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.[1][2][3][4] It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.[5]

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)}, \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{\alpha\lambda^\alpha}{(x + \lambda)^{\alpha+1}}.

Non-central moments

The \nuth 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)}.

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

The Lomax distribution is a Pareto Type II distribution with xm = λ and μ = 0:[6]

\text{If } X \sim \operatorname{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 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 X \sim \operatorname{Exp}(1) and Y \sim \operatorname{Gamma}(\alpha,1), then X/Y \sim \operatorname{Lomax}(\alpha,1).[4]

See also

References

  1. ^ Lomax, K. S. (1954) "Business Failures; Another example of the analysis of failure data". Journal of the American Statistical Association, 49, 847–852. JSTOR 2281544
  2. ^ Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "20 Pareto distributions". Continuous univariate distributions. Vol. 1. 2nd ed. New York: Wiley. p. 573.
  3. ^ 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.
  4. ^ Angervuori, Ilari; Haenggi, Martin; Wichman, Risto (2025). "Meta Distribution of the SIR in a Narrow-Beam LEO Uplink". IEEE Transactions on Communications. 73 (9): Eq. (27)
  5. ^ Van Hauwermeiren M and Vose D (2009). A Compendium of Distributions [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com.
  6. ^ Kleiber, Christian & Kotz, Samuel (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Vol. 470, Wiley Series in Probability and Statistics, John Wiley & Sons, p. 60, ISBN 9780471457169.