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The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution often used in business, economics, and actuarial 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

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

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:

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

Differential equation

The pdf of the Lomax distribution is a solution to the following differential equation:

$\left\{\begin{array}{l}(\gamma +x)p^{\prime }(x)+(\alpha +1)p(x)=0,\\ p(0)=\frac{\alpha }{\gamma }\end{array}\right\}$

Relation to the Pareto distribution

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

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

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

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

Relation to 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 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:

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

Non-central moments

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

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

See also

• Power law

References

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