<figure><img src="image_9.png" /><figcaption>Figure D.1: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\xi_{\tau'_n}^{\star,\text{BR}}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\hat{s}_n^2(1)$ and the average values of $\hat{s}_n^{2,\text{BR}}$ (dashed curve), $\hat{s}_n^2(1)$ (green curve), $\hat{s}_n^2$ (dashed-dotted curve) and $\bar{s}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\hat{I}_{\tau'_n}^{(1)}(\alpha)$, $\hat{I}_{\tau'_n}^{(2)}(\alpha; 1)$, $\hat{I}_{\tau'_n}^{(3)}(\alpha)$ and $\hat{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Burr distribution with $\rho = -5$ and $\gamma = 0.1$ (first row), $\gamma = 0.2$ (second row), $\gamma = 0.3$ (third row), $\gamma = 0.4$ (fourth row). The sample size is $n = 200$ and the target expectile level is $\tau'_n = 1 - 1/n = 0.995$.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure D.2: As in Figure D.1, with $n = 1,000$ and $\tau'_n = 1 - 1/n = 0.999$.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure D.3: As in Figure D.1, with $n = 5,000$ and $\tau'_n = 1 - 1/n = 0.9998$.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure D.4: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\tilde{\xi}_{\tau'_n}^{*,\text{BR}}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\tilde{\sigma}_n^2(2)$ and the average values of $\tilde{\sigma}_n^{2,\text{BR}}$ (dashed curve), $\tilde{\sigma}_n^2(2)$ (green curve), $\tilde{\sigma}_n^2$ (dashed-dotted curve) and $\bar{\sigma}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\tilde{I}_{\tau'_n}^{(1)}(\alpha)$, $\tilde{I}_{\tau'_n}^{(2)}(\alpha; 2)$, $\tilde{I}_{\tau'_n}^{(3)}(\alpha)$ and $\tilde{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Burr distribution with $\rho = -5$ and $\gamma = 0.1$ (first row), $\gamma = 0.3$ (second row), $\gamma = 0.5$ (third row), $\gamma = 0.7$ (fourth row). The sample size is $n = 200$ and the target expectile level is $\tau'_n = 1 - 1/n = 0.995$.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure D.5: As in Figure D.4, with $n = 1,000$ and $\tau'_n = 1 - 1/n = 0.999$.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure D.6: As in Figure D.4, with $n = 5,000$ and $\tau'_n = 1 - 1/n = 0.9998$.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure D.7: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\hat{\xi}_{\tau'_n}^{\star,\text{BR}}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\hat{s}_n^2(1)$ and the average values of $\hat{s}_n^{2,\text{BR}}$ (dashed curve), $\hat{s}_n^2(1)$ (green curve), $\hat{s}_n^2$ (dashed-dotted curve) and $\bar{s}_n^2$ (blue curve) as functions of $k = k_n = n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\hat{I}_{\tau'_n}^{(1)}(\alpha)$, $\hat{I}_{\tau'_n}^{(2)}(\alpha; 1)$, $\hat{I}_{\tau'_n}^{(3)}(\alpha)$ and $\hat{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Burr distribution with $\rho = -1$ and $\gamma = 0.1$ (first row), $\gamma = 0.2$ (second row), $\gamma = 0.3$ (third row), $\gamma = 0.4$ (fourth row). The sample size is $n = 200$ and the target expectile level is $\tau'_n = 1 - 1/n = 0.995$.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure D.8: As in Figure D.7, with $n = 1,000$ and $\tau'_n = 1 - 1/n = 0.999$.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure D.9: As in Figure D.7, with $n = 5,000$ and $\tau'_n = 1 - 1/n = 0.9998$.