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Gumbel Distribution The Gumbel distribution is a special case of the Generalized Extreme Value distribution. It used in industry in QA/QC applications and in the environmental sciences to model extreme values associated with flooding and rainfall. The Gumbel distribution takes its name from Emil J. Gumbel. An attractive feature of the Gumbel distribution is that the parameter equations provide an estimate of the mode. Profile 
Parameters
Range and Tails The theoretical range is from -∞ to +∞, however in practice it is classed as a thin/light tailed distribution. As a form of comparison, the Pareto and Cauchy distributions are classed as having thick/heavy tails and the Beta distribution is bounded by its maximum and minimum values. Functions Properties The profile of the probability density function is independent of the mode and scale factor, thus skewness and kurtosis are constants. Example The example shows the distribution of the largest monthly rainfall over a period of 291 years at Kew Gardens. This type of model might be used for modelling the flow of rivers, the effectiveness of flood defences and systems which are designed on the basis of extreme values, rather than modal ones. It would be interesting to see if the maximum range of trading values in financial markets produced a statistically significant fit. Parameter Estimation The matching moment equations for the mode and scale factor are: Random Number Generation Random number generation (referred to as r) for a Gumbel distribution can be performed by transforming a continuous uniform variable in the range 0 to 1 (referred to as u) with the distribution's inverse probability function:
Using Basic style code, the function would be similar to:
Page updated: 05-May-2005 |
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