, n) = +\infty$. $\square$

**Proof of Proposition 28.** Let $p \in (1-\alpha, 1)$. Let us note $Z_p$ a random variable whose distribution follows $U([s, s+1])$ with probability $p$ and $U([s - \eta - 1, s - \eta])$ with probability $1-p$. Then $Z_p \in V_{\text{unsafe}}$. Moreover, $P(\hat{q}_{\alpha,n}(Z_p) + \eta \le s) = P(\hat{p}_{s-\eta,n}(Z_p) \le 1-\alpha) = F_{\mathcal{B}(n,p_{s-\eta}(Z_p))}((1-\alpha)n)$. Moreover, $p_{s-\eta}(Z_p) = p$. Hence, $\sup_{Z \in V_{\text{unsafe}}}{P(\hat{q}_{\alpha,n}(Z) + \eta \le s)} \ge P(\hat{q}_{\alpha,n}(Z_p) + \eta \le s) = F_{\mathcal{B}(n,p)}((1-\alpha)n)$. By going to the limit when $p \to 1-\alpha$, we obtain $\sup_{Z \in V_{\text{unsafe}}}{P(\hat{q}_{\alpha,n}(Z) + \eta \le s)} \ge F_{\mathcal{B}(n,1-\alpha)}((1-\alpha)n)$. Last, $(1-\alpha)n$ is the average of the binomial distribution $\mathcal{B}(n, 1-\alpha)$, which is by assumption greater or equal than its median. Thus, $F_{\mathcal{B}(n,1-\alpha)}((1-\alpha)n) \ge F_{\mathcal{B}(n,1-\alpha)}(q_{\frac{1}{2}}(\mathcal{B}(n, 1-\alpha))) \ge \frac{1}{2}$. This concludes the proof. $\square$

**Remark 27.** The assumption for Proposition 28 and Corollary 4 on the median and expectation of binomial distribution $\mathcal{B}(n, p)$, where $p = 1 - \alpha$, may be unusual. The median of a binomial distribution is close to its average, but it can be slightly above or below. More precisely, we always have $|q_{\frac{1}{2}}(\mathcal{B}(n,p)) - np| < \ln 2$. This is proven in [Hamza, 1995]. Thus, the median is either equal to the upper or lower floor of the average. In the proof of Corollary 4, we have to find a lower bound for the CDF of $\mathcal{B}(n, p)$ applied in its expectation np, i.e. $F_{\mathcal{B}(n,p)}(np)(q_{\frac{1}{2}}(\mathcal{B}(n,p)))$. The assumption on the median $q_{\frac{1}{2}}(\mathcal{B}(n,p)) \le np$ allows us to obtain $F_{\mathcal{B}(n,p)}(np)(q_{\frac{1}{2}}(\mathcal{B}(n,p))) \ge \frac{1}{2}$. However, without this assumption, there is no trivial lower bound in general. This is illustrated in Figure 23.

<figure><img src="image_30.png" /><figcaption>Figure 23: Probability that a binomial distribution $\mathcal{B}(n, p)$ is below its average np, with respect to p. Their is no simple lower bound in general.</figcaption></figure>

# F Computation of the risk measure of the academic test cases

We provide here the computation of risk measures for the three academic test cases provided in Section 5. For the margin and the margin-dual, we provide just a formula for the expectation and the standard-deviation, and using Definition 3 and Equation 8, it is straightforward to compute those two risk measures. The computation of some of those risk measures is already known. For instance, the superquantile of a normal distribution is provided in [Rockafellar and Royset, 2015]. We nonetheless remind these results.
**Proposition 29.** Let $Y \sim \mathcal{N}(\mu, \sigma)$ where $\mu \in \mathbb{R}$ and $\sigma > 0$. For any $s \in \mathbb{R}$ and any $\alpha \in (0, 1)$, we have $\mathbb{E}(Y) = 0$, $\sigma(Y) = 1$, $$ \begin{cases} q_{\alpha}(Y) = \mu + \sigma\Phi^{-1}(\alpha), \\ p_s(Y) = 1 - \Phi\left(\frac{s-\mu}{\sigma}\right), \end{cases} $$ and $\bar{q}_{\alpha}(Y) = \mu + \frac{\sigma}{1-\alpha}\phi(\Phi^{-1}(\alpha))$, where $\phi$ and $\Phi$ denote respectively the PDF and the CDF of a standard normal distribution $\mathcal{N}(0, 1)$.

*Proof.* Let prove the results in the case where $G \sim \mathcal{N}(0, 1)$. Indeed, if $Y \sim \mathcal{N}(\mu, \sigma)$, then $\mathcal{R}_{\alpha}(Y) = \mu + \sigma\mathcal{R}(G)$ and $p_s(Y) = \mathcal{P}_{\frac{s-\mu}{\sigma}}(G)$, where $\mathcal{R}$ is the quantile or the superquantile, because they respect the constant equivalence and the positive homogeneity, or respectively their dual properties.

The computation of the expectation, the standard-deviation, the quantile and the failure probability is immediate.

According to Equation (3), we have $(1-\alpha)\bar{q}_{\alpha}(G) = \mathbb{E}(G1_{\{G \ge q_{\alpha}(G)\}})$. By integration, we find $(1-\alpha)\bar{q}_{\alpha}(G) = \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{q_{\alpha}(G)^2}{2}\right) = \phi(q_{\alpha}(G))$. Since $q_{\alpha}(G) = \Phi^{-1}(\alpha)$, we have $\bar{q}_{\alpha}(G) = \frac{1}{1-\alpha}\phi(\Phi^{-1}(\alpha))$. This concludes the proof. $\square$

**Remark 28.** The computation of the buffered failure probability of a normal distribution is not explicit, except when $s < \mu$ where it is equal to 1. Because the buffered failure probability respects the dual properties of the constant equivalence and the positive homogeneity, it is sufficient to be able to compute $\bar{p}_s(\mathcal{N}(0, 1))$ for any $s \in \mathbb{R}$ to be able to compute the buffered failure probability of any normal distribution.

Let $G \sim \mathcal{N}(0, 1)$. Then, according to Definition 11, the buffered failure probability $\bar{p}_s(G)$ can be computed by solving the equation $\bar{q}_{\alpha}(G) = s$, i.e. $\frac{1}{1-\alpha}\phi(\Phi^{-1}(\alpha))$. This equation can be solved numerically to be able to compute the buffered failure probability of a normal distribution.

**Proposition 30.** Let $Y \sim \mathcal{P}(a, k)$ where $a > 0$ and $k > 2$. For any $s \in \mathbb{R}$ and any $\alpha \in (0, 1)$, we have $\mathbb{E}(Y) = \frac{k}{k-1}a$ and $\sigma(Y) = \frac{\sqrt{k}}{(k-1)\sqrt{k-2}}a$. Moreover, we have:

$$ \begin{cases} q_{\alpha}(Y) = a(1-\alpha)^{-\frac{1}{k}}, \\ p_s(Y) = \left(\frac{a}{s}\right)^k \text{ if } s \ge a \text{ and } 1 \text{ else,} \end{cases} \quad \text{and} \quad \begin{cases} \bar{q}_{\alpha}(Y) = \frac{(1-\alpha)^{-\frac{1}{k}}}{1-\frac{1}{k}}a, \\ \bar{p}_s(Y) = \left(\frac{\mathbb{E}(Y)}{s}\right)^k \text{ if } s \ge \mathbb{E}(Y) \text{ and } 1 \text{ else.} \end{cases} $$

*Proof.* The computation of the moments of the Pareto distribution is left to the reader.

Let $y \in \mathbb{R}$, and let compute the CDF of $Y$. If $y \le a$, then $F_Y(y) = 0$. If $y > a$, then by integration we have $F_Y(y) = 1 - \left(\frac{a}{y}\right)^k$. This gives the failure probability. Let compute the quantile: if $y < a$ then $F_Y(y) = 0 < \alpha$, else, we have $F_Y(y) \ge \alpha \iff y \ge a(1-\alpha)^{-\frac{1}{k}}$. This concludes the computation of the quantile.

