**3. Threshold models and the generalised Pareto distribution**

Modelling using block maxima is inefficient as it is wasteful of the data, considering that the complete dataset is available. An alternative approach is to model all the data above some high threshold, in what is commonly referred to as threshold modelling or Excess over Threshold (EOT) modelling.

Given a set of independent and identically distributed random variables *Z*1, … , *Zn*, having a common distribution function,F, we are interested in estimating the conditional excess distribution function, *Fv*, of random variable *X* above a high threshold *v*:

$$F\_u(y) = P(Z - \nu \le y | Z > \nu), \quad 0 \le y \le z\_+ - \nu \tag{10}$$

where *y* ¼ *z* � *v* are the exceedances and *z*<sup>þ</sup> is the right endpoint of F. We can express *Fv*ð Þ*y* in terms of *z* as:

$$F\_{\boldsymbol{v}}(\boldsymbol{y}) = \frac{F(\boldsymbol{v} + \boldsymbol{y}) - F(\boldsymbol{v})}{\mathbf{1} - F(\boldsymbol{v})} = \frac{F(\boldsymbol{v} + \boldsymbol{z} - \boldsymbol{v}) - F(\boldsymbol{v})}{\mathbf{1} - F(\boldsymbol{v})} = \frac{F(\boldsymbol{z}) - F(\boldsymbol{v})}{\mathbf{1} - F(\boldsymbol{v})} \tag{11}$$

Piklands [23] posed that if the underlying distribution F(z) is in the maximum domain of attraction of extreme value distribution, then the conditional excess distribution function *Fv*ð Þ*z* for a large *v*, can be approximated by:

$$F\_v(\mathbf{z}) \approx F\_{(\zeta, \nu, \sigma)}(\mathbf{z}), v \to \infty \tag{12}$$

where *F*ð Þ *<sup>ζ</sup>*,*ν*,*<sup>σ</sup>* ð Þ*z* is the Generalised Pareto Distribution (GPD).
