Generalized Pareto distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ generalized Pareto(δ,κ,γ) is used to indicate that the random variable X has the generalized Pareto distribution with shape parameters δ,κ and γ. A generalized Pareto random variable X has probability density function

f(x) = ( γ + \frac{κ}{x + δ} ) (1 + x/δ)^{−κ} e^{−γx} δ > 0; γ ≥ 0; κ ≥ −δγ

for x > 0. The probability density function with two different parameter combinations is illustrated below.

PSfrag replacements δ = 1, γ = 1, κ = 1 δ = 10, γ = .5, κ = 1

x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = 1 − e^{−γx} (1 + x/δ)^{−κ} x > 0.

The survivor function on the support of X is

S(x) = P(X ≥ x) = e^{−γx} (1 + x/δ)^{−κ} x > 0.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = γ + \frac{κ}{x + δ} x > 0.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = γx + κln(1 + x/δ) x > 0.

The moment generating function and characteristic function of X, as well as the population mean, variance, skewness and kurtosis of X, are mathematically intractable.