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Estimation of the Adjusted Standard-deviatile for Extreme Risks

Haoyu Chen, Tiantian Mao, Fan Yang

Abstract

In this paper, we modify the Bayes risk for the expectile, the so-called variantile risk measure, to better capture extreme risks. The modified risk measure is called the adjusted standard-deviatile. First, we derive the asymptotic expansions of the adjusted standard-deviatile. Next, based on the first-order asymptotic expansion, we propose two efficient estimation methods for the adjusted standard-deviatile at intermediate and extreme levels. By using techniques from extreme value theory, the asymptotic normality is proved for both estimators. Simulations and real data applications are conducted to examine the performance of the proposed estimators.

Estimation of the Adjusted Standard-deviatile for Extreme Risks

Abstract

In this paper, we modify the Bayes risk for the expectile, the so-called variantile risk measure, to better capture extreme risks. The modified risk measure is called the adjusted standard-deviatile. First, we derive the asymptotic expansions of the adjusted standard-deviatile. Next, based on the first-order asymptotic expansion, we propose two efficient estimation methods for the adjusted standard-deviatile at intermediate and extreme levels. By using techniques from extreme value theory, the asymptotic normality is proved for both estimators. Simulations and real data applications are conducted to examine the performance of the proposed estimators.

Paper Structure

This paper contains 15 sections, 11 theorems, 93 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on RV and 2RV
  3. Asymptotic Expansions of the Deviatile
  4. Estimation
  5. Independence
  6. Intermediate level
  7. Extreme level
  8. Serial dependence
  9. Simulation
  10. Independence
  11. Intermediate level
  12. Extreme level
  13. Serial dependence
  14. Real Data Applications
  15. Conclusion

Key Result

Theorem 1

Assume that $U(t)\in \mathrm{RV}_{\gamma}$ with $0<\gamma<1/2$. Then as $\tau \uparrow1$, where $\beta_{\gamma}=\frac{(\gamma^{-1}-1)^{-\gamma}}{\sqrt{1-2\gamma}}$.

Figures (5)

  • Figure 1: The value of the variantile defined in \ref{['var_tau']} with $X$ following a Pareto($\alpha,\theta$) or Student's $t_\alpha$-distribution.
  • Figure 2: Asymptotic expansions of the deviatile for the Pareto($\alpha,1$) distribution
  • Figure 3: Asymptotic expansions of the deviatile for the Student's $t_\alpha$-distribution
  • Figure 4: Daily log-losses of the S$\&$P 500 index.
  • Figure 5: $\widehat{dev}_{0.95}(H)$ and $\widehat{dev}^*_{0.999}(H)$ for the S& P 500 index daily log-losses are plotted in yellow curves in the left and right panel, respectively. The lower and upper bounds of the corresponding 0.95 confidence interval are plotted in blue and red curves.

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Corollary 2
  • Remark 1
  • Theorem 4
  • Corollary 3
  • ...and 10 more