# Generalized Pareto distribution (GPD)

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.

• @DJohnson That comment confuses statistics with parameters. As explained in the accepted answer, $\sigma$ is a scale parameter and $\mu$ is a location parameter, period. There is no "Gaussian thinking" lurking here.
– whuber
Feb 19 '18 at 16:02
• @DJ There is nothing in this question that even refers to "normal deviates"!
– whuber
Feb 19 '18 at 17:20
• The book concerned may not be widely accessible but the point is just pure mathematical statistics with no underlying ideology, let alone any fallacy. This can be seen from en.wikipedia.org/wiki/Generalized_Pareto_distribution -- which uses the same notation where $\mu$ and $\sigma$ are location and scale parameters and $z$ is just a linear rescaling so that discussion can focus on shape parameters. Choosing any other parameterisation (using other location or scale parameters) would be a matter of taste or convenience alone with no other advantages or disadvantages. cc:@whuber Feb 19 '18 at 18:26
• @DJohnson The mathematical and statistical argument is there for anyone willing to follow it. This is nothing to do with anything but the mathematics of rescaling to scales free of units and dimensions. Feel free to show us a GPD parameterised in any way you prefer; the difference will be purely one of notation. How different summary statistics means behave with highly non-Gaussian data is interesting and important but not germane to the point being made. Feb 19 '18 at 19:35
• Your three comments address a quite different broad question that you find very interesting. I agree with some of what you say, but no matter. It is irrelevant to the point made first by whuber and then myself. The point we make is a matter of understanding what is (more crucially what is not) entailed by a certain mathematical manipulation. No more, no less Feb 20 '18 at 16:56

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.

• Thanks a lot @Easymode44! Nice intuitive examples and a graph! Feb 19 '18 at 13:44

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 $$X$$ $$$$\tag{1} X - u \, \vert \, X > u \quad \overset{\text{dist}}{=} \quad a(u) X$$$$ where $$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 $$S(x) := \text{Pr}\{X > x\}$$ $$$$\tag{2} \frac{S(x + u)}{S(u)} = S[x/a(u)] \quad \text{for all }u, \, x \text{ with } u \in [0,\,\omega) \text{ and } x \geq 0.$$$$ Interestingly, the functional equation (2) nearly characterises the GPD survival. Consider a continuous probability distribution on $$\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)$$.