Application
Gumbel has shown that the maximum value (or last order statistic) in a sample of a random variable following an exponential distribution approaches the Gumbel distribution closer with increasing sample size.
In hydrology, therefore, the Gumbel distribution is used to analyze such variables as monthly and annual maximum values of daily rainfall and river discharge volumes, and also to describe droughts.
Gumbel has also shown that the estimator r / (n+1) for the probability of an event - where r is the rank number of the observed value in the data series and n is the total number of observations - is an unbiased estimator of the cumulative probability around the mode of the distribution. Therefore, this estimator is often used as a plotting position.
The blue picture illustrates an example of fitting the Gumbel distribution to ranked maximum one-day October rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by the plotting position r / (n+1) as part of the cumulative frequency analysis.
Read more about this topic: Gumbel Distribution
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