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
:$p(x) = \frac\alpha\lambda \left(1 + \frac x\lambda \right)^, \qquad x \geq 0,$

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:
:$p(x) = \frac.$

Non-central moments

The $\nu$th non-central moment E\left ^\nu\right/math> exists only if the shape parameter $\alpha$ strictly exceeds $\nu$, when the moment has the value
:$E\left(X^\nu\right) = \frac.$

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:
:$\text Y \sim \operatorname(x_m = \lambda, \alpha), \text Y - x_m \sim \operatorname(\alpha,\lambda).$

The Lomax distribution is a Pareto Type II distribution with ''x''_''m'' = ''λ'' and ''μ'' = 0:.
:$\text X \sim \operatorname(\alpha, \lambda) \text X \sim \text\left(x_m = \lambda, \alpha, \mu = 0\right).$

Relation to the generalized Pareto distribution

The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:
:$\mu = 0,~ \xi = ,~ \sigma = .$

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 $\frac \sim \beta^\prime(1, \alpha)$.

Relation to the F distribution

The Lomax distribution with shape parameter ''α'' = 1 and scale parameter ''λ'' = 1 has density $f(x) = \frac$, 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:
:$\alpha = , ~ \lambda = .$

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).

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)

References

{{ProbDistributions, continuous-semi-infinite
Continuous distributions
Compound probability distributions
Probability distributions with non-finite variance