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Extreme Value Theory

4. Block maxima results – examples of different limiting behaviour

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The limiting behaviours of many traditional distributions are well-known. For example, block maxima values for the normal, log-normal and exponential distributions converge to the Gumbel distribution. Those for the Student’s t, inverse gamma, F and Pareto distributions converge to the Fréchet distribution.

 

Two examples where it is easy to derive the limiting distributions are the exponential and the Pareto distribution:

 

Exponential

 

Suppose the underlying distribution is an exponential distribution with distribution function  for  and . Then by choosing normalising sequences  and  we have:

 

 

So .

 

Pareto

 

Suppose the underlying distribution is a Pareto distribution  with distribution function   for  and . Then by choosing the normalising sequences  and  we have:

 

 

So .

 


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