Equation (2) gives then the expression of the superquantile. The expression of the buffered failure probability follows then using Definition 11 and by solving the equation $\bar{q}_{\gamma} = s$ with unknown $\gamma \in (0, 1)$. $\square$

**Proposition 31.** Let $Y$ be a "switch on-switch off" random variable presented in Subsection 5.1 and represented in Figure 13, with parameters $p \in (0, 1]$ and $a \le -1$. Let $s \in \mathbb{R}$ and $\alpha \in (0, 1)$. Then $\mathbb{E}(Y) = (1-p)a$ and $\text{Var}(Y) = \frac{p}{3} + p(1-p)a^2$. The quantile and failure probability are respectively:

$$ q_{\alpha}(Y) = \begin{cases} a & \text{if } 0 < \alpha \le 1-p, \\ 1-\frac{2}{p}(1-\alpha) & \text{if } 1-p < \alpha < 1, \end{cases} \quad \text{and} \quad p_s(Y) = \begin{cases} 1 & \text{if } s < a, \\ \frac{p}{2}(1-s) & \text{if } a \le s \le -1, \\ 0 & \text{if } -1 \le s \le 1, \\ 0 & \text{if } 1 \le s. \end{cases} $$

The superquantile and buffered failure probability are respectively:

$$ \bar{q}_{\alpha}(Y) = \begin{cases} \frac{1-p-\alpha}{1-\alpha}a & \text{if } 0 < \alpha \le 1-p, \\ \frac{1}{p}(\alpha - (1-p)) & \text{if } 1-p \le \alpha < 1, \end{cases} \quad \text{and} \quad \bar{p}_s(Y) = \begin{cases} 1 & \text{if } s \le (1-p)a, \\ \frac{p-a}{a-s} & \text{if } (1-p)a \le s \le 0, \\ p(1-s) & \text{if } 0 \le s \le 1, \\ 0 & \text{if } 1 \le s. \end{cases} $$
*Proof.* Let compute the moments of order $k \in \mathbb{N}$. We have $E(Y^k) = (1-p)a + \frac{1-(-1)^{k+1}}{2} \frac{p}{2}$. This allows to compute the expectation and the variance.

The CDF can be computed by case disjonction. Figure 13a gives an illustration of the CDF. The failure probability follows then directly. Let us compute the quantile, first when $0 < \alpha \le p - 1$. Let $y \in \mathbb{R}$, then $F_Y(y) \ge \alpha \iff y \ge a$. This gives $q_\alpha(Y) = a$. Then if $p-1 < \alpha < 1$, then if $y \ge 1$ we have $F_Y(y) \ge \alpha$, and if $y < -1$ we have $F_Y(y) < \alpha$. Else if $-1 \le y < 1$, we have $F_Y(y) \ge \alpha \iff y \ge 1 - \frac{p}{2}(1-\alpha)$. Hence, $q_\alpha(Y) = 1 - \frac{p}{2}(1-\alpha)$.

Then, we can compute the superquantile using Equation (2) and by direct integration. The buffered failure probability can finally be computed using Definition 11, by solving the equation $\bar{q}_\gamma(Y) = s$. $\square$

## Data availability statement

The research code associated with this article are available in RiskMeasure, under the reference [Temple-Boyer, 2025].

## Acknowledgment

The authors would like to thank Pr. Julien Reygner for his valuable assistance with the statistical formalization of the decision-making problem and Dr. Franck Delcoigne (project manager at EDF R&D) for the financial and organizational support.

## References

[Ajenjo et al., 2023] Ajenjo, A., Ardillon, E., Chabridon, V., Cogan, S., and Sadoulet-Reboul, E. (2023). Robustness evaluation of the reliability of penstocks combining line sampling and neural networks. *Reliability Engineering and System Safety*, 234:109192.

