Article/Chapter Title :
Statistical Analysis of the Gumbel Model
Statistical Theory of Extremes
93-120
Tolerance intervals , Test of hypotheses , BLIE , ME , MLE , Statistical inference
It is assumed that the samples of maxima come from a Gumbel distribution. Gumbel characteristics are studied: moment generating and characteristic functions, mean value, variance, skewness, kurtosis, quantiles and mode. Return periods and return levels are considered. Statistical inference for location/scale parameters: Maximum Likelihood Estimators (MLE), Moments Estimators (ME), Best Linear Unbiased Estimators (BLUE), Best Linear Invariant Estimators (BLIE), confidence interval, tests of hypotheses, point and interval prediction, tolerance intervals and overpassing probability of some threshold. A worked example on maximal yearly precipitations is presented.
Gumbel distribution plays a central role in extreme value theory, being sometimes called the extreme value distribution.
The return period T(a), for overpassing the level (or threshold) “a” means that on average T(a) trials will be made until the maximum overpasses “a” , after the last overpassing.
In a sense, with “a” the largest flood in 100 years, or the 100-years flood, on average, we expect to have to wait 100 years to observe a maximum yearly flood greater than or equal to “a”.