*Tid:***5 februari 2007 kl 15.15-16.00 **

*Plats :***Seminarierummet 3733**, Institutionen för
matematik, KTH, Lindstedts väg 25, plan 7. Karta!

*Föredragshållare:***
Klara Persson
**

* Titel: *
The exponentiated Gumbel distribution: an alternative to the generalised extreme value in wave-length modeling. (Examensarbete)

* Sammanfattning: *
The Gumbel distribution is one of the statistical distributions mostly
applied in extreme value analysis of environmental data. A
generalisation of the Gumbel distribution was recently introduced
referred to as the exponentiated Gumbel distribution (EGD), with the
cumulative distribution function

F(x) = 1 - (1-exp(-e^{-(x-μ)/σ)})^{α}
where σ>0 and α>0

and its use was exemplified by modeling rainfall. The aim of this study was to investigate whether the new distribution is a suitable model for extreme wave heights and compare its fitness with earlier used models, such as the Gumbel distribution and the generalised extreme value distribution (GEV).

The data analysed were obtained from two buoys in the north-east Pacific. Each sample contained yearly measures of the average of the wave heights in the highest one-third in a series of waves during a period of 21 years. The plots of the EGD and the empirical distribution showed that the model fitted data well. The quantile-quantile (QQ) plots confirmed this observation.

Simulations from a known EGD with different sample sizes showed that the exponential factor is very volatile. Even with a sample size of 500 the estimate of α greatly differed from the theoretical value. This difference did not affect the value of the 100-year return wave at the same extent. One explanation is that the shape of the tail in the cumulative distribution is more dependent on how large the estimates of σ and μ are and less affected by the size of α. The estimates of σ and μ were very close to their theoretical values in every sample. Hence, the tail of the distribution was stable. In conclusion, the EGD is a possible model for extreme wave heights. But it has a few disadvantages. The parameter α is very much affected of the dataset used and it is hard to get a feeling for which values of the parameters that are reasonable. The confidence intervals for the estimates are extremely large, especially for α. The model is also very complicated and, the parameters are not easily estimated. Even if the estimate of α is very volatile, the estimates of the extreme values, such as the 100-year return wave are very accurate. This indicates that the distribution is suitable for calculating extreme events and could be a complement to the generalised extreme value distribution in modeling wave heights.