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

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

for λ > 0 and κ > 0. The probability density function with two different parameterizations is illustrated below:

PSfrag replacements λ = 1, κ = 1

λ = 1, κ = 5
x
f(x)

Using the original parameterization, the cumulative distribution function on the support of X is

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

The survivor function on the support of X is

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

The hazard function on the support of X is

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

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = κ ln(1 + λx) x > 0.

The inverse distribution function of X is

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

The median of X is

\frac{2^{1/κ} − 1}{λ} .

The moment generating function of X is mathematically intractable. The population mean of X is

E[X] = \frac{λ}{κ − 1}

provided κ > 1. The population variance, skewness, and kurtosis of X are mathematically intractable. For X_{1}, X_{2}, ..., X_{n} mutually independent Lomax(λ) random variables, the maximum likelihood estimator for α is

α̂ = \frac{n}{∑ _{i=1}^{n} ln( 1 + x_{i}/λ̂)} .

An iteration procedure must be used to solve the following equation for λ̂, and then substitute in the previous to obtain α̂.

\frac{n}{λ̂( ∑ {n \atop i=1} \frac{x_{i}}{λ̂^{2}+λ̂x_{i}} )} − 1 = \frac{n}{∑ _{i=1}^{n} ln( 1 + \frac{x_{i}}{λ̂})}