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

f(x) = δκ^{x}e^{−δ(κ^{x}−1) /ln(κ)} x > 0,

for all δ > 0 and κ > 1. The Gompertz distribution is used to model adult lifetimes by actuaries. The probability density function for three parameter combinations is illustrated below.

PSfrag replacements δ = 1,κ = 2

δ = 0.5,κ = 3
δ = 0.1,κ = 2
x
f(x)

The cumulative distribution function on the support of X is

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

The survivor function on the support of X is

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

The hazard function on the support of X is

h(x) = δκ^{x} x > 0.

The cumulative hazard function on the support of X is

H(x) = \frac{δ (κ^{x}− 1)}{ln(κ)} x > 0.

The inverse distribution function of X is

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

The median of X is

−\frac{ln(δ)− ln(δ + ln(2)ln(κ))}{ln(κ)} .

The moment generating function and the characteristic function of X cannot be expressed in closed form. The population mean, variance, skewness, and kurtosis of X are mathematically intractable.

APPL verification: The APPL statements

assume(delta¿0); assume(kappa¿1); X := [[x -¿ delta * kappa ˆ x * exp(-delta * (kappa ˆ x-1) / ln(kappa))],

[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.