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Risk Modelling and Survival Analysis · Introduction to extreme value theory

Generalised Extreme Value Distribution: Gumbel, Fréchet and Weibull Types

Updated 11 October 2026 · Fact-checked

The generalised extreme value (GEV) distribution is the family of limit laws for suitably normalised block maxima. One formula covers three types, set by the shape parameter ξ: ξ > 0 gives Fréchet, ξ = 0 gives Gumbel and ξ < 0 gives Weibull. You solve questions by finding ξ, then using the GEV distribution function.

Understand Generalised Extreme Value Distribution

Take n independent, identically distributed losses and record the maximum, Mₙ. As n grows, Mₙ alone has a degenerate limit. So we rescale it: (Mₙ − bₙ) ÷ aₙ, with aₙ > 0. The Fisher-Tippett-Gnedenko theorem says that if this rescaled maximum converges to a non-degenerate distribution, the limit must be a GEV distribution.

The GEV has three parameters. The location μ and scale σ > 0 move and stretch the distribution. The shape ξ controls the tail and decides the type. You can think of the three types as three tail behaviours.

  • Fréchet (ξ > 0): heavy tail that decays like a power. The upper endpoint is infinite. Moments of high order do not exist. Examples of parent distributions: Pareto, Cauchy.
  • Gumbel (ξ = 0): the tail is lighter than any power law. Many members, such as the exponential and the normal, decay roughly exponentially or faster. The lognormal also lies in this domain: its tail is heavier than the exponential, but it is still lighter than any power law. The upper endpoint is usually infinite, as for the exponential, normal, gamma and lognormal. The domain also includes some distributions with a finite endpoint. One example is a distribution on (0, 1) whose survival function is exp(−1/(1 − x)) near x = 1.
  • Weibull (ξ < 0): the distribution has a finite upper endpoint. Examples: uniform, beta.

The domain of attraction of a type is the set of parent distributions whose maxima converge to that type. A Pareto parent is in the Fréchet domain. An exponential parent is in the Gumbel domain. A uniform parent is in the Weibull domain. Note that this Weibull is the extreme value type for maxima, with a finite upper bound. It is not the same as the Weibull loss distribution used for claim sizes.

The practical point: you do not need to know the exact parent distribution to model maxima. You only need the tail type. That is why the GEV is used to fit annual maximum losses, such as the largest flood or the largest single claim in a year.

Key rules to remember

GEV distribution function (ξ ≠ 0)
G(x) = exp{ −[1 + ξ(x − μ) ÷ σ]^(−1/ξ) }, for 1 + ξ(x − μ) ÷ σ > 0
μ is location, σ > 0 is scale, ξ is shape. If 1 + ξ(x − μ) ÷ σ ≤ 0, then G(x) = 0 when ξ > 0 (x is below the lower endpoint) and G(x) = 1 when ξ < 0 (x is above the upper endpoint).
GEV distribution function (ξ = 0, Gumbel)
G(x) = exp{ −exp[ −(x − μ) ÷ σ ] }, for all real x
This is the limit of the general form as ξ → 0.
Type by shape parameter
ξ > 0: Fréchet; ξ = 0: Gumbel; ξ < 0: Weibull
The sign of ξ is the quickest way to name the type.
Support endpoints
ξ > 0: x > μ − σ/ξ; ξ < 0: x < μ − σ/ξ
For ξ < 0 the upper endpoint is μ − σ/ξ, which is finite. For ξ > 0 there is a finite lower bound and no upper bound.
Quantile function (ξ ≠ 0)
x_p = μ + (σ ÷ ξ) × [ (−ln p)^(−ξ) − 1 ]
Solves G(x_p) = p. Gumbel case: x_p = μ − σ ln(−ln p).
Normalised maxima
P[(Mₙ − bₙ) ÷ aₙ ≤ x] = Fⁿ(aₙx + bₙ) → G(x)
This is the Fisher-Tippett-Gnedenko statement. It needs the maxima to be of iid variables and a non-degenerate limit to exist.
Maximum of n iid variables
P(Mₙ ≤ x) = [F(x)]ⁿ
Starting point for all block-maxima work.

How to solve Generalised Extreme Value Distribution questions

Use this method for most GEV questions, whether you are asked to name the type, find a probability or find a quantile.

  1. 1Write down the parameters μ, σ and ξ, or identify them from the information given.
  2. 2Use the sign of ξ to name the type: positive is Fréchet, zero is Gumbel, negative is Weibull.
  3. 3If you are given a parent distribution, look at its upper tail. A power-law tail points to Fréchet. An exponential-type tail points to Gumbel. A finite upper endpoint points to Weibull.
  4. 4Check the support. For ξ ≠ 0, confirm that 1 + ξ(x − μ) ÷ σ > 0 for the x you are using.
  5. 5Substitute into G(x) for a probability. Use the quantile formula for a percentile or return level.
  6. 6Interpret the result in context. State what the block is, for example one year, and what the maximum represents.
  7. 7State any assumptions: iid observations, blocks large enough for the limit to be a good approximation.

Quickest way: Sign of ξ and straight substitution

When to use it: Use this for multiple-choice questions and short written parts where you must name the type or compute one probability.

  1. Read off ξ. Positive means Fréchet, zero Gumbel, negative Weibull.
  2. For a parent distribution, ask: does it have a finite maximum (Weibull)? A polynomial tail (Fréchet)? Otherwise, a tail that decays faster than any power (Gumbel).
  3. Compute z = 1 + ξ(x − μ) ÷ σ first. If z ≤ 0, then G(x) = 0 when ξ > 0 (below the lower endpoint) and G(x) = 1 when ξ < 0 (above the upper endpoint).
  4. Then G(x) = exp(−z^(−1/ξ)). Keep the calculator in memory to avoid rounding errors.

