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1 Markets I: The Ecosystem and Exchange-Traded Marketsالأسواق عبر الإنترنت 2 Markets II: Rates, FX and Creditالأسواق عبر الإنترنت 3 Markets III: Commodities, Energy and Cryptoالأسواق عبر الإنترنت 4 Quantitative Methodsالأساليب عبر الإنترنت 5 Derivatives and Volatilityالمشتقات عبر الإنترنت 6 Rates, Credit, XVA and Riskالفائدة والائتمان والمخاطر عبر الإنترنت 7 Research Craft: Predictors, Backtests, Measurement, Portfoliosالبحث عبر الإنترنت 8 Strategies I: Equities and Futuresالاستراتيجيات عبر الإنترنت 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desksالاستراتيجيات عبر الإنترنت 10 Microstructure and Executionالتنفيذ عبر الإنترنت 11 Market Making and High-Frequency Tradingصناعة السوق عبر الإنترنت 12 Machine Learning for Marketsتعلم الآلة عبر الإنترنت 13 Low-Latency Softwareالتكنولوجيا عبر الإنترنت 14 Networks, Hardware and Trading Infrastructureالتكنولوجيا عبر الإنترنت 15 Research, Data and Risk Platformsالتكنولوجيا عبر الإنترنت 16 The Desk and the Firmالشركة عبر الإنترنت 17 The Industry: Firms, Roles and Careersالمسارات المهنية عبر الإنترنت 18 The Interview Bookالمسارات المهنية عبر الإنترنت
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Quantitative Finance · المسرد

ما معنى Hill estimator؟

Definition 15.14 Quantitative Methods · الفصل 15 — Robust Statistics and Heavy Tails

With the order statistics X(1)≥X(2)≥…X_{(1)} \ge X_{(2)} \ge \dots of a positive sample, the Hill estimator of the tail index from the kk largest is α^k=(1k∑i=1kln⁡X(i)−ln⁡X(k+1))−1\hat\alpha_k = \bigl(\frac1k\sum_{i=1}^k\ln X_{(i)} - \ln X_{(k+1)}\bigr)^{-1} (Hill, 1975).

Hill plot of the daily losses of EUR/USD, 1999–2026: the tail index estimated from the k largest losses, with two asymptotic standard errors. A plateau near 3.9 for k between 100 and 200; beyond, the estimate drifts as the body of the distribution enters. Data: ECB euro reference rates (source: ECB statistics).
Figure 15.4. Hill plot of the daily losses of EUR/USD, 1999–2026: the tail index estimated from the kk largest losses, with two asymptotic standard errors. A plateau near 3.9 for kk between 100 and 200; beyond, the estimate drifts as the body of the distribution enters. Data: ECB euro reference rates (source: ECB statistics).
Exceedance probabilities of daily EUR/USD losses, 1999–2026, against three fits: the normal law, a Student t (= 4.7) and a generalised Pareto law above the 95% quantile. The grey line is the one-in-a-thousand level, crossed at 1.80%, 2.74% and 2.47%. Data: ECB euro reference rates (source: ECB statistics).
Figure 15.5. Exceedance probabilities of daily EUR/USD losses, 1999–2026, against three fits: the normal law, a Student tt (ν=4.7\nu = 4.7) and a generalised Pareto law above the 95% quantile. The grey line is the one-in-a-thousand level, crossed at 1.80%, 2.74% and 2.47%. Data: ECB euro reference rates (source: ECB statistics).
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