keisoku

Generalized Pareto Distribution Percent Point

Find the percent point (quantile) of the generalized Pareto distribution for a probability p from the location μ, scale σ, and shape ξ using a closed form. Supports lower and upper tail input.

Input

Enter probability p, the input mode, location μ, scale σ, and shape ξ to compute the percent point (quantile) of the generalized Pareto distribution in closed form.

A value greater than 0 and less than 1 (e.g. 0.95)

Input mode

Choose whether p is treated as a lower or upper tail probability.

Lower bound or threshold of the distribution. Any real number.

A positive value that sets the spread of the tail.

Tail heaviness. Positive gives a heavy tail, 0 is exponential, negative gives a finite upper bound.

Result

Percent point x with lower probability 0.95

4.10282102

Location μ

0

Scale σ

1

Shape ξ

0.2

Lower probability F(x)

0.95

Upper probability 1 minus F(x)

0.05

Density f(x)

0.02746401

Mean

1.25

Variance

2.60416667

Support upper bound

Infinity

Probability density function f(x)

Cumulative distribution function F(x)

How it works

  • The generalized Pareto distribution is defined by a location μ, a positive scale σ, and a shape ξ. Its support is x at or above μ, and when ξ is negative it is the finite interval from μ up to μ minus σ divided by ξ.
  • The lower tail percent point has a closed form. When ξ is not zero, x equals μ plus σ times ((1−p) to the power of −ξ minus 1) divided by ξ. When ξ is zero, x equals μ minus σ times ln(1−p).
  • Choosing the upper tail returns the point whose upper probability equals p, which is the same as the percent point of lower probability 1−p.
  • The CDF is F(x) equals 1 minus (1 plus ξz) to the power of −1 divided by ξ when ξ is not zero, and 1 minus exp(−z) when ξ is zero, where z equals (x−μ) divided by σ.
  • The mean is μ plus σ divided by (1−ξ) when ξ is below 1, and the variance is σ squared divided by ((1−ξ) squared times (1−2ξ)) when ξ is below 0.5. Beyond those limits they diverge to infinity.
  • It is widely used in extreme value statistics to model exceedances over a threshold (the peaks over threshold method). It includes the exponential distribution when ξ is zero and Pareto type tails when ξ is positive.

Reviews

Tell us what you think of this calculator.

Write a review

  1. Home

  2. Generalized Pareto Distribution Percent Point