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.
- 1 Characterization
- 2 Related distributions
- 3 See also
- 4 References
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:
The pdf of the Lomax distribution is a solution to the following differential equation:
The th non-central moment exists only if the shape parameter strictly exceeds , when the moment has the value
Relation to the Pareto distribution
The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:
Relation to generalized Pareto distribution
The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:
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:
The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution. If λ ~ Gamma(scale=k, shape=θ) and X ~ Exponential(rate=λ) then the marginal distribution of X is Lomax(scale=1/k, shape=θ).
- 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
- Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "20 Pareto distributions". Continuous univariate distributions. 1 (2nd ed.). New York: Wiley. p. 573.
- 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.
- Van Hauwermeiren M and Vose D (2009). A Compendium of Distributions [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com. Accessed 07/07/11
- Kleiber, Christian; Kotz, Samuel (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley Series in Probability and Statistics, 470, John Wiley & Sons, p. 60, ISBN 9780471457169.