Fit Distribution (EVA Theory)

EVA in IMS fits a Gumbel distribution to the data. This a special form of the GEV distribution with the shape parameter taken to be 0. The Gumbel distribution is preferred to the generalized GEV as it gives narrower confidence bounds (one parameter less to be estimated).

The standardized GEV distribution is given by:

where:

• μ is the location parameter;
• σ is the scale parameter; and
• ξ is the shape parameter (ξ = 0 for Gumbel).

The following plots show the PDF (Probability Density Function) of a Gumbel distribution for different values of the location and scale parameters. The location parameter influences the center of the support of the distribution while the scale parameter influences the spread of the distribution. The location parameter is comparable to the normal distribution’s mean, while the scale parameter is comparable to the standard deviation.

PDF of the Gumbel distribution with different scale parameters.

 

PDF of the Gumbel distribution with different location parameters.

The parameter estimates  of the Gumbel distribution are obtained by maximizing the log likelihood function:

Here x_i (1…m) is the observed data. There is no analytical solution for this optimization problem but with numerical methods the solution can be provided.