The generalized Pareto distribution with location 
, scale 
, and shape 
has distribution function
![F(x)=1-[1+xi(x-mu)/sigma]^(-1/xi),](/images/equations/GeneralizedParetoDistribution/NumberedEquation1.svg) |
on the set where 
and 
. The limiting case 
is the exponential
distribution ![F(x)=1-exp[-(x-mu)/sigma]](/images/equations/GeneralizedParetoDistribution/Inline7.svg)
.
The generalized Pareto distribution arises in extreme-value theory as the limiting distribution of threshold exceedances under broad conditions. Positive, zero, and
negative 
give heavy-tailed, exponential-type, and bounded-upper-tail cases, respectively.
In the last case, the upper endpoint is 
.
