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

f(x) = \frac{e^{βx}e^{−e^{x}/α}}{α^{β}Γ(β)} − ∞ < x < ∞.

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

PSfrag replacements α = 1, β = 1

α = 5, β = 1
α = 1, β = 2
x
f(x)

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

APPL failure: The APPL statements

X := [[exp(beta * x) * exp(-exp(x) / alpha) / (alpha ˆ beta * GAMMA(beta))],

[-infinity,infinity],[”Continuous”, ”PDF”]]; Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

fails to return the population mean, variance, skewness, kurtosis, and moment generating function.