Exponential power distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ exponential power(λ, κ) is used to indicate that the random variable X has the exponential power distribution with positive scale parameter λ and positive shape parameter κ. An exponential power random variable X has probability density function

f(x) = ( e^{1−e^{λ x^{κ}}} ) e^{λx^{κ}}λκx^{κ−1} x > 0.

The exponential power distribution is one of the few two-parameter distributions that can achieve a bathtub-shaped hazard function. The probability density function for three different parameter settings is illustrated below.

PSfrag replacements λ = 1,κ = 1

λ = 2,κ = 1
λ = 1,κ = 2
x
f(x)

The cumulative distribution function on the support of X is

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

The survivor function on the support of X is

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

The hazard function on the support of X is

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

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = e^{λx^{κ}} − 1 x > 0.

The inverse distribution function of X is

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

The median of X is

F^{−1} (1/2) = (λln(1 − ln(1/2)))^{−1/κ} .

The moment generating function of X is

M(t) = E [e^{tX}] = ∫_{0}^{∞} λκx^{κ−1} e^{tx+1+λx^{κ}−e^{(λx^{κ})}} dx.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = ∫_{0}^{∞} λκx^{κ−1} e^{itx+1+λx^{κ}−e^{(λx^{κ})}} dx.

The population mean, variance, skewness, and kurtosis of X are mathematically intractable.

APPL verification: The APPL statements

X := ExponentialPowerRV(lambda, kappa); CDF(X); SF(X); HF(X); IDF(X); MGF(X);

verify the cumulative distribution function, hazard function, inverse distribution function, and moment generating function.