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Quantitative Finance · शब्दावली

Hill estimator क्या है?

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

With the order statistics $X_{(1)} \ge X_{(2)} \ge \dots$ of a positive sample, the Hill estimator of the tail index from the $k$ largest is $\hat\alpha_k = \bigl(\frac1k\sum_{i=1}^k\ln X_{(i)} - \ln X_{(k+1)}\bigr)^{-1}$ (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 $k$ 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).

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 $t$ ( $\nu = 4.7$ ) 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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