Common mistakes in Generalised Extreme Value Distribution

  • Reversing the sign convention so that ξ < 0 is called Fréchet.

    Some texts use α = 1/ξ or write the Weibull type with a different sign, so memory gets mixed.

    Fix: Use the IAI form with ξ. Remember: positive ξ means a positive, heavy tail, which is Fréchet.

  • Treating the Weibull extreme value type as the Weibull claim-size distribution.

    The names are the same, so students assume the two are the same object.

    Fix: The extreme value Weibull type is for maxima with a finite upper endpoint (ξ < 0). Say 'for maxima' in your answer.

  • Using the Gumbel formula with ξ ≠ 0, or the general formula with ξ = 0.

    The general form has ξ in the denominator of the exponent, so it is undefined at zero.

    Fix: Check ξ first. If ξ = 0 use exp{−exp[−(x − μ) ÷ σ]}.

  • Ignoring the support condition 1 + ξ(x − μ) ÷ σ > 0.

    Students substitute straight away and raise a negative number to a fractional power.

    Fix: Compute z first. If z ≤ 0, then G(x) = 0 when ξ > 0 (below the lower endpoint) and G(x) = 1 when ξ < 0 (above the upper endpoint).

  • Thinking the normal distribution is in the Fréchet domain because it has an infinite range.

    An infinite range is confused with a heavy tail.

    Fix: Normal, exponential and lognormal distributions all lie in the Gumbel domain. The lognormal has a heavier tail than the exponential but is still not power-law. Only power-law tails give Fréchet.

  • Forgetting that the theorem applies to normalised maxima, not to raw maxima.

    The limit of Fⁿ(x) without scaling is degenerate, but this is skipped in short notes.

    Fix: Always mention the constants aₙ > 0 and bₙ in a written answer.

Worked examples

Example 1

Annual maximum losses for a portfolio are modelled by a GEV distribution with μ = 50, σ = 10 and ξ = 0.2 (in ₹ lakh). (a) Name the type. (b) Find the probability that the annual maximum is at most ₹80 lakh.

Show the solution
  1. ξ = 0.2 > 0, so the type is Fréchet.
  2. Compute z = 1 + ξ(x − μ) ÷ σ = 1 + 0.2 × (80 − 50) ÷ 10 = 1 + 0.2 × 3 = 1.6. Since z > 0, x = 80 is inside the support.
  3. The exponent is −1/ξ = −5. So z^(−5) = 1.6^(−5).
  4. 1.6² = 2.56, 1.6⁴ = 6.5536, 1.6⁵ = 10.48576. So 1.6^(−5) = 1 ÷ 10.48576 = 0.095367.
  5. G(80) = exp(−0.095367) = 0.9091.

Answer: (a) Fréchet. (b) P(maximum ≤ ₹80 lakh) ≈ 0.909.

Example 2

A Gumbel distribution has μ = 100 and σ = 20. Find the 95th percentile and the value exceeded by the annual maximum with probability 0.01.

Show the solution
  1. Gumbel quantile: x_p = μ − σ ln(−ln p).
  2. For p = 0.95: −ln 0.95 = 0.051293. ln(0.051293) = −2.97020.
  3. x_0.95 = 100 − 20 × (−2.97020) = 100 + 59.404 = 159.40.
  4. Exceeded with probability 0.01 means p = 0.99. −ln 0.99 = 0.010050. ln(0.010050) = −4.60015.
  5. x_0.99 = 100 + 20 × 4.60015 = 100 + 92.003 = 192.00.

Answer: 95th percentile ≈ 159.4. The value exceeded with probability 0.01 is ≈ 192.0.

Exam tips

  • Learn the three types by the sign of ξ and by a parent example for each: Pareto for Fréchet, exponential or normal for Gumbel, uniform for Weibull.
  • In written answers, state the Fisher-Tippett-Gnedenko result in words and mention the normalising constants and the iid assumption.
  • Always check the support before you substitute. Examiners like to set values near the endpoint when ξ < 0.
  • Show the formula, the substituted values and the final number. Method marks are given even if arithmetic slips.
  • In the computer-based paper, state which block size you used and what the fitted ξ implies about the tail.

Practice questions from Introduction to extreme value theory

Generalised Extreme Value Distribution: frequently asked questions

What is the difference between Gumbel, Fréchet and Weibull distributions in extreme value theory?

They are the three types of the GEV family, set by the shape parameter ξ. Fréchet (ξ > 0) has a heavy, power-law tail. Gumbel (ξ = 0) has an exponential-type tail. Weibull (ξ < 0) has a finite upper endpoint.

What does the Fisher-Tippett-Gnedenko theorem say?

If normalised maxima of iid random variables converge to a non-degenerate limit, that limit must be a GEV distribution. It plays a role for maxima similar to the central limit theorem for sums.

How do I identify the domain of attraction for a maximum?

Look at the upper tail of the parent distribution. A power-law tail such as Pareto gives Fréchet. An exponential-type tail such as the exponential or normal gives Gumbel. A finite upper endpoint such as the uniform gives Weibull.

Is the GEV Weibull the same as the Weibull loss distribution?

No. The extreme value Weibull type describes maxima of variables with a finite upper endpoint and has ξ < 0. The Weibull loss distribution is a model for claim sizes.