Introduction to extreme value theory: formula sheet

Key formulas

Distribution of the maximum

P(Mₙ ≤ x) = [F(x)]ⁿ, where Mₙ = max(X₁, …, Xₙ)

Holds for independent, identically distributed X with distribution function F.

Return period

T = 1 ÷ p

p is the probability of exceeding the level in one period. A level exceeded with probability p = 0.01 per year has a return period of 100 years.

Probability of at least one exceedance in n periods

1 − (1 − p)ⁿ

Assumes independent periods with the same exceedance probability p.

GEV distribution function (ξ ≠ 0)

H(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. For ξ = 0: H(x) = exp{ −exp[−(x − μ) ÷ σ] }.

GPD distribution function (ξ ≠ 0)

G(y) = 1 − (1 + ξy ÷ σ)^(−1/ξ), for y ≥ 0

y = X − u is the excess over threshold u. For ξ = 0: G(y) = 1 − exp(−y ÷ σ).

Meaning of the shape parameter ξ

ξ > 0: heavy tail; ξ = 0: exponential-type tail; ξ < 0: finite upper endpoint

Learn the interpretation as well as the formula. It is a common written-question point.

CDF of the maximum

Fₘ(x) = P(Mₙ ≤ x) = [F(x)]ⁿ

Needs independence and identical distribution. Without independence you cannot multiply.

Density of the maximum

fₘ(x) = n [F(x)]ⁿ⁻¹ f(x)

Differentiate [F(x)]ⁿ using the chain rule.

Survival function of the maximum

P(Mₙ > x) = 1 − [F(x)]ⁿ

Use this for the chance that at least one value exceeds x.

CDF of the minimum

P(min ≤ x) = 1 − [1 − F(x)]ⁿ

Condition for the minimum to exceed x: all variables exceed x.

Normalised maximum

(Mₙ − bₙ) ÷ aₙ → GEV in distribution, with aₙ > 0

The limit exists only under suitable conditions on F. The constants depend on F.

Exceedance in n blocks from one block

If a block maximum has P(M ≤ x) = p, then P(M ≤ x in all k independent blocks) = pᵏ

Follows from the same product rule applied to block maxima.

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.

Excess over threshold

Y = X − u, given X > u

Only observations above u are used. Subtract u before fitting.

GPD distribution function

H(y) = 1 − (1 + ξy/σ)^(−1/ξ) for ξ ≠ 0; H(y) = 1 − e^(−y/σ) for ξ = 0

Valid for y ≥ 0 and 1 + ξy/σ > 0. If ξ < 0, y is bounded above by −σ/ξ.

Tail probability of the original variable

P(X > x) ≈ (N_u ÷ n) × (1 + ξ(x − u)/σ)^(−1/ξ), for x > u

N_u is the number of exceedances and n the total sample size. N_u ÷ n estimates P(X > u).

High quantile (value at risk)

x_p = u + (σ/ξ) × [ ((n ÷ N_u) × (1 − p))^(−ξ) − 1 ], for ξ ≠ 0

Valid for p high enough that x_p > u. For ξ = 0 use x_p = u + σ ln[(N_u ÷ n) ÷ (1 − p)].

Mean of the GPD

E[Y] = σ ÷ (1 − ξ), for ξ < 1

The mean does not exist for ξ ≥ 1.

Mean excess function

e(u) = E[X − u | X > u]; for a GPD, e(u) = (σ + ξu) ÷ (1 − ξ) when ξ < 1

For GPD data this is linear in u. A positive slope suggests ξ > 0. Here σ is the scale at threshold zero; the scale at u is σ + ξu.

Sample mean excess

ê(u) = Σ(xᵢ − u) over xᵢ > u, ÷ N_u

Plot ê(u) against u to choose the threshold.

GEV distribution function

G(x) = exp{ −[1 + ξ(x − μ)/σ]^(−1/ξ) } for ξ ≠ 0, where 1 + ξ(x − μ)/σ > 0

For ξ = 0 (Gumbel): G(x) = exp{ −exp[−(x − μ)/σ] }.

GPD distribution function

H(y) = 1 − (1 + ξy/σ)^(−1/ξ) for ξ ≠ 0; H(y) = 1 − exp(−y/σ) for ξ = 0

y = x − u ≥ 0, σ > 0. If ξ < 0, also y ≤ −σ/ξ.

GPD log-likelihood

l(ξ, σ) = −Nu ln σ − (1 + 1/ξ) Σ ln(1 + ξ yᵢ/σ), for ξ ≠ 0

yᵢ are the Nu excesses. For ξ = 0: l(σ) = −Nu ln σ − Σyᵢ/σ, so σ̂ = ȳ.

