Threshold Stability of Generalized Pareto Tails

Why Threshold Stability Matters

A threshold model describes the amount by which an observation exceeds a chosen level. If that level is raised, the relevant excess is smaller and is conditioned on a rarer event. The generalized Pareto distribution (GPD) has a useful exact property: after raising the threshold, the excess remains generalized Pareto with the same shape parameter, while its scale changes in a specific way.

This is a statement about a tail that follows an exact GPD model. It does not say that every distribution has an exactly generalized Pareto tail, nor that raising a threshold leaves all parameters unchanged. The scale update is essential for calculating conditional probabilities correctly.

GPD Excesses and Valid Thresholds

Definition: A random variable $Y$ has a generalized Pareto distribution with scale $\sigma_u>0$ and shape $\xi$, written $Y\sim \operatorname{GPD}(\sigma_u,\xi)$, if its survival function is $$ P(Y>y)= \begin{cases} (1+\xi y/\sigma_u)^{-1/\xi}, & \xi\ne 0,\\ \exp(-y/\sigma_u), & \xi=0, \end{cases} $$ for $y\ge 0$ where $1+\xi y/\sigma_u>0$. If $\xi<0$, the support has finite upper endpoint $-\sigma_u/\xi$, and the survival probability is zero at and above that endpoint.

Let $X$ be the variable of interest and $u$ an initial threshold with $P(X>u)>0$. The excess over $u$, conditional on exceeding it, is $Y=X-u\mid X>u$. Suppose this excess has a GPD distribution with scale $\sigma_u$ and shape $\xi$. Raising the threshold by $d\ge 0$ means considering $v=u+d$. A nonzero conditional excess above $v$ is possible only if $P(X>v)>0$. For $\xi<0$, this requires $d<-\sigma_u/\xi$; for $\xi\ge0$, any finite $d\ge0$ is within the modeled support.

Above the raised threshold, the excess is $Z=X-v=Y-d$, conditional on $Y>d$. The event $Y>d$ is exactly the event $X>v$, so this conditioning is what turns the original excess model into the new one.

The Scale Update

Theorem (threshold stability of the GPD): Suppose $Y=X-u\mid X>u$ has a GPD distribution with scale $\sigma_u>0$ and shape $\xi$. For a shift $d\ge0$ such that $P(Y>d)>0$, the conditional excess $Z=Y-d\mid Y>d$ has a GPD distribution with shape $\xi$ and scale $$ \sigma_{u+d}=\sigma_u+\xi d. $$ In particular, $\sigma_{u+d}=\sigma_u$ when $\xi=0$.

To prove the result, calculate the survival function of the excess above the raised threshold. For any $z\ge0$ within the new support,

P(Z>z) =P(Y>d+z\mid Y>d) =\frac{P(Y>d+z)}{P(Y>d)}.

For $\xi\ne0$, substituting the GPD survival function and simplifying gives

\begin{aligned} P(Z>z) &=\left( \frac{1+\xi(d+z)/\sigma_u}{1+\xi d/\sigma_u} \right)^{-1/\xi}\\ &=\left(1+\frac{\xi z}{\sigma_u+\xi d}\right)^{-1/\xi}. \end{aligned}

The denominator $\sigma_u+\xi d$ is positive under the stated condition: this is automatic for $\xi\ge0$, and for $\xi<0$ it follows from $d<-\sigma_u/\xi$. The final expression is the GPD survival function with scale $\sigma_u+\xi d$ and shape $\xi$. When $\xi=0$, the ratio instead gives

P(Z>z)=\frac{\exp(-(d+z)/\sigma_u)}{\exp(-d/\sigma_u)} =\exp(-z/\sigma_u),

which is the GPD survival function for shape zero and scale $\sigma_u$. Thus the shape stays fixed and the scale has the stated update. $\square$

Consequences for Conditional Probabilities

The theorem gives probabilities for further exceedances above the raised level without returning to the original threshold each time. If $X>u$ follows the GPD excess model, then

P(X>u+d)=P(X>u) \begin{cases} (1+\xi d/\sigma_u)^{-1/\xi}, & \xi\ne0,\\ \exp(-d/\sigma_u), & \xi=0. \end{cases}

Conditional on $X>u+d$, the probability of exceeding that threshold by a further amount $z$ is

P(X>u+d+z\mid X>u+d) = \begin{cases} \left(1+\dfrac{\xi z}{\sigma_u+\xi d}\right)^{-1/\xi}, & \xi\ne0,\\[6pt] \exp(-z/\sigma_u), & \xi=0, \end{cases}

for $z$ within the support. The first formula updates the probability of reaching the new threshold; the second describes the distribution of additional excess once that threshold has been reached. Keeping those two quantities separate avoids a common conditional-probability error.

Worked Example: Raising a Threshold in a Heavy-Tailed Model

Suppose the excess above $u$ has scale $\sigma_u=12$ and shape $\xi=0.2$. Raise the threshold by $d=10$. Find the new scale and the probability of exceeding the new threshold by more than a further $z=7$, conditional on exceeding it.

The updated scale is $$ \sigma_{u+10}=12+0.2(10)=14. $$

Using the GPD survival function at the new threshold, $$ P(X>u+17\mid X>u+10) =\left(1+\frac{0.2(7)}{14}\right)^{-1/0.2} =(1.1)^{-5} \approx 0.6209. $$

The same result follows by conditioning at the original threshold: $$ \frac{P(Y>17)}{P(Y>10)} =\left(\frac{1+0.2(17)/12}{1+0.2(10)/12}\right)^{-5} =\left(\frac{77/60}{70/60}\right)^{-5} =(1.1)^{-5}. $$

Thus, among observations already above $u+10$, about 62.1% exceed it by more than another 7 units under this model.

