Lomax distribution

Lomax

Parameters	$\lambda >0$ scale (real) $\alpha > 0$ shape (real)
Support	$x \ge 0$
PDF	${\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}$
CDF	$1- \left[{1+ {x \over \lambda}}\right]^{-\alpha}$
Mean	${\lambda \over {\alpha -1}} \text{ for } \alpha > 1$ Otherwise undefined
Median	$\lambda (\sqrt[\alpha]{2} - 1)$
Mode	0
Variance	${{\lambda^2 \alpha} \over {(\alpha-1)^2(\alpha-2)}} \text{ for } \alpha > 2$ $\infty \text{ for } 1 < \alpha \le 2$ Otherwise undefined
Skewness	$\frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3\,$
Ex. kurtosis	$\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4\,$

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

Characterization

Probability density function

The probability density function (pdf) for the Lomax distribution is given by

$p(x) = {\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}, \qquad x \geq 0,$

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:

$p(x) = {{\alpha \lambda^\alpha} \over { (x+\lambda)^{\alpha+1}}}$.

Differential equation

The pdf of the Lomax distribution is a solution to the following differential equation:

$\left\{\begin{array}{l} (\lambda +x) p'(x)+(\alpha +1) p(x)=0, \\ p(0)=\frac{\alpha}{\lambda} \end{array}\right\}$

Relation to the Pareto distribution

The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:

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

The Lomax distribution is a Pareto Type II distribution with x_m=λ and μ=0:^[5]

$\text{If } X \sim \mbox{Lomax}(\lambda,\alpha) \text{ then } X \sim \text{P(II)}(x_m = \lambda, \alpha, \mu=0).$

Relation to 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 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:

$\alpha = { {2-q} \over {q-1}}, ~ \lambda = {1 \over \lambda_q (q-1)} .$

Non-central moments

The $\nu$th non-central moment $E[X^\nu]$ exists only if the shape parameter $\alpha$ strictly exceeds $\nu$, when the moment has the value

$E(X^\nu) = \frac{ \lambda^\nu \Gamma(\alpha-\nu)\Gamma(1+\nu)}{\Gamma(\alpha)}$

See also

• Power law

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) Continuous Univariate Distributions, Volume 1, 2nd Edition, Wiley. ISBN 0-471-58495-9 (pages 575, 602)
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. 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
5. 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.