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure D.10: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\tilde{\xi}_{\tau'_n}^{*,BR}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\tilde{\sigma}_n^2(2)$ and the average values of $\tilde{\sigma}_n^{2,BR}$ (dashed curve), $\tilde{\sigma}_n^2(2)$ (green curve), $\tilde{\sigma}_n^2$ (dashed-dotted curve) and $\bar{\sigma}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\tilde{I}_{\tau'_n}^{(1)}(\alpha)$, $\tilde{I}_{\tau'_n}^{(2)}(\alpha; 2)$, $\tilde{I}_{\tau'_n}^{(3)}(\alpha)$ and $\tilde{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Burr distribution with $\rho = -1$ and $\gamma = 0.1$ (first row), $\gamma = 0.3$ (second row), $\gamma = 0.5$ (third row), $\gamma = 0.7$ (fourth row). The sample size is $n = 200$ and the target expectile level is $\tau'_n = 1 - 1/n = 0.995$.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure D.11: As in Figure D.10, with $n = 1,000$ and $\tau'_n = 1 - 1/n = 0.999$.</figcaption></figure>
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<figure><img src="image_31.png" /><figcaption>Figure D.12: As in Figure D.10, with $n = 5,000$ and $\tau'_n = 1 - 1/n = 0.9998$.</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>Figure D.13: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\xi_{\tau'_n}^{\star,\text{BR}}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\hat{s}_n^2(1)$ and the average values of $\hat{s}_n^{2,\text{BR}}$ (dashed curve), $\hat{s}_n^2(1)$ (green curve), $\hat{s}_n^2$ (dashed-dotted curve) and $\bar{s}_n^2$ (blue curve) as functions of $k = k_n = n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\hat{I}_{\tau'_n}^{(1)}(\alpha)$, $\hat{I}_{\tau'_n}^{(2)}(\alpha; 1)$, $\hat{I}_{\tau'_n}^{(3)}(\alpha)$ and $\hat{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Burr distribution with $\rho = -1/2$ and $\gamma = 0.1$ (first row), $\gamma = 0.2$ (second row), $\gamma = 0.3$ (third row), $\gamma = 0.4$ (fourth row). The sample size is $n = 200$ and the target expectile level is $\tau'_n = 1 - 1/n = 0.995$.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>Figure D.14: As in Figure D.13, with $n = 1,000$ and $\tau'_n = 1 - 1/n = 0.999$.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>Figure D.15: As in Figure D.13, with $n = 5,000$ and $\tau'_n = 1 - 1/n = 0.9998$.</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>Figure D.16: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\tilde{\xi}_{\tau'_n}^{*,BR}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\tilde{\sigma}_n^2(2)$ and the average values of $\tilde{\sigma}_n^{2,\text{BR}}$ (dashed curve), $\tilde{\sigma}_n^2(2)$ (green curve), $\tilde{\sigma}_n^2$ (dashed-dotted curve) and $\bar{\sigma}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\tilde{I}_{\tau'_n}^{(1)}(\alpha)$, $\tilde{I}_{\tau'_n}^{(2)}(\alpha; 2)$, $\tilde{I}_{\tau'_n}^{(3)}(\alpha)$ and $\tilde{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Burr distribution with $\rho = -0.5$ and $\gamma = 0.1$ (first row), $\gamma = 0.3$ (second row), $\gamma = 0.5$ (third row), $\gamma = 0.7$ (fourth row). The sample size is $n = 200$ and the target expectile level is $\tau'_n = 1 - 1/n = 0.995$.</figcaption></figure>
<figure><img src="image_36.png" /><figcaption>Figure D.17: As in Figure D.16, with $n = 1,000$ and $\tau'_n = 1 - 1/n = 0.999$.</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>Figure D.18: As in Figure D.16, with $n = 5,000$ and $\tau'_n = 1 - 1/n = 0.9998$.</figcaption></figure>
<figure><img src="image_38.png" /><figcaption>Figure D.19: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\hat{\xi}_{\tau'_n}^{\star,\text{BR}}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\hat{s}_n^2(1)$ and the average values of $\hat{s}_n^{2,\text{BR}}$ (dashed curve), $\hat{s}_n^2(1)$ (green curve), $\hat{s}_n^2$ (dashed-dotted curve) and $\bar{s}_n^2$ (blue curve) as functions of $k = k_n = n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\hat{I}_{\tau'_n}^{(1)}(\alpha)$, $\hat{I}_{\tau'_n}^{(2)}(\alpha; 1)$, $\hat{I}_{\tau'_n}^{(3)}(\alpha)$ and $\hat{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Fréchet distribution with $\gamma = 0.1$ (first row), $\gamma = 0.2$ (second row), $\gamma = 0.3$ (third row), $\gamma = 0.4$ (fourth row). The sample size is $n = 200$ and the target expectile level is $\tau'_n = 1 - 1/n = 0.995$.</figcaption></figure>
