Estimation of the Adjusted Standard-deviatile for Extreme Risks

Paper: arXiv 2411.07203

Authors: 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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a mathematically rigorous adjustment to the variantile risk measure to properly evaluate extreme risks in heavy-tailed settings. It establishes first- and second-order asymptotic expansions and proves asymptotic normality under both i.i.d. and β-mixing assumptions, supported by simulation and empirical applications.

Research Flowchart

  flowchart TD
    A[Goal: Capture Extreme Risks via Adjusted Standard-Deviatile] --> B[Modify Expectile Bayes Risk & Derive Asymptotic Expansions]
    B --> C[Develop Estimators for Intermediate & Extreme Levels]
    C --> D[Inputs: Theoretical Models, Simulated Data & Real Financial Data]
    D --> E[Prove Asymptotic Normality via Extreme Value Theory & Run Simulations]
    E --> F[Outcomes: Efficient Risk Estimators with Verified Performance]