This statistical function with ξ shape parameter = 0 is an exponential distribution about a mean of zero. However, the same statistical function with ξ shape parameter > 0, and at a location where the mean = the σ/ξ, is a Pareto distribution. One can see that the generalized Pareto distribution at a low ξ provide for more extreme value distribution - note that the ξ parameter can supported at < 0 in some instances.
Put another way, one can drive their location to 0 by setting a very high ξ. And this would therefore show that at the highest shape parameters, the generalized Pareto distribution would converge to a Pareto distribution. But how can one match the α shape parameter for the former distribution? One can do this by setting the first raw moments equal to one another. The mean of the generalized Pareto, though at a ξ of just less than 1, would be ∞. This would imply by the parametric qualities of a Pareto distribution, that the α shape parameter would be less than 1.
One can see this supported by the initial deviations from the zero average location value of the charts below, by observing the values of the standardized risk axis:
Put another way, one can drive their location to 0 by setting a very high ξ. And this would therefore show that at the highest shape parameters, the generalized Pareto distribution would converge to a Pareto distribution. But how can one match the α shape parameter for the former distribution? One can do this by setting the first raw moments equal to one another. The mean of the generalized Pareto, though at a ξ of just less than 1, would be ∞. This would imply by the parametric qualities of a Pareto distribution, that the α shape parameter would be less than 1.
One can see this supported by the initial deviations from the zero average location value of the charts below, by observing the values of the standardized risk axis:
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| PDF of generalized Pareto |
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| CDF of generalized Pareto |
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