The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling.

It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.

Characterization

Probability density function

The probability density function (pdf) for the Lomax distribution is given by

\[{\displaystyle p(x)={\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)},\qquad x\geq 0,}\] with shape parameter \({\displaystyle \alpha >0}\) and scale parameter \({\displaystyle \lambda >0}\).

shape <- 5; scale <- 10
xl <- qlomax(c(0.00, 0.99), scale = scale, shape = shape)
x <- seq(from = xl[1], to = xl[2], length.out = 200)
f <- dlomax(x, scale = scale, shape = shape)
plot(x, f, type = "l", main = "Lomax density")

The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:

\[{\displaystyle p(x)={{\alpha \lambda ^{\alpha }} \over {(x+\lambda )^{\alpha +1}}}}.\]

F <- plomax(x, scale = scale, shape = shape)
plot(x, F, type ="l", main ="Lomax distribution function")

Non-central moments

The \({\displaystyle \nu }\) th non-central moment \({\displaystyle E[X^{\nu }]}\) exists only if the shape parameter \({\displaystyle \alpha }\) strictly exceeds \({\displaystyle \nu }\) , when the moment has the value

\[{\displaystyle E(X^{\nu })={\frac {\lambda ^{\nu }\Gamma (\alpha -\nu )\Gamma (1+\nu )}{\Gamma (\alpha )}}}\]

Random Numbers

Lomax <- data.frame(
  shape12 = rlomax(1000, scale=5,shape=12),
  shape15 = rlomax(1000, scale=5,shape=15),
  shape10 = rlomax(1000, scale=5,shape=10)
)
ggplot() + geom_density(data = Lomax, aes(shape12),colour = "black") + 
         geom_density(data = Lomax, aes(shape15),colour = "blue") + 
         geom_density(data = Lomax, aes(shape10),colour = "red") + 
         theme_classic() + 
         xlab("Lomax Distribution")