Generalized Pareto Distribution Calculator

Enter location mu, scale sigma, and shape xi to compute the probability density, cumulative distribution (lower probability), upper tail probability, mean, and variance of the generalized Pareto distribution, with PDF and CDF charts. Useful for extreme value theory.

Input

Enter location mu, scale sigma, and shape xi to compute the probability density, cumulative distribution (lower probability), upper tail probability, mean, and variance at the point x.

x (evaluation point)

The value at which to evaluate the probability

Location mu

Lower endpoint of the distribution (threshold)

Scale sigma

Enter a positive value

Shape xi

xi equal to 0 gives the exponential distribution; negative is bounded; positive has a heavy tail

Result

Lower probability F(x) at x equal to 2

0.81406557

Upper tail probability Q(x)

0.18593443

Probability density f(x)

0.13281031

Mean

1.25

Variance

2.60416667

Probability density function (PDF)

Cumulative distribution function (CDF)

How it works

The generalized Pareto distribution is defined by three parameters: location mu, scale sigma (positive), and shape xi. Let z equal (x minus mu) divided by sigma.
When the shape xi is not zero, the cumulative distribution is F(x) equals 1 minus (1 plus xi times z) raised to the power minus 1 over xi, and the density is f(x) equals (1 over sigma) times (1 plus xi times z) raised to the power minus 1 over xi minus 1.
When the shape xi is zero, the distribution reduces to the exponential distribution: F(x) equals 1 minus e to the minus z, and f(x) equals (1 over sigma) times e to the minus z.
The support is x at least mu when xi is zero or positive, and mu up to mu minus sigma over xi when xi is negative. Outside the support the density is zero.
The upper tail probability is Q(x) equals 1 minus F(x), representing the chance of exceeding a threshold.
The mean is mu plus sigma over (1 minus xi) when xi is below 1, and the variance is sigma squared over ((1 minus xi) squared times (1 minus 2 xi)) when xi is below one half. Otherwise these quantities diverge.
In extreme value theory, exceedances over a high threshold are approximately generalized Pareto distributed, which is the basis of the peaks-over-threshold method.

Reviews

Tell us what you think of this calculator.