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

f(x) = \frac{α β (e^{α x}− 1) ^{β−1} e^{α x}}{( 1 + (e^{α x}− 1) ^{β}) ^{2}} x > 0,

for all α > 0 and for β > 0. The probability density function for two different parameter settings is illustrated below.

PSfrag replacements α = 5, β = 1

α = 1, β = 2
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{(e^{α x}− 1) ^{β}}{1 + (e^{α x}− 1) ^{β}} x > 0.

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{1}{1 + (e^{α x}− 1) ^{β}} x > 0.

The hazard function on the support of X is

h(x) = \frac{αβ (e^{α x}− 1) ^{β−1} e^{α x}}{1 + (e^{α x}− 1) ^{β}} x > 0.

The cumulative hazard function on the support of X is mathematically intractable.

H(x) = ln( 1 + (e^{α x}− 1) ^{β}) x > 0.

The inverse distribution function of X is

F^{−1} (u) = ln( ( \frac{u}{1 − u} ) ^{1/β} + 1) / α 0 < u < 1.

The median of X is

\frac{ln(2)}{α}.

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

APPL verification: The APPL statements

X := [[x -¿ alpha * beta * (exp(alpha * x) - 1) ˆ (beta - 1) * exp(alpha * x)/ (1 + (exp(alpha * x) - 1) ˆ beta) ˆ 2],[0, infinity], [”Continuous”, ”PDF”]];

CDF(X); SF(X); HF(X); CHF(X); IDF(X);

verify the cumulative distribution function, survivor function, hazard function, cumulative hazard function, and inverse distribution function.