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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

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

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:

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

Differential equation

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

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

Relation to the Pareto distribution

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

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

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

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

Relation to generalized Pareto distribution

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

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

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:

$\alpha ={{2-q} \over {q-1}},~\lambda ={1 \over \lambda _{q}(q-1)}.$

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

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

See also

• Power law

References

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