# GPD: Introduction

After sampling the extreme observations in our time series using POT, we have a sequence of independent and identically distributed excesses. Now, we want to model their statistical behavior.

Let $F_{th}$ be the distribution function of the excesses over a threshold, $th$,

$
F_{th} = P[X-th \leq x|X>th], \hspace{0.5cm} x  \geq 0
$

Let the excesses be $Y=X-th$ for $X>th$ for a sequence of $n$ independent random variables $X = X_1,...,X_n$. 

We already saw that given:
- A sequence of independent random variables $X = X_1,...,X_n$,
- The maxima of the sequence $M_n = max(X_1,..., X_n)$, where $n$ is the number of observations in a given block,
- $M_n$ are independent and identically distributed.

For large $n$ the cumulative distribution function of those maxima tend to the Generalized Extreme Value ($G(x)$, GEV) family of distributions, regardless the distribution of $X$. If that is true, **the distribution of the excesses can be approximated by a Generalized Pareto distribution** [^gpd_ref].

## Definition and parameters of GPD distribution

The Generalized Pareto distribution is defined as

$
H(y) = \left\{
    \begin{array}{ll}
        1 - \left(1+\frac{\xi y}{\sigma_{th}}\right)^{-1/\xi} \hspace{1cm} for \ \xi \neq 0\\
        1 - exp\left(-\frac{y}{\sigma_{th}}\right)\hspace{1.3cm} for \ \xi = 0
    \end{array}
\right.
$

where $y\geq0$ if $\xi \geq 0$, and $0\leq y\leq -\frac{\sigma_{th}}{\xi}$ if $\xi<0$.

When applied to model the excesses over a threshold, we are calculating conditional probabilities to $X>th$. This distribution function can be computed as function of the random variable $X$ and the threshold $th$ as

$
 P[X<x|X>th] = \left\{
    \begin{array}{ll}
        1 - \left(1+\frac{\xi (x-th)}{\sigma_{th}}\right)^{-1/\xi} \hspace{1cm} for \ \xi \neq 0\\
        1 - exp\left(-\frac{x-th}{\sigma_{th}}\right)\hspace{1.6cm} for \ \xi = 0
    \end{array}
\right.
$

If our previous assumption holds, the parameters of the GPD can be determined based on those of the GEV: the shape parameter ($\xi$) is the same for both distributions, and $\sigma_{th}$ is function of the location and scale parameters of the GEV as

$
\sigma_{th} = \sigma + \xi (th-\mu)
$

Note that $th$ acts like a location parameter for the GPD distribution. In the plot below, this effect is illustrated.

```{figure} https://files.mude.citg.tudelft.nl/GPD_loc.png

---

---
Influence of shape parameter on pareto distribution.
```

Similarly to the GEV, $\xi$ determines the behavior of the tail of the distribution. If $\xi<0$, the distribution presents an upper bound at $th-\sigma_{th}/\xi$. If $\xi>0$, the distribution has a heavy upper tail that behaves like a power function of the exponent $-1/\xi$. If both $\xi=0$ and $th=0$, GPD reduces to the Exponential distribution and if $\xi=-1$, to the uniform distribution. We can see it in the following plot.

```{figure} https://files.mude.citg.tudelft.nl/GPD_shape_2.png

---

---
Influence of shape parameter on pareto distribution.
```

In the above plot, we have the cdf of four GPDs with a $th=0$, $\sigma_{th}=1$ and $\xi=-1.5, \ -1, \ 0 \ and \ 3$. We can see the cdfs of the GPD with $\xi=-1.5 \ and \ -1$ have a clear upper bound in $x=0.67 \ and \ 1$, respectively. If we apply the previous formula $th-\sigma_{th}/\xi = 0-1/(-1.5)=0.67$ and $0-1/(-1)=1$, which fit our visual observation.

Regarding the scale parameter $\sigma_{th}$, it presents a behavior equivalent to that observed on $\sigma$ for the GEV distribution. The higher $\sigma_{th}$, the wider the pdf of the distribution.

```{figure} https://files.mude.citg.tudelft.nl/GPD_scale.png

---

---
Influence of scale parameter on pareto distribution.
```

[^gpd_ref]: J. Pickands. Statistical inference using extreme order statistics. *the Annals of Statistics*, 119-131, 1975