Worked Example: A Finite Upper Endpoint

Suppose $\sigma_u=8$ and $\xi=-0.25$. Raise the threshold by $d=12$, then calculate the probability of a further excess greater than 8, conditional on exceeding the raised threshold.

The original excess has endpoint $$ -\frac{\sigma_u}{\xi}=-\frac{8}{-0.25}=32. $$

Since $12<32$, the raised threshold is valid. The remaining scale is $$ \sigma_{u+12}=8+(-0.25)(12)=5. $$

The new excess endpoint is $-5/(-0.25)=20$. A further excess of 8 is within that endpoint, and its conditional survival probability is $$ P(X>u+20\mid X>u+12) =\left(1+\frac{(-0.25)(8)}{5}\right)^{-1/(-0.25)} =(0.6)^4 =0.1296. $$

For a check from the original threshold, the ratio is $$ \frac{P(Y>20)}{P(Y>12)} =\frac{(1-20/32)^4}{(1-12/32)^4} =\left(\frac{0.375}{0.625}\right)^4 =(0.6)^4. $$

The endpoint also explains why a threshold shift of 32 or more would not be allowed: there would be no positive probability of exceeding that level under this model.

Worked Example: The Exponential Case and Tail Probability

Suppose the excess above $u$ is exponential, so $\xi=0$, with scale $\sigma_u=5$. Let $P(X>u)=0.04$. Raise the threshold by 10. Find the probability of reaching the new threshold, and then the conditional probability of exceeding it by a further 7.

For shape zero the scale does not change. The probability of reaching the raised threshold is $$ P(X>u+10)=0.04\exp(-10/5) =0.04e^{-2} \approx 0.00541. $$

Conditional on reaching $u+10$, the probability of a further excess greater than 7 is $$ P(X>u+17\mid X>u+10) =\exp(-7/5) \approx 0.2466. $$

These are different answers to different questions: about 0.541% of the population exceeds $u+10$, while about 24.66% of those above $u+10$ exceed it by a further 7 units.

Consistency Across Successive Thresholds

The scale update is consistent if the threshold is raised in stages. Starting at $u$, raise it first by $d_1$ and then by $d_2$, with both shifts valid. Applying the theorem twice gives

(\sigma_u+\xi d_1)+\xi d_2 =\sigma_u+\xi(d_1+d_2).

This is exactly the scale obtained by raising the threshold once by $d_1+d_2$. The conditional survival probabilities agree as well. For $\xi\ne0$,

\left(1+\frac{\xi d_1}{\sigma_u}\right)^{-1/\xi} \left(1+\frac{\xi d_2}{\sigma_u+\xi d_1}\right)^{-1/\xi} = \left(1+\frac{\xi(d_1+d_2)}{\sigma_u}\right)^{-1/\xi}.

Indeed, the product inside the powers simplifies because $$ \left(1+\frac{\xi d_1}{\sigma_u}\right) \left(1+\frac{\xi d_2}{\sigma_u+\xi d_1}\right) =1+\frac{\xi(d_1+d_2)}{\sigma_u}. $$ For $\xi=0$, the corresponding identity is $e^{-d_1/\sigma_u}e^{-d_2/\sigma_u}=e^{-(d_1+d_2)/\sigma_u}$. These equalities show that successive conditioning and a single conditioning over the total shift give the same result. $\square$

Common Pitfalls and Interpretation

Keeping the scale fixed for every shape. The scale stays fixed only when $\xi=0$. For positive shape it increases as the threshold rises; for negative shape it decreases toward zero as the finite endpoint is approached.
Confusing the additional excess with the original excess. Above $u+d$, an excess $z$ means $X>u+d+z$, not merely $X>u+z$.
Using the conditional probability as an unconditional tail probability. The GPD survival at the new threshold is conditional on exceeding that threshold. To get $P(X>u+d)$, multiply the conditional survival from $u$ by $P(X>u)$.
Ignoring the finite endpoint when $\xi<0$. The new scale must remain positive, and the raised threshold must still lie below the endpoint. At or beyond the endpoint, the conditional excess model is not defined because the exceedance event has probability zero.

Threshold stability matters because it lets a single exact GPD tail description be expressed at different valid thresholds. Conditional excess calculations can then be made at the threshold of interest using the updated scale, while the unchanged shape records the same tail form. In applications, the property is an implication of the GPD model; it is not by itself proof that observed data follow an exact GPD tail.

Check Your Understanding

Use the scale-update theorem and distinguish carefully between conditional and unconditional probabilities.

If $\sigma_u=10$, $\xi=0.3$, and the threshold is raised by 5, what is the new scale?
For $\sigma_u=6$ and $\xi=-0.2$, what is the original excess endpoint, and which shifts $d$ give a positive probability of exceeding the raised threshold?
When $\xi=0$, how does the scale change after a threshold increase, and why?
Suppose $P(X>u)=0.02$. Which additional quantity is needed to calculate $P(X>u+d)$ from a GPD excess model?
Explain why $P(X>u+d+z\mid X>u+d)$ uses $z$ as an additional excess rather than the total excess above $u$.