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.Lomax, K. S. (1954) "Business Failures; Another example of the analysis of failure data". Journal of the American Statistical Association, 49, 847–852. J. Chen, J., Addie, R. G., Zukerman. M., Neame, T. D. (2015) "Performance Evaluation of a Queue Fed by a Poisson Lomax Burst Process", IEEE Communications Letters, 19, 3, 367–370. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.Van Hauwermeiren M and Vose D (2009). A Compendium of Distributions [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com.
Characterization Probability density function The probability density function (pdf) for the Lomax distribution is given by
with shape parameter and scale parameter . The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:
Non-central moments The th non-central moment exists only if the shape parameter strictly exceeds , when the moment has the value
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:
The Lomax distribution is a Pareto Type II distribution with xm = λ and μ = 0:.
Relation to the generalized Pareto distribution The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:
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 .
Relation to the F distribution The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density , 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:
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 and , then
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)
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