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.[1] [2] [3] It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.[4]
The probability density function (pdf) for the Lomax distribution is given by
p(x)={\alpha\overλ}\left[{1+{x\overλ}}\right]-(\alpha+1), x\geq0,
with shape parameter
\alpha>0
λ>0
p(x)={{\alphaλ\alpha}\over{(x+λ)\alpha+1
The
\nu
E\left[X\nu\right]
\alpha
\nu
E\left(X\nu\right)=
| λ\nu\Gamma(\alpha-\nu)\Gamma(1+\nu) | |
| \Gamma(\alpha) |
The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:
IfY\simPareto(xm=λ,\alpha),thenY-xm\simLomax(\alpha,λ).
The Lomax distribution is a Pareto Type II distribution with xm=λ and μ=0:[5]
IfX\simLomax(\alpha,λ)thenX\simP(II)\left(xm=λ,\alpha,\mu=0\right).
The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:
\mu=0,~\xi={1\over\alpha},~\sigma={λ\over\alpha}.
The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then
| X | |
| λ |
\sim\beta\prime(1,\alpha)
The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density
f(x)=
| 1 | |
| (1+x)2 |
The Lomax distribution is a special case of the q-exponential distribution. The q-exponential extends this distribution to support on a bounded interval. The Lomax parameters are given by:
\alpha={{2-q}\over{q-1}},~λ={1\overλq(q-1)}.
The logarithm of a Lomax(shape = 1.0, scale = λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0.This implies that a Lomax(shape = 1.0, scale = λ)-distribution equals a log-logistic distribution with shape β = 1.0 and scale α = log(λ).
The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution.If λ|k,θ ~ Gamma(shape = k, scale = θ) and X|λ ~ Exponential(rate = λ) then the marginal distribution of X|k,θ is Lomax(shape = k, scale = 1/θ).Since the rate parameter may equivalently be reparameterized to a scale parameter, the Lomax distribution constitutes a scale mixture of exponentials (with the exponential scale parameter following an inverse-gamma distribution).