**2.1 The generalised extreme value distribution**

Von Misses [4] and Jenkinson [5] combined the three types of extreme value distributions leading to the generalised extreme value distribution (GEV).

Theorem 1.2 If there exists sequences of constants *cn* and *dn*, such that

$$P\left(\frac{M\_n - d\_n}{c\_n} \le y\right) \quad \xrightarrow{n \to \infty} \quad G(y) \tag{8}$$

**3.1 Generalised Pareto distribution (GPD)**

*DOI: http://dx.doi.org/10.5772/intechopen.94578*

cumulative distribution function of Z is given by:

8 >><

*On Modelling Extreme Damages from Natural Disasters in Kenya*

>>:

*σ*

*F*ð Þ *<sup>ζ</sup>*,*ν*,*<sup>σ</sup>* ð Þ¼ *z*

for *<sup>z</sup>*>*<sup>ν</sup>* and <sup>1</sup> <sup>þ</sup> *<sup>ζ</sup>*ð Þ *<sup>z</sup>* � *<sup>ν</sup>*

continuous distributions:

The mean of the GPD is:

The Variance of the GPD is:

**3.2 Relationship between GEV and GPD**

shape *ζ* ∈ .

*<sup>ν</sup>* � *<sup>σ</sup><sup>v</sup> ζ* .

for large n:

**279**

Definition 1The random variable Z has a Generalised Pareto Distribution if the

*σ* � ��1*=<sup>ζ</sup>*

*σ*

Remark 1 (Special Cases) Under specific conditions, the GPD simplifies to other

� � <sup>&</sup>gt; 0 and parameters: location *<sup>ν</sup>*>0, scale *<sup>σ</sup>* <sup>&</sup>gt;0,

� � for *<sup>ζ</sup>* <sup>¼</sup> <sup>0</sup>

for *ζ* 6¼ 0

, for *ζ* <1 (14)

1

z � *ν σ*

h i � � �1*=<sup>ζ</sup>* � � (16)

*r* .

<sup>2</sup> (15)

(13)

<sup>1</sup> � <sup>1</sup> <sup>þ</sup> *<sup>ζ</sup>*ð Þ *<sup>z</sup>* � *<sup>ν</sup>*

<sup>1</sup> � exp � *<sup>z</sup>* � *<sup>ν</sup>*

• When *ζ* ¼ 0, the GDP simplifies to the exponential distribution

• when *ζ* <0, the GDP simplifies to a Pareto type II distribution.

*E Z*ð Þ¼ *<sup>ν</sup>* <sup>þ</sup> *<sup>σ</sup>*

*Var Z*ð Þ¼ *<sup>ν</sup>* <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>

In general, the *<sup>r</sup>* � *th* moment of the GPD only exists if *<sup>ζ</sup>* <sup>&</sup>lt; <sup>1</sup>

*P M*f g *<sup>N</sup>* ≤*z* ¼ *G z*ð Þ, where G zð Þ¼ exp � 1 þ *ζ*

ð Þ *Z* � *v* , conditioned to *Z* > *v* is approximately given by

1 � *ζ*

The shape parameter, *ζ*, determines the tail distribution of the GPD as indicated

Theorem 1.3 Let *Z*1, … , *Zn* be a sequence of independent and identically distributed random variables with a common cumulative distribution function F, and let *Mn* ¼ max f g *Z*1, … , *Zn* satisfying the conditions to be approximated by GEV, i.e.,

Then, for a sufficiently high threshold *v*, the conditional distribution function of

ð Þ 1 � 2*ζ*

, for *ζ* <

ð Þ <sup>1</sup> � *<sup>ζ</sup>* <sup>2</sup>

in remark 1. When *ζ* ¼ 0, there exists a decreasing exponential tail, when *ζ* >0, there is a heavy tail and when *ζ* < 0, the tail is short, with finite upper end point

• When *ζ* >0, the GDP becomes an ordinary Pareto distribution, and

where G is a non-degenerate distribution function, then G is a member of the GEV family:

$$G(\boldsymbol{y}) = \exp\left\{-\left[\mathbf{1} + \zeta\left(\frac{\boldsymbol{\mathcal{Y}} - \boldsymbol{\nu}}{\sigma}\right)\right]^{-1/\zeta}\right\} \tag{9}$$

defined on y such that 1 <sup>þ</sup> *<sup>ζ</sup> <sup>y</sup>* � *<sup>ν</sup> σ* � � <sup>&</sup>gt;0 and with parameters: scale *<sup>σ</sup>* <sup>&</sup>gt;0, location *ν*∈ and scale *ζ* ∈ .

The shape parameter determines the tail behaviour under the same values of location and scale, and thus indicates the type of extreme value distribution:

