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

f(x) = \frac{λκ(λx)^{κ−1}}{(1 + (λx)^{κ})^{2}} x > 0

for λ > 0, κ > 0. The log logistic distribution can be used to model the lifetime of an object, the lifetime of a organism, or a service time. The probability density function with three different parameter settings is illustrated below.

PSfrag replacements λ = 1,κ = 2

λ = 5,κ = 1
λ = 1,κ = 5
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{(λx)^{κ}}{1 + (λx)^{κ}} x > 0.

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{1}{1 + (λx)^{κ}} x > 0.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{λκ(λx)^{κ−1}}{1 + (λx)^{κ}} x > 0.

The cumulative hazard function on the support of X is

H(x) = −ln(S(x)) = ln[1 + (λx)^{κ}] x > 0.

The inverse distribution function of X is

F^{−1} (u) = \frac{1}{λ} ( \frac{u}{1 − u} ) ^{1/κ} 0 < u < 1.

The median of X is

\frac{1}{λ}.

The moment generating function of X is

M(t) = E [e^{tX}] = ∫_{0}^{∞} \frac{e^{tx}λ^{κ}κx^{κ−1}}{(1 + (λx)^{κ})^{2}}dx t > 0.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = ∫_{0}^{∞} \frac{e^{itx}λ^{κ}κx^{κ−1}}{(1 + (λx)^{κ})^{2}}dx t > 0.

The population mean and variance are

E[X] = \frac{π}{κλ(sin( \frac{π}{κ}))} V [X] = \frac{π( 2κ( 1 − cos( \frac{π}{κ})^{2}) + πsin( \frac{π(κ+2)}{κ} ) )}{( sin( \frac{π(κ+2)}{κ} ) ) ( (cos( \frac{π}{κ}))^{2} − 1) (λκ)^{2}}

APPL verification: The APPL statements

X := LogLogisticRV(lambda, kappa); CDF(X); SF(X); HF(X); CHF(X); IDF(X); MGF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the cumulative distribution function, survivor function, hazard function, cumulative hazard function, inverse, moment generating function, population mean, variance, skewness, and kurtosis.