<figure><img src="image_39.png" /><figcaption>Figure D.20: As in Figure D.19, with $n = 1,000$ and $\tau'_n = 1 - 1/n = 0.999$.</figcaption></figure>
<figure><img src="image_40.png" /><figcaption>Figure D.21: As in Figure D.19, with $n = 5,000$ and $\tau'_n = 1 - 1/n = 0.9998$.</figcaption></figure>
<figure><img src="image_41.png" /><figcaption>Figure D.22: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}}/\log((1-\tau_n)/(1-\tau'_n))) \log(\tilde{\xi}_{\tau'_n}^{*,\text{BR}}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\tilde{\sigma}_n^2(2)$ and the average values of $\tilde{\sigma}_n^{2,\text{BR}}$ (dashed curve), $\tilde{\sigma}_n^2(2)$ (green curve), $\tilde{\sigma}_n^2$ (dashed-dotted curve) and $\bar{\sigma}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\tilde{I}_{\tau'_n}^{(1)}(\alpha)$, $\tilde{I}_{\tau'_n}^{(2)}(\alpha; 2)$, $\tilde{I}_{\tau'_n}^{(3)}(\alpha)$ and $\tilde{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Fréchet distribution with $\gamma = 0.1$ (first row), $\gamma = 0.3$ (second row), $\gamma = 0.5$ (third row), $\gamma = 0.7$ (fourth row). The sample size is $n = 200$ and the target expectile level is $\tau'_n = 1 - 1/n = 0.995$.</figcaption></figure>
<figure><img src="image_42.png" /><figcaption>Figure D.23: As in Figure D.22, with $n = 1,000$ and $\tau'_n = 1 - 1/n = 0.999$.</figcaption></figure>
<figure><img src="image_43.png" /><figcaption>Figure D.24: As in Figure D.22, with $n = 5,000$ and $\tau'_n = 1 - 1/n = 0.9998$.</figcaption></figure>
<figure><img src="image_44.png" /><figcaption>Figure D.25: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}}/\log((1-\tau_n)/(1-\tau'_n))) \log(\xi_{\tau'_n}^{\times,\text{BR}}/\xi_{\tau_n})$ (red curve) with boxplots of the asymptotic variance estimate $\hat{s}_n^2(1)$ and the average values of $\hat{s}_n^{2,\text{BR}}$ (dashed curve), $\hat{s}_n^2(1)$ (green curve), $\hat{s}_n^2$ (dashed-dotted curve) and $\bar{s}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\hat{I}_{\tau'_n}^{(1)}(\alpha)$, $\hat{I}_{\tau'_n}^{(2)}(\alpha; 1)$, $\hat{I}_{\tau'_n}^{(3)}(\alpha)$ and $\hat{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Student distribution with $\nu = 10/3$ degrees of freedom, resulting in $\gamma = 0.3$. The sample size is $n = 200$ (first row), 1,000 (second row) and 5,000 (third row) and the target expectile level is $\tau'_n = 1 - 1/n$.</figcaption></figure>
<figure><img src="image_45.png" /><figcaption>Figure D.26: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}}/\log((1-\tau_n)/(1-\tau'_n))) \log(\tilde{\xi}_{\tau'_n}^{\star,\text{BR}}/\tilde{\xi}_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\tilde{\sigma}_n^2(2)$ and the average values of $\tilde{\sigma}_n^{2,\text{BR}}$ (dashed curve), $\tilde{\sigma}_n^2(2)$ (green curve), $\tilde{\sigma}_n^2$ (dashed-dotted curve) and $\bar{\sigma}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\tilde{I}_{\tau'_n}^{(1)}(\alpha)$, $\tilde{I}_{\tau'_n}^{(2)}(\alpha; 2)$, $\tilde{I}_{\tau'_n}^{(3)}(\alpha)$ and $\tilde{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Student distribution with $\nu=10/3$ degrees of freedom, resulting in $\gamma=0.3$. The sample size is $n=200$ (first row), 1,000 (second row) and 5,000 (third row) and the target expectile level is $\tau'_n = 1 - 1/n$.</figcaption></figure>
