Lomax distribution

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

$p(x)=\frac{\alpha }{\lambda }{(1+\frac{x}{\lambda })}^{-(\alpha +1)},x\ge 0,$

with shape parameter $\alpha >0$ and scale parameter $\lambda >0$. The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:

$p(x)=\frac{\alpha {\lambda }^{\alpha }}{(x+\lambda {)}^{\alpha +1}}.$

Non-central moments

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

$E\left(X^{\nu }\right)=\frac{{\lambda }^{\nu }\mathrm{\Gamma }(\alpha -\nu )\mathrm{\Gamma }(1+\nu )}{\mathrm{\Gamma }(\alpha )}.$

Related distributions

Relation to the Pareto distribution

The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:

$\text{If }Y\sim \mathrm{Pareto}(x_m=\lambda ,\alpha ),\text{ then }Y-x_m\sim \mathrm{Lomax}(\alpha ,\lambda ).$

The Lomax distribution is a Pareto Type II distribution with x_m = λ and μ = 0:

$\text{If }X\sim \mathrm{Lomax}(\alpha ,\lambda )\text{ then }X\sim \text{P(II)}(x_m=\lambda ,\alpha ,\mu =0).$

Relation to the generalized Pareto distribution

The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:

$\mu =0,\xi =\frac{1}{\alpha },\sigma =\frac{\lambda }{\alpha }.$

Relation to the beta prime distribution

The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then $\frac{X}{\lambda }\sim {\beta }^{\prime }(1,\alpha )$.

Relation to the F distribution

The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density $f(x)=\frac{1}{(1+x{)}^2}$, the same distribution as an F(2,2) distribution. This is the distribution of the ratio of two independent and identically distributed random variables with exponential distributions.

Relation to the q-exponential distribution

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:

$\alpha =\frac{2-q}{q-1},\lambda =\frac{1}{{\lambda }_q(q-1)}.$

Relation to the logistic distribution

The logarithm of a Lomax(shape = 1.0, scale = λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0.

Gamma-exponential (scale-) mixture connection

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).

Relation to the gamma distribution

Let $X\sim \mathrm{Exp}(1)$ and $Y\sim \mathrm{Gamma}(\alpha ,1)$, then $X/Y\sim \mathrm{Lomax}(\alpha ,1).$

See also

• Power law
• Compound probability distribution
• Hyperexponential distribution (finite mixture of exponentials)
• Normal-exponential-gamma distribution (a normal scale mixture with Lomax mixing distribution)