The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution often used in business, economics, and actuarial 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.
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:
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 th non-central moment exists only if the shape parameter strictly exceeds , when the moment has the value
- 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) Continuous Univariate Distributions, Volume 1, 2nd Edition, Wiley. ISBN 0-471-58495-9 (pages 575, 602)
- 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.