Lomax distribution
# 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

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

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:

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

### 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

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

## 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:

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

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

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

### Relation to the 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 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:

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

### Relation to the (log-) 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. 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).

### 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).