<figure><img src="image_46.png" /><figcaption>Figure D.27: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\xi_{\tau'_n}^{\times,\text{BR}}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\hat{s}_n^2(1)$ and the average values of $\hat{s}_n^{2,\text{BR}}$ (dashed curve), $\hat{s}_n^2(1)$ (green curve), $\hat{s}_n^2$ (dashed-dotted curve) and $\bar{s}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\hat{I}_{\tau'_n}^{(1)}(\alpha)$, $\hat{I}_{\tau'_n}^{(2)}(\alpha; 1)$, $\hat{I}_{\tau'_n}^{(3)}(\alpha)$ and $\hat{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Student distribution with $\nu = 10/3$ degrees of freedom and location parameter 1 (i.e. distributed as $Z+1$, where $Z$ is Student distributed with $\nu = 10/3$ degrees of freedom), resulting in $\gamma = 0.3$. The sample size is $n = 200$ (first row), 1,000 (second row) and 5,000 (third row) and the target expectile level is $\tau'_n = 1 - 1/n$.</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>Figure D.28: Left panels: Comparison of the empirical variance of $(\sqrt{n(1-\tau_n)}/\log((1-\tau_n)/(1-\tau'_n))) \log(\xi_{\tau'_n}^{\star,\text{BR}}/\xi_{\tau'_n})$ (red curve) with boxplots of the asymptotic variance estimate $\tilde{\sigma}_n^2(2)$ and the average values of $\tilde{\sigma}_n^{2,\text{BR}}$ (dashed curve), $\tilde{\sigma}_n^2(2)$ (green curve), $\tilde{\sigma}_n^2$ (dashed-dotted curve) and $\bar{\sigma}_n^2$ (blue curve) as functions of $k=k_n=n(1-\tau_n)$. Middle panels: coverage probabilities of the associated 95% confidence intervals $\tilde{I}_{\tau'_n}^{(1)}(\alpha)$, $\tilde{I}_{\tau'_n}^{(2)}(\alpha; 2)$, $\tilde{I}_{\tau'_n}^{(3)}(\alpha)$ and $\tilde{I}_{\tau'_n}^{(4)}(\alpha)$ (respectively in black dashed line, green line, black dotted line and blue line) with the nominal level 0.95 in red. Right panels: Median length of these confidence intervals (same color code as in the middle panels). The underlying distribution is a Student distribution with $\nu = 10/3$ degrees of freedom and location parameter 1 (i.e. distributed as $Z+1$, where $Z$ is Student distributed with $\nu = 10/3$ degrees of freedom), resulting in $\gamma = 0.3$. The sample size is $n = 200$ (first row), 1,000 (second row) and 5,000 (third row) and the target expectile level is $\tau'_n = 1 - 1/n$.</figcaption></figure>
<figure><img src="image_48.png" /><figcaption>Figure D.29: For each cluster in Table 2, the corresponding plot of the bias-reduced Hill estimator versus *k* (red line), along with the asymptotic Gaussian 95% confidence interval (lighter red bands), the value of *k* selected by the path stability procedure (vertical dotted line) and its associated pointwise estimate (horizontal dashed line).</figcaption></figure>
<figure><img src="image_49.png" /><figcaption>Figure D.30: For each cluster in Table 2, the corresponding plot of the Weissman quantile estimator $\hat{q}_{1-1/n}^{BR}$ (calculated by making use of the bias-reduced Hill estimator) versus $k$ (red line), along with its associated asymptotic Gaussian 95% confidence interval (lighter red bands), the value of $k$ selected by the path stability procedure (vertical dotted line) with its corresponding pointwise estimate (horizontal dashed blue line), and the sample maximum (horizontal dashed black line).</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>Figure D.31: For each cluster in Table 2, the corresponding plot of the quantile-based bias-reduced expectile estimator $\tilde{\xi}_{1-1/n}^{BR}$ versus $k$ (red line), along with the bias- and variance-corrected asymptotic Gaussian 95% confidence interval (lighter red bands), the value of $k$ selected by the path stability procedure (vertical dotted line) with its associated pointwise estimate (horizontal dashed blue line), and the sample maximum (horizontal dashed black line).</figcaption></figure>