Generalized Pareto distribution (GPD)

Question

I would like to understand the functional form of the Generalized Pareto distribution (GPD) presented in Wikipedia. My questions are:

1. what is the rationale for replacing $z$ with $\frac{x-\mu}{\sigma}$. Also, in Picture 2 in the last paragraph it is mentioned " ... extend the family by adding the location parameter $\mu$".
2. what is the interpretation of the location parameter $\mu$, and scale parameter $\sigma$ when we are dealing with the distribution of threshold excesses.

Picture 3 features an extract from Extreme Value Modeling and Risk Analysis: Methods and Applications, Dipak K. Dey, Jun Yan

Picture 1. Picture 2. Picture 3.

Answer 1

The replacement of $z$ with $\frac{x-\mu}{\sigma}$ allows the generalization to a "location-scale family". This is common when dealing with continuous distributions. That is, tweaking $\mu$ and $\sigma$ you can center the distribution and spread the distribution as you please.

Check out what happens to the distribution yourself, remembering parameter bounds.

When it comes to tweaking the location parameter, you might want your distribution centered on certain values according to your data. If you are talking about yearly rain maxima, your location parameter might be in the hundreds range, if you are measuring temperatures in a combustion chamber, your distribution will inevitably be centered at higher levels. Similar considerations go for the scale parameter.

Answer 2

The max-stability property of the GEV distribution is quite well known in relation with the Fisher-Tippett-Gnedenko theorem. The GPD has the following remarkable property which can be named threshold stability and relates to the Pickands-Balkema-de Haan theorem. It helps to understand the relation between the location $\mu$ and the scale $\sigma$.

Assume that $X \sim \text{GPD}(0,\,\sigma,\,\xi)$, and let $\omega$ be the upper end-point. Then for each threshold $u \in [0,\, \omega)$, the distribution of the excess $X-u$ conditional on the exceedance $X>u$ is the same, up to a scaling factor, as the distribution of $X$ \begin{equation} \tag{1} X - u \, \vert \, X > u \quad \overset{\text{dist}}{=} \quad a(u) X \end{equation} where $a(u) = 1+ \xi u / \sigma> 0$. So, conditional on $X >u$, the excess $X-u$ is GPD with location $0$ and shape $\sigma_u := a(u) \times \sigma = \sigma + \xi u$.

An appealing interpretation is when $X$ is the lifetime of an item. If the item is alive at time $u$, then the property tells that it will behave as if it was a new one and if the time clock was changed with the new unit $1 / a(u)$. See Figure, where a positive value of $\xi$ is used, implying a rejuvenation and a thick tail.

It seems that in most applications of the GPD the parameter $\mu$ is fixed, and is not estimated. The scale parameter $\sigma$ should then be thought of as related to $\mu$ because the tail remains identical when $\sigma^\star := \sigma - \xi \mu$ is constant.

The relation (1) writes as a functional equation for the survival function $S(x) := \text{Pr}\{X > x\}$ \begin{equation} \tag{2} \frac{S(x + u)}{S(u)} = S[x/a(u)] \quad \text{for all }u, \, x \text{ with } u \in [0,\,\omega) \text{ and } x \geq 0. \end{equation} Interestingly, the functional equation (2) nearly characterises the GPD survival. Consider a continuous probability distribution on $\mathbb{R}$ with end-points $0$ and $\omega >0$ possibly infinite. Assume that the survival function $S(x)$ is strictly decreasing and smooth enough on $[0, \,\omega)$. If (2) holds for a function $a(u) > 0$ which is smooth enough on $[0,\,\omega)$, then $S(x)$ must be the survival function of a $\text{GPD}(0, \, \sigma,\,\xi)$.