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

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

for all δ > 0, κ > 1, and γ > 0. The Makeham distribution is used to model adult lifetimes by actuaries. The probability density function with δ = 1, κ = 2, and γ = 1 is illustrated below.

PSfrag replacements δ = 1, κ = 2, γ = 1

x
f(x)

The cumulative distribution function on the support of X is

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

The survivor function on the support of X is

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

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = γ + δκ^{x} x > 0.

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

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

The inverse distribution function, 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

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

[0,infinity],[”Continuous”, ”PDF”]]; CDF(X); SF(X); HF(X); CHF(X);

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