Hill estimator: Definition & Examples

The Hill estimator estimates the heaviness of a positive upper tail by averaging the logarithms of the largest observations relative to a high threshold. A risk analyst can use it to estimate a power-law tail index before extrapolating rare losses beyond the observed sample. That extrapolation depends on both the tail assumption and the observations admitted to the fit. Hill's original approach models the tail without specifying the whole distribution.

Count the observations above the threshold

Write the ordered sample in ascending order:

X_{(1)}\le\cdots\le X_{(n)}.

Choose an integer $1\le k<n$ and a positive threshold $u=X_{(n-k)}$ . The Hill estimator is

\widehat\gamma_{k,n} =\frac{1}{k}\sum_{j=1}^{k} \log\!\left(\frac{X_{(n-j+1)}}{X_{(n-k)}}\right), \qquad \widehat\alpha_{k,n}=\frac{1}{\widehat\gamma_{k,n}}.

Here,

k

counts the upper observations used in the average. With no ties, all

k

are strictly above

u

; the threshold is a separate observation. In descending order, the same threshold is the

(k+1)

st largest observation. Check ordering and counting conventions when comparing formulas or software. Resnick and Stărică, equation (1.1) use that descending convention.

The two parameters have opposite interpretations. The extreme-value index

\gamma>0

is also commonly written

\xi

; larger

\gamma

means a heavier tail. The Pareto exponent is

\alpha=1/\gamma

; smaller

\alpha

means a heavier tail. A reported “Hill tail index” is ambiguous unless its convention is stated. If all selected values equal the threshold, the log average is zero and its reciprocal is undefined.

For the illustrative sorted sample $[1,2,4,8,16]$ and $k=2$ , the threshold is $X_{(3)}=4$ . Thus

\widehat\gamma =\frac{\log(8/4)+\log(16/4)}{2} =\frac{\log 2+\log 4}{2} \approx1.0397, \qquad \widehat\alpha\approx0.9618.

This five-point example checks the arithmetic only. It does not establish a power-law tail or justify a conclusion about the population's moments.

Sorted values 1, 2, 4, 8, 16 with threshold 4 and top two observations 8 and 16. Their log ratios are 0.6931 and 1.3863, averaging to Hill gamma 1.0397 and reciprocal alpha 0.9618.

Original arithmetic diagram. The threshold anchors both ratios and is excluded from the two-observation average; values are in arbitrary loss units.

Why logarithms recover tail heaviness

The model is a regularly varying upper tail:

\mathbb{P}(X>x)=x^{-1/\gamma}L(x),\qquad \gamma>0, \qquad \frac{L(tx)}{L(x)}\longrightarrow1 \quad(x\to\infty)

for every fixed

t>0

. The slowly varying factor

L

allows departures from an exact Pareto law. This is a particular tail class; the informal description “heavy-tailed” alone is insufficient. Resnick's definitions 4–5 connect this assumption to the Hill estimator.

For an exact Pareto distribution, at a fixed threshold

u

within its Pareto region,

\mathbb{P}(X/u>t\mid X>u)=t^{-\alpha},\qquad t\ge1.

Consequently $\log(X/u)$ conditional on $X>u$ is exponential with mean $1/\alpha=\gamma$ . Averaging log ratios estimates that mean. In the more general regularly varying case, the Pareto relation is a high-threshold approximation rather than an exact identity at every $u$ .

Choosing k is part of the estimate

A smaller

k

raises the threshold and uses fewer extremes: sampling variability can be severe. Increasing

k

lowers the threshold and supplies more observations, but may admit values where the Pareto approximation is poor. This creates a bias–variance tradeoff. Under iid sampling and a regularly varying tail, consistency uses an intermediate sequence:

k\to\infty

while

k/n\to0

. These limits do not supply a universal finite-sample choice. Resnick and Stărică's threshold discussion explains why choosing

k

remains difficult.

A Hill plot graphs

\widehat\gamma_{k,n}

against

k

. Look for sensitivity across a defensible threshold range, rather than treating one attractive plateau as proof of the model. Adjacent estimates reuse most of the same observations. A stable-looking segment can therefore be misleading. Bias need not rise with

k

for an exact Pareto model; the issue is departure from the Pareto approximation at the thresholds used.

Paired simulated Hill plots for 2000 exact Pareto observations X with alpha 2 and shifted values Y equals X plus 5. Both have limiting gamma 0.5. At k 200, the Hill estimates are 0.5007 for X and 0.2614 for Y; the shifted curve moves farther below 0.5 as k rises.

Original seeded simulation: 2,000 paired observations, exact Type I Pareto exponent 2 and the same draws plus 5. Both models have limiting gamma 0.5; one simulated Hill plot is neither a confidence band nor a recommended choice of k.

The shifted model still has a power-law tail with exponent 2, but finite-threshold ratios change. At $k=200$ , the estimates in this simulation are $0.5007$ for $X$ and $0.2614$ for $X+5$ . Adding observations to a misspecified finite-threshold approximation can make its answer look less noisy without bringing it closer to the limiting index.

Keep the tail and the data convention aligned

Multiplying every observation by the same positive constant leaves the ratios unchanged. Adding a constant does not. Do not shift signed returns merely to make their logarithms legal. For downside returns $R$ , define the loss variable $X=-R$ and use its upper observations above a positive threshold. Using $|R|$ instead pools both return tails and answers a different question.

Likewise, the Hill calculation uses

X/u

, not the additive excess

X-u

. The latter belongs to a different modelling construction. The Generalised Pareto and Peaks-Over-Threshold lesson develops that excess distribution and threshold diagnostics (public preview; full lesson requires membership).

For financial time series, volatility clustering makes iid uncertainty calculations questionable. Dependence does not automatically invalidate Hill estimation, but validity and uncertainty need assumptions suited to the process. Resnick and Stărică prove consistency for specified dependent models under explicit conditions; this is not a guarantee for every dependent series. Ties and rounding also change log spacings. Inspect those features alongside the Hill plot before interpreting a fitted tail index. Belzile's tutorial illustrates rounding artefacts and explains the estimator's lack of location invariance.

Sources

Hill (1975), A Simple General Approach to Inference About the Tail of a Distribution, Annals of Statistics 3(5), 1163–1174. Original conditional-tail-likelihood motivation.
Resnick and Stărică (1998), Tail Index Estimation for Dependent Data, introduction, equation (1.1), and section 2. Order-statistic convention and conditions for dependent-data consistency.
Resnick, Modeling Data Network Sessions, sections 8.1–8.2, slides 43–44. Regularly varying tails and Hill estimation.
Resnick and Stărică (1997), Smoothing the Hill Estimator, abstract. Intermediate sequences and the difficulty of choosing $k$ .
Belzile, Semiparametric methods and Hill estimation, EVA 2023 tutorial. Finite-threshold bias, Hill plots, rounding and location sensitivity.

Count the observations above the threshold

Why logarithms recover tail heaviness

Choosing k is part of the estimate

Keep the tail and the data convention aligned

Sources

Related terms

Bias-variance tradeoff

Pareto distribution

Volatility clustering