Error distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ error(a,b,c) is used to indicate that the random variable X has the error distribution with location parameter a, scale parameter b, and shape parameter c. An error random variable X with parameters a, b, and c has probability density function

f(x) = \frac{e^{(−|x−a|/b) ^{2/c}/2}}{b(2^{c/2+1} )Γ(c/2 + 1)} − ∞ < x < ∞

for all real values a and for b > 0, c > 0. The probability density function with three different parameterizations is illustrated below.

PSfrag replacements a = −1, b = 1, c = 0.25 a = −1, b = 0.5, c = 1

a = 0, b = 1, c = 2

x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = ∫_{−∞}^{x} \frac{e^{(−|t−a|/b)^{2/c}/2}}{b(2^{c/2+1} )Γ(c/2 + 1)} dt − ∞ < x < ∞.

The survivor function, hazard function, inverse distribution function, moment generating function, and characteristic functions are all mathematically intractable. The population median and mode of X occurs at x = a. The population mean, variance, skewness, and kurtosis of X are

E[X] = a V [X] = \frac{2^{c}b^{2}Γ(3c/2)}{Γ(c/2)}.

E [( \frac{X − μ}{σ} ) ^{3}] = 0 E [( \frac{X − μ}{σ} ) ^{4}] = \frac{Γ(5c/2)Γ(c/2)}{(Γ(3c/2))^{2}} .

APPL verification: The APPL statements

assume(b ¿ 0); assume(c ¿ 0); X := [[x -¿ exp((-abs(x - a) / b) ˆ (2 / c) / 2) / (b * (2 ˆ (c / 2 + 1)) *

GAMMA(c / 2 + 1))], [-infinity, infinity], [”Continuous”, ”PDF”]]; CDF(X); SF(X); HF(X); CHF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

fail due to integration problems.