Tail probability (POT)

P(X > x) = (Nu ÷ n) × [1 + ξ(x − u)/σ]^(−1/ξ), for x > u

Nu = number of exceedances, n = total sample size.

High quantile (POT)

x_p = u + (σ/ξ) × { [(n ÷ Nu)(1 − p)]^(−ξ) − 1 }

Valid for p high enough that x_p > u. For ξ = 0: x_p = u + σ ln[Nu ÷ (n(1 − p))].

Return level (GEV, block maxima)

z_T = μ − (σ/ξ){ 1 − [−ln(1 − 1/T)]^(−ξ) } for ξ ≠ 0

T is the return period in blocks. Gumbel case: z_T = μ − σ ln[−ln(1 − 1/T)].

Mean excess function (GPD)

E[X − v | X > v] = [σ + ξ(v − u)] ÷ (1 − ξ), for v ≥ u and ξ < 1

Linear in v. A rising mean excess plot suggests ξ > 0.

Expected shortfall (GPD tail)

ES_p = x_p ÷ (1 − ξ) + (σ − ξu) ÷ (1 − ξ), for ξ < 1

The mean exists only if ξ < 1. State this condition.

Quick revision

• Extreme value theory models the tail, not the centre, of a distribution.
• For independent, identically distributed variables, P(Mₙ ≤ x) = [F(x)]ⁿ.
• Block maxima: split the data into blocks and model the maximum of each block.
• Under suitable conditions, the scaled and shifted maximum converges to a GEV distribution.
• The GEV has location, scale and shape parameters.
• The shape parameter's sign gives the tail type: positive is heavy-tailed, zero is light-tailed with no finite upper end, negative is bounded above.
• Peaks over threshold: keep observations above a high threshold and model the excesses.
• Excesses over a high threshold are modelled by the generalised Pareto distribution.
• Block size and threshold are the key choices: too low a threshold adds bias, too high leaves little data.
• Check the fit with diagnostic plots before you trust any tail estimate.
• A return level is a high quantile expressed in terms of an event expected once in a stated period.
• Always state your assumptions, especially independence and a high enough threshold.

Common mistakes

• Treating a return period of T years as an event that happens exactly every T years. Fix: Say it is the average waiting time. In each year the probability of exceedance is 1 ÷ T, and a 100-year event can occur in consecutive years.
• Using a normal distribution for tail quantiles of a heavy-tailed risk. Fix: Remember that fit in the bulk says little about the tail. Use a tail model such as the GPD for high quantiles.
• Writing P(Mₙ ≤ x) = n F(x) or F(x) ÷ n. Fix: Remember that the maximum is at most x only if all are at most x, so you multiply: F(x)ⁿ.
• Using 1 − F(x)ⁿ for P(Mₙ ≤ x). Fix: Ask whether the event is 'all below x' (F(x)ⁿ) or 'at least one above x' (1 − F(x)ⁿ).
• Reversing the sign convention so that ξ < 0 is called Fréchet. 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. Fix: The extreme value Weibull type is for maxima with a finite upper endpoint (ξ < 0). Say 'for maxima' in your answer.
• Fitting the GPD to the raw losses instead of the excesses. Fix: Always subtract u first. Add u back when you convert to a quantile of X.
• Using the GPD tail probability without multiplying by N_u ÷ n. Fix: Multiply by P(X > u), estimated as N_u ÷ n, to get the unconditional tail probability.
• Using 1 − p directly in the quantile formula without the factor n ÷ Nu. Fix: Always form r = (n ÷ Nu)(1 − p). The factor n ÷ Nu converts a tail probability on the full data into one on the exceedances.
• Mixing up the GEV return level formula with the GPD quantile formula. Fix: Block maxima and return period T use the GEV with −ln(1 − 1/T). Threshold data use the GPD with Nu ÷ n. Decide the method first.

Exam tips

• Write the method name and the matching distribution together: block maxima with GEV, peaks over threshold with GPD.
• For written questions, explain why tails need separate modelling. Mention that the bulk-fit may not describe extremes.
• Always interpret ξ in words as well as stating its sign.
• In calculations, show the formula [F(x)]ⁿ or 1 − (1 − p)ⁿ before substituting numbers.
• Add a short comment on data scarcity and uncertainty in tail estimates. It earns marks in discussion parts.
• In MCQs, first decide whether the event is 'all below x' or 'at least one above x'. This picks the formula in seconds.
• In written answers, always state the iid assumption before using F(x)ⁿ. Markers look for it.
• When asked to derive the density, show the differentiation step and keep the factor n and the power n − 1.