[Arnold et al., 2008] Arnold, B., Balakrishnan, N., and Nagaraja, H. (2008). *A first course in order statistics*. SIAM.

[Artzner et al., 1999] Artzner, P., Delbaen, F., Eber, J., and Heath, D. (1999). Coherent Measures of Risk. *Mathematical Finance*, 9(3):203–228.

[Aven, 2012] Aven, T. (2012). *Foundations of risk analysis*. John Wiley & Sons.

[Baudin et al., 2017] Baudin, M., Dutfoy, A., Iooss, B., and Popelin, A.-L. (2017). *OpenTurns: An Industrial Software for Uncertainty Quantification in Simulation*, pages 2001–2038. Springer International Publishing, Cham.

[Beaucourt et al., 2025] Beaucourt, J., Dubois, F., Fouet, F., and Ruyer, P. (2025). The Role of BEPU Methodology in Nuclear Safety Demonstration. NURETH-21.

[Bedford and Cooke, 2001] Bedford, T. and Cooke, R. (2001). *Probabilistic risk analysis: foundations and methods*. Cambridge University Press.

[Betz et al., 2022] Betz, W., Papaioannou, I., and Straub, D. (2022). Bayesian post-processing of Monte Carlo simulation in reliability analysis. *Reliability Engineering and System Safety*, 227:108731.

[Blanchard et al., 2023] Blanchard, J.-B., Chocat, R., Damblin, G., Baudin, M., Bousquet, N., Chabridon, V., Iooss, B., Keller, M., Pelamatti, J., and Sueur, R. (2023). Fiches pédagogiques sur le traitement des incertitudes dans les codes de calcul. Technical report, EDF.

[Candelpergher, 2013] Candelpergher, B. (2013). *Théorie des probabilités*. Calvage & Mounet.

[De Rocquigny, 2012] De Rocquigny, E. (2012). *Modelling under risk and uncertainty: an introduction to statistical, phenomenological and computational methods*. John Wiley & Sons.

[De Rocquigny et al., 2008] De Rocquigny, E., Devictor, N., and Tarantola, S. (2008). *Uncertainty in industrial practice: a guide to quantitative uncertainty management*. John Wiley & Sons.
[Delmas, 2009] Delmas, J.-F. (2009). Chapitre 2 - Lois de valeurs exprhêmes.

[Feller, 1991] Feller, W. (1991). *An introduction to probability theory and its applications, Volume 1*. John Wiley & Sons.

[Filippi et al., 2020] Filippi, C., Guastaroba, G., and Speranza, M. (2020). Conditional value-at-risk beyond finance: a survey. *International Transactions in Operational Research*, 27(3):1277-1319.

[Hafsa, 2015] Hafsa, H. (2015). CVaR in portfolio optimization: An essay on the French market. *International Journal of Financial Research*, 6(2):101-111.

[Hamza, 1995] Hamza, K. (1995). The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions. *Statistics & Probability Letters*, 23(1):21-25.

[Iooss and Marrel, 2019] Iooss, B. and Marrel, A. (2019). Advanced Methodology for Uncertainty Propagation in ComputerExperiments with Large Number of Inputs. *Nuclear Technology*, 205(12):1588-1606.

[Kang et al., 2018] Kang, S., Heo, J., Choi, C., Ha, K.-S., and Bae, S.-W. (2018). Assessment study about the risk of Wilks' formula for uncertainty quantification of design extension condition scenarios in prototype Gen-IV sodium fast reactor. *Journal of Nuclear Science and Technology*, 55(7):746-755.

[Komuro, 1989] Komuro, N. (1989). On basic properties of convex functions and convex integrands. *Hokkaido Mathematical Journal*, 18(1):1-30.

[Labopin-Richard, 2016] Labopin-Richard, T. (2016). *Méthodes statistiques et d'apprentissage pour l'estimation de quantiles et de superquantiles dans des modèles de codes numériques ou stochastiques*. PhD thesis, Université Toulouse 3 Paul Sabatier.

