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

f(x) = \frac{γx^{γβ−1} e^{−(x/α)^{γ}}}{α^{γβ}Γ(β)} x > 0.

The probability density function with three different parameter combinations is illustrated below.

PSfrag replacements α = 1, β = 2, γ = 1 α = 3, β = 2, γ = 1 α = 1, β = 1, γ = 3

x
f(x)

The cumulative distribution, survivor, hazard, cumulative hazard, inverse distribution, moment generating, and characteristic functions on the support of X are mathematically intractable. The population mean and variance of X are

E[X] = \frac{αΓ(β + 1/γ)}{Γ(β)} V [X] = α^{2}( \frac{Γ(β + 2/γ)}{Γ(β)} − ( \frac{Γ(β + 1/γ)}{Γ(β)} ) ^{2}) .

APPL verification: The APPL statements

assume(alpha ¿ 0); assume(beta ¿ 0); assume(y ¿ 0); X := [[x -¿ y * x ˆ (y * beta - 1) * exp(-(x / alpha) ˆ y) /

(alpha ˆ (y * beta) * GAMMA(beta))], [0, infinity], [”Continuous”, ”PDF”]];

Mean(X); Variance(X);

verify the population mean and variance. APPL also calculates the skewness and kurtosis, but the expressions are long.