Deriving the GPD Approximation from a GEV Tail

Why a GEV Tail Leads to a GPD Approximation

A generalized extreme-value (GEV) distribution describes the behavior of suitably normalized block maxima. A generalized Pareto distribution (GPD) describes the excess above a high threshold. These models are linked: when the tail above a threshold is represented by a GEV model, the conditional excess distribution has a GPD form in the high-threshold limit. The GPD shape is the GEV shape, while its scale changes with the threshold.

It is important to distinguish the limit from the finite-threshold calculation. The conditional excess survival function under a GEV model is generally not exactly a GPD survival function at a fixed threshold. We will derive that exact conditional probability first, then show why it approaches a GPD form as the threshold moves into the tail.

GEV Model and Conditional Excesses

Let \(X\) have a GEV distribution with location \(\mu\), scale \(\sigma>0\), and shape \(\xi\). For \(\xi\ne 0\), its cumulative distribution function on the support where \(1+\xi(x-\mu)/\sigma>0\) is

For \(\xi=0\), the GEV is the Gumbel distribution, with \(F(x)=\exp\{-\exp[-(x-\mu)/\sigma]\}\). Fix a threshold \(u\) with \(P(X>u)>0\), and define the excess \(Y_u=X-u\), conditional on \(X>u\). Its survival probability at \(y\ge 0\) is

The support must also be respected: both \(u\) and \(u+y\) must be in the GEV support. When \(\xi<0\), the upper endpoint is \(x_F=\mu-\sigma/\xi\), so a positive excess must satisfy \(y<x_F-u\).

Deriving the Threshold Scale

Suppose first that \(\xi\ne 0\). Define the GEV support factor at the threshold by \(t_u=1+\xi(u-\mu)/\sigma\), which is positive for an admissible threshold. The same factor at \(u+y\) can be expressed relative to \(t_u\):

This identifies the natural threshold-dependent scale \(\sigma_u=\sigma t_u=\sigma+\xi(u-\mu)\). Define \(r=1+\xi y/\sigma_u\); the excess is admissible exactly when \(r>0\), in addition to \(y\ge 0\). Also let \(q_u=t_u^{-1/\xi}\). Substituting the factored support term into the GEV survival function gives the exact conditional survival:

This expression contains an exponential transformation of \(q_u\), so it is not generally the GPD survival \(r^{-1/\xi}\). To see the approximation, write \(1-e^{-v}=v h(v)\), where \(h(v)=(1-e^{-v})/v\) for \(v>0\) and \(h(0)=1\). Since \(h(v)\to 1\) as \(v\to 0\), the exact ratio becomes

Thus the GPD term is \(r^{-1/\xi}=(1+\xi y/\sigma_u)^{-1/\xi}\). The remaining factor measures the finite-threshold difference between the exact conditional tail and its GPD approximation.

High-Threshold Limit

The scaled excess \(z\sigma_u\) in the nonzero-shape cases matters. It keeps the GPD argument \(1+\xi z\) fixed as the threshold changes, and stays within the limiting excess support. The zero-shape limit instead uses a fixed excess \(y\), because its threshold scale is the constant \(\sigma\).

Proof. For \(\xi\ne 0\), \(q_u=t_u^{-1/\xi}\to 0\) as \(u\to\infty\) when \(\xi>0\), since then \(t_u\to\infty\). When \(\xi<0\), \(t_u\to 0\) as \(u\) increases to \(x_F\), and the exponent \(-1/\xi\) is positive, so again \(q_u\to 0\). Set \(y=z\sigma_u\). Then \(r=1+\xi z\) is fixed and positive under the theorem's conditions. The exact expression is therefore

Both arguments of \(h\) tend to zero, and \(h(v)\to 1\); the ratio of the \(h\) terms tends to one. This proves the asserted limit for \(\xi\ne 0\).

For \(\xi=0\), put \(q_u=\exp[-(u-\mu)/\sigma]\), which tends to zero as \(u\to\infty\). The Gumbel survival at \(u+y\) uses \(q_u\exp(-y/\sigma)\), so the conditional survival is

For fixed \(y\), both arguments of \(h\) tend to zero. The ratio tends to one, leaving \(\exp(-y/\sigma)\), as claimed. \(\square\)

Worked Examples

Interpreting the Result and Avoiding Pitfalls

For \(\xi\ne 0\), the threshold scale can be written \(\sigma_u=\sigma+\xi(u-\mu)\). It increases with \(u\) when \(\xi>0\), decreases when \(\xi<0\), and remains positive for admissible thresholds. In the negative-shape case, \(\sigma_u\to0\) as \(u\to x_F\); this shrinking scale reflects the shrinking range of possible excesses. For \(\xi=0\), the scale remains \(\sigma\).

A common mistake is to claim that the conditional excess survival converges to a fixed GPD survival at fixed positive \(y\) for every nonzero \(\xi\). The GPD expression contains \(\sigma_u\), which depends on the threshold, so it is not a fixed function of \(y\). More importantly, when \(\xi<0\), a fixed positive excess eventually lies outside the support as \(u\to x_F\). The appropriate nondegenerate limit holds \(z=y/\sigma_u\) fixed and uses \(y=z\sigma_u\); then \(1+\xi z\) is fixed and positive. For \(\xi=0\), fixed \(y\) is valid and gives the exponential limit.

The result explains why GEV and GPD models use the same shape parameter when they describe related extreme-value behavior. It also gives a principled threshold-specific scale for excesses. In applications, the GPD is used as a tail approximation at high thresholds; the exact finite-threshold ratio derived here helps clarify what that approximation does and does not assert.