[Lemaire, 2013] Lemaire, M. (2013). *Structural reliability*. John Wiley & Sons.

[Mafusalov et al., 2018] Mafusalov, A., Shapiro, A., and Uryasev, S. (2018). Estimation and asymptotics for buffered probability of exceedance. *European Journal of Operational Research*, 270(3):826-836.

[Mafusalov and Uryasev, 2018] Mafusalov, A. and Uryasev, S. (2018). Buffered Probability of Exceedance: Mathematical Properties and Optimization. *SIAM Journal on Optimization*, 28(2):1077-1103.

[Modarres, 2006] Modarres, M. (2006). *Risk Analysis in Engineering: Techniques, Tools, and Trends*. CRC press.

[Morio and Balesdent, 2015] Morio, J. and Balesdent, M. (2015). *Estimation of Rare Event Probabilities in Complex Aerospace and Other Systems: A Practical Approach*. Woodhead Publishing.

[Norton et al., 2021] Norton, M., Khokhlov, V., and Uryasev, S. (2021). Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation. *Annals of Operations Research*, 299(1):1281-1315.

[OpenTURNS, 2024] OpenTURNS (2024). OpenTURNS – A flood model.

[Perrin et al., 2024] Perrin, G., Reygner, J., and Chabridon, V. (2024). Enhancing Reliability Analysis with Limited Observations: A Statistical Framework for System Safety Margins.

[Porter, 2019] Porter, N. W. (2019). Wilks' formula applied to computational tools: A practical discussion and verification. *Annals of Nuclear Energy*, 133:129-137.

[Powell and Allen, 2009] Powell, R. and Allen, D. (2009). *CVaR and Credit Risk Measurement*. School of Accounting, Finance and Economics, Edith Cowan University.

[Rockafellar and Royset, 2015] Rockafellar, R. and Royset, J. (2015). Engineering Decisions under Risk Averseness. *ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering*, 1(2):04015003.
[Rockafellar and Uryasev, 2000] Rockafellar, R. T. and Uryasev, S. (2000). Optimization of conditional value-at-risk. *Journal of Risk*, 2:21–41.

[Rockafellar and Uryasev, 2002] Rockafellar, R. T. and Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. *Journal of Banking & Finance*, 26(7):1443–1471.

[Rockafellar and Uryasev, 2013] Rockafellar, R. T. and Uryasev, S. (2013). The fundamental risk quadrangle in risk management, optimization and statistical estimation. *Surveys in Operations Research and Management Science*, 18(1-2):33–53.

[Rubino and Tuffin, 2007] Rubino, G. and Tuffin, B. (2007). *Simulations et méthodes de Monte Carlo*. Ed. Techniques Ingénieur.

[Shapiro et al., 2021] Shapiro, A., Dentcheva, D., and Ruszczyński, A. (2021). *Lectures on stochastic programming: modeling and theory*. SIAM.

[Stenger et al., 2021] Stenger, J., Gamboa, F., and Keller, M. (2021). Optimization of quasi-convex function over product measure sets. *SIAM Journal on Optimization*, 31(1):425–447.

[Temple-Boyer, 2025] Temple-Boyer, M. (2025). Python code for “Risk Measures in Reliability Engineering: Comparison, Duality and Decision-Making”.

[Van Dorp et al., 2023] Van Dorp, J. R., Shittu, E., and Rabiti, C. (2023). An enhanced two-quantile Wilks methodology for engineering uncertainty analysis. *Applied Stochastic Models in Business and Industry*, 39(1):88–113.

[Wikipedia, 2024] Wikipedia (2024). Convergence of random variables — Wikipedia, the free encyclopedia.

[Wilks, 1941] Wilks, S. S. (1941). Determination of Sample Sizes for Setting Tolerance Limits. *The Annals of Mathematical Statistics*, 12(1):91–96.

[Wilks, 1948] Wilks, S. S. (1948). Order statistics. *Bulletin of the American Mathematical Society*, 54(1):6–50.