Generalized Extreme-Value Distributions

Why the Shape Parameter Matters

The generalized extreme-value (GEV) family uses one shape parameter to describe distributions with different support and tail behavior. Its location and scale parameters set the distribution’s position and spread; its shape parameter determines whether the support has a finite lower endpoint, a finite upper endpoint, or neither. The zero-shape case is defined separately, but it is also the continuous limit of the nonzero-shape formulas.

This tutorial develops the GEV cumulative distribution function (CDF) and quantile function, checks how the support changes with the sign of the shape parameter, and proves the limit at zero shape. The earlier material on extreme-value limits for normalized maxima motivates this family; here the focus is on using its parameterization precisely.

Definition, Support, and Density

Definition: A random variable $X$ has a generalized extreme-value distribution with location $\mu\in\mathbb{R}$, scale $\sigma>0$, and shape $\xi\in\mathbb{R}$ if its CDF is given as follows. For $\xi\ne 0$, put $z=(x-\mu)/\sigma$. On the set where $1+\xi z>0$, $$ F(x)=\exp\!\left\{-(1+\xi z)^{-1/\xi}\right\}. $$ For $\xi=0$, $$ F(x)=\exp\!\left\{-\exp(-z)\right\},\qquad z=\frac{x-\mu}{\sigma},\quad x\in\mathbb{R}. $$ When $\xi>0$, set $F(x)=0$ for $x\leq \mu-\sigma/\xi$. When $\xi<0$, set $F(x)=1$ for $x\geq \mu-\sigma/\xi$.

The boundary values in this definition make the CDF continuous at each finite endpoint. For $\xi>0$, the support is $x>\mu-\sigma/\xi$, with a finite lower endpoint. For $\xi<0$, it is $x<\mu-\sigma/\xi$, with a finite upper endpoint. When $\xi=0$, the support is the whole real line. The endpoint $\mu-\sigma/\xi$ is finite only for nonzero $\xi$, and its location relative to $\mu$ changes with the sign of $\xi$.

For $\xi\ne0$, the expression $1+\xi z$ must be positive wherever the formula is used. The separate definitions outside that range are essential: extending the power expression without checking its base can produce an undefined or invalid CDF.

Theorem: On the interior of its support, the GEV distribution has density $$ f(x)=\frac{1}{\sigma}\exp\!\left\{-(1+\xi z)^{-1/\xi}\right\}(1+\xi z)^{-1/\xi-1},\qquad \xi\ne0, $$ and, for $\xi=0$, $$ f(x)=\frac{1}{\sigma}\exp(-z)\exp\!\left\{-\exp(-z)\right\}. $$

To derive the nonzero-shape formula, write $t=(1+\xi z)^{-1/\xi}$. Since $dz/dx=1/\sigma$, $$ \frac{dt}{dx}=-\frac{1}{\sigma}(1+\xi z)^{-1/\xi-1}. $$ Differentiating $F(x)=\exp(-t)$ gives $f(x)=-\exp(-t)\,dt/dx$, which is the stated density. For $\xi=0$, differentiating $\exp\{-\exp(-z)\}$ with respect to $x$ gives the second formula. These densities apply only on the support’s interior; outside it, the density is zero.

Worked Example: Reading the Support from the Shape

Let $\mu=10$, $\sigma=2$, and $\xi=0.5$. The finite endpoint is $$ \mu-\frac{\sigma}{\xi}=10-\frac{2}{0.5}=6. $$ Because $\xi>0$, this is a lower endpoint. The CDF is zero for $x\leq6$, and the formula applies for $x>6$. At $x=8$, $z=(8-10)/2=-1$, so $1+\xi z=1-0.5=0.5$. Therefore $$ F(8)=\exp\!\left\{-(0.5)^{-2}\right\}=e^{-4}\approx0.0183. $$ At $x=5$, which lies below the endpoint, $F(5)=0$; the power formula is not used there.

Worked Example: A CDF Value with Negative Shape

Let $\mu=10$, $\sigma=2$, and $\xi=-0.5$. The finite endpoint is $$ \mu-\frac{\sigma}{\xi}=10-\frac{2}{-0.5}=14. $$ Since the shape is negative, this is an upper endpoint. At $x=12$, $z=1$, and $1+\xi z=1-0.5=0.5>0$. The exponent is $-1/\xi=2$, so $$ F(12)=\exp\!\left\{-(0.5)^2\right\}=e^{-0.25}\approx0.7788. $$ At the endpoint and above it, the CDF is 1 by definition. In particular, the probability $P(X>14)$ is zero.

