Generalized Pareto Distribution

In statistics the Generalized Pareto Distribution (GPD) is a set of continuous probability distributions. It is often used for modeling of other distributions tails.

Assume that the random variable X is distributed by the rule, described by the function:

For x ≥ μ when ξ ≥ 0 and μ ≤ x ≤ μ - σ / ξ when ξ < 0, where:

. Layout parameter.
σ > 0. Scale parameter.
. Form parameter.

Then random value X has the Generalized Pareto Distribution or .

Distribution density function:

If μ ≤ x ≤ μ - σ / ξ when ξ < 0.

Distribution Characteristics

Mean:

Variance:

Median:

Generalized Pareto distribution does not have skewness and kurtosis coefficients.

Distribution Parameters Estimation

Maximum likelihood estimation

Log-likelihood function for GPD:

For x ≥ μ when ξ ≥ 0 and μ ≤ x ≤ μ - σ / ξ when ξ < 0.

To find optimal parameters estimation, maximize ln L(ξ, μ, σ).