Quantiles

For $0<p<1$, the $p$-quantile $Q(p)$ is the value satisfying $F(Q(p))=p$. The GEV CDF is continuous and strictly increasing on the interior of its support, so this equation has a unique solution there.

Theorem: For $0<p<1$, the GEV quantile is $$ Q(p)= \begin{cases} \displaystyle \mu+\frac{\sigma}{\xi}\left[(-\log p)^{-\xi}-1\right],&\xi\ne0,\\[8pt] \displaystyle \mu-\sigma\log(-\log p),&\xi=0. \end{cases} $$

For $\xi\ne0$, let $Q(p)=\mu+\sigma z$. Taking logarithms of $p=\exp\{-(1+\xi z)^{-1/\xi}\}$ yields $$ -\log p=(1+\xi z)^{-1/\xi}. $$ Both sides are positive. Raising them to the power $-\xi$ gives $(-\log p)^{-\xi}=1+\xi z$. Solving for $z$ gives the first quantile formula. For $\xi=0$, taking logarithms of $p=\exp\{-\exp(-z)\}$ gives $-\log p=\exp(-z)$. Taking logarithms again yields $z=-\log(-\log p)$, giving the second formula.

The quantile formula also verifies that the solution lies in the support. For $\xi\ne0$, let $A=(-\log p)^{-\xi}$, which is positive. Then $1+\xi z=A>0$. In the negative-shape case, a useful endpoint comparison is $$ Q(p)-\left(\mu-\frac{\sigma}{\xi}\right)=\frac{\sigma A}{\xi}<0,\qquad \xi<0. $$ Thus the quantile is below the finite upper endpoint. This argument does not assume that $A<1$, which is not true for every $p$.

Worked Example: A Median with Positive Shape

Let $\mu=10$, $\sigma=2$, $\xi=0.5$, and $p=0.5$. The quantile formula gives $$ Q(0.5)=10+4\left[(-\log 0.5)^{-0.5}-1\right]. $$ Using the exact value inside the expression, $-\log(0.5)=0.693147\ldots$, gives $$ Q(0.5)\approx10.8045. $$ This exceeds the lower endpoint 6, as required. Equivalently, $A=(-\log 0.5)^{-0.5}>0$, so $1+\xi z=A>0$ for the quantile obtained by inversion.

The Three Shape Cases and the Zero-Shape Limit

For positive shape, the lower endpoint is finite and the CDF rises from zero there toward one as $x$ increases. For negative shape, the upper endpoint is finite and the CDF reaches one there. These conclusions follow directly from the formula: as the positive base $1+\xi z$ approaches zero from above, the transformed quantity $(1+\xi z)^{-1/\xi}$ tends to infinity for $\xi>0$, and tends to zero for $\xi<0$. Substitution into $\exp\{-(1+\xi z)^{-1/\xi}\}$ gives the respective boundary values.

At zero shape, the GEV CDF becomes the Gumbel form $\exp\{-\exp(-z)\}$. This is not merely a convention for filling in a missing parameter value; it is the pointwise limit of the nonzero-shape CDF at every fixed real $x$.

Theorem: Fix $\mu\in\mathbb{R}$, $\sigma>0$, and $x\in\mathbb{R}$, and let $z=(x-\mu)/\sigma$. As $\xi\to0$, the nonzero-shape GEV CDF at $x$ converges to $\exp\{-\exp(-z)\}$. For every fixed $p\in(0,1)$, its quantile converges to $\mu-\sigma\log(-\log p)$.

To prove the CDF limit, fix $z$. For all sufficiently small $|\xi|$, $1+\xi z>0$, so the nonzero-shape formula applies. The elementary logarithm limit gives $$ \lim_{\xi\to0}\frac{\log(1+\xi z)}{\xi}=z. $$ Consequently, $$ (1+\xi z)^{-1/\xi} =\exp\!\left\{-\frac{\log(1+\xi z)}{\xi}\right\} \longrightarrow \exp(-z). $$ Continuity of the exponential function now implies $$ \exp\!\left\{-(1+\xi z)^{-1/\xi}\right\} \longrightarrow \exp\{-\exp(-z)\}. $$ This proves pointwise convergence at the fixed $x$, including when $z=0$.

For the quantile limit, put $b=-\log p>0$. Then $b^{-\xi}=\exp(-\xi\log b)$, and $$ \frac{b^{-\xi}-1}{\xi} =\frac{\exp(-\xi\log b)-1}{\xi} \longrightarrow-\log b. $$ Multiplying by $\sigma$ and adding $\mu$ gives $Q(p)\to\mu-\sigma\log(-\log p)$. This establishes both limits without interchanging a limit and an integral. $\square$

Worked Example: Gumbel CDF and Median

Let $\mu=3$, $\sigma=2$, and $\xi=0$. At $x=4$, $z=(4-3)/2=0.5$, so $$ F(4)=\exp\{-\exp(-0.5)\} =\exp(-0.606530\ldots)\approx0.5452. $$ For the median, $$ Q(0.5)=3-2\log(-\log 0.5)\approx3.7330. $$ The approximation is calculated from the exact expression; rounding the intermediate value $-\log(0.5)$ too early can change the last reported digit.

Worked Example: A Median with Negative Shape

Let $\mu=10$, $\sigma=2$, $\xi=-0.5$, and $p=0.5$. The quantile is $$ Q(0.5)=10-4\left[(-\log 0.5)^{0.5}-1\right]\approx10.6698. $$ To check its position relative to the upper endpoint 14, use $A=(-\log 0.5)^{0.5}>0$: $$ Q(0.5)-14=\frac{2A}{-0.5}=-4A<0. $$ Thus the median lies below the endpoint. The sign of this difference, rather than an assumption that $A$ is less than one, establishes the support check.

Common Pitfall and Practical Use

A common mistake is to infer whether a quantile lies above or below the location just by looking at the sign of $\xi$, or to assume the power term $(-\log p)^{-\xi}$ is always below one. Neither shortcut is reliable: that power depends on both $\xi$ and $p$. For example, with $\xi=-0.5$ and $p=0.1$, it equals $\sqrt{-\log(0.1)}$, which is approximately $1.517>1$. To check a finite endpoint, subtract the endpoint algebraically as above; the sign then follows from $\sigma>0$, $A>0$, and the sign of $\xi$.

The GEV parameterization is useful because the same CDF and quantile formulas cover three support patterns, while the shape parameter identifies which pattern applies. A practical calculation should therefore proceed in a disciplined order: check the sign of $\xi$, identify any finite endpoint, verify that the power expression has a positive base, and only then evaluate the CDF or quantile. This avoids treating an out-of-support value as though the interior formula were valid there.

Check the shape.
If $\xi>0$, calculate the lower endpoint; if $\xi<0$, calculate the upper endpoint; if $\xi=0$, use the Gumbel formulas.

Check the input.
For nonzero shape, verify $1+\xi(x-\mu)/\sigma>0$ before using the interior CDF expression.

Evaluate or invert.
Use the CDF for probabilities and the appropriate quantile formula for a specified probability; compare quantiles with the endpoint when one exists.

Key takeaway: The GEV CDF is $\exp\{-(1+\xi z)^{-1/\xi}\}$ on its valid support when $\xi\ne0$, and $\exp\{-\exp(-z)\}$ when $\xi=0$. Positive shape gives a finite lower endpoint, negative shape a finite upper endpoint, and the nonzero-shape formulas converge to the zero-shape formulas as $\xi\to0$.

Check Your Understanding

Use the support and formulas to answer these questions.

For $\mu=4$, $\sigma=3$, and $\xi=0.6$, calculate the finite endpoint and state whether it is a lower or upper endpoint.
For $\mu=0$, $\sigma=1$, and $\xi=-0.25$, what is the finite endpoint, and what is the CDF at and above it?
Starting from $F(Q(p))=p$, show how the nonzero-shape quantile formula follows by taking logarithms and raising positive quantities to a power.
Why is $1+\xi(x-\mu)/\sigma>0$ required before using the nonzero-shape CDF formula?
State the limiting CDF and quantile as $\xi\to0$, and identify which elementary limit gives each result.