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{{Probability distribution | |||
| name =Lomax | |||
| type =density | |||
| pdf_image = | |||
| cdf_image = | |||
| parameters = | |||
<math>\lambda >0 </math> [[scale parameter|scale]] (real)<br /> | |||
<math>\alpha > 0 </math> [[shape parameter|shape]] (real) | |||
| support =<math> x \ge 0 </math> | |||
| pdf =<math> {\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}</math> | |||
| cdf =<math> 1- \left[{1+ {x \over \lambda}}\right]^{-\alpha}</math> | |||
| mean =<math> {\lambda \over {\alpha -1}} \text{ for } \alpha > 1</math><br /> Otherwise undefined | |||
| median =<math>\lambda (\sqrt[\alpha]{2} - 1)</math> | |||
| mode = 0 | |||
| variance =<math> {{\lambda^2 \alpha} \over {(\alpha-1)^2(\alpha-2)}} \text{ for } \alpha > 2 </math><br /><math> \infty \text{ for } 1 < \alpha \le 2 </math> <br /> Otherwise undefined | |||
| skewness =<math>\frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3\,</math> | |||
| kurtosis =<math>\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4\,</math> | |||
| entropy = | |||
| mgf = | |||
| char = | |||
}} | |||
The '''Lomax distribution''', conditionally also called the '''[[Pareto_distribution#Pareto types I–IV|Pareto Type II distribution]]''', is a [[heavy tail|heavy-tail]] [[probability distribution]] often used in business, economics, and actuarial modeling.<ref>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}}</ref><ref>Johnson, N.L., Kotz, S., Balakrishnan, N. (1994) ''Continuous Univariate Distributions, Volume 1'', 2nd Edition, Wiley. ISBN 0-471-58495-9 (pages 575, 602)</ref> It is named after K. S. Lomax. It is essentially a [[Pareto distribution]] that has been shifted so that its support begins at zero.<ref>Van Hauwermeiren M and Vose D (2009). [http://www.vosesoftware.com/content/ebook.pdf '' A Compendium of Distributions''] [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com. Accessed 07/07/11</ref> | |||
The | |||
== Characterization == | |||
=== Probability density function === | |||
The [[probability density function]] (pdf) for the Lomax distribution is given by | |||
:<math> p(x) = {\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}, \qquad x \geq 0, | |||
</math> | |||
with shape parameter <math>\alpha>0</math> and scale parameter <math>\lambda>0</math>. The density can be rewritten in such a way that more clearly shows the relation to the [[Pareto distribution|Pareto Type I distribution]]. That is: | |||
:<math> p(x) = {{\alpha \lambda^\alpha} \over { (x+\lambda)^{\alpha+1}}}</math>. | |||
== | ===Differential equation=== | ||
The pdf of the Lomax distribution is a solution to the following [[differential equation]]: | |||
:<math>\left\{\begin{array}{l} | |||
:<math>\ | (\gamma +x) p'(x)+(\alpha +1) p(x)=0, \\ | ||
p(0)=\frac{\alpha}{\gamma} | |||
\end{array}\right\} | |||
</math> | </math> | ||
== Relation to the Pareto distribution == | |||
The Lomax distribution is a [[Pareto distribution|Pareto Type I distribution]] shifted so that its support begins at zero. Specifically: | |||
:<math>\text{If } Y \sim \mbox{Pareto}(x_m = \lambda, \alpha), \text{ then } Y - x_m \sim \mbox{Lomax}(\lambda,\alpha).</math> | |||
The Lomax distribution is a [[Pareto_distribution#Pareto types I–IV|Pareto Type II distribution]] with ''x''<sub>m</sub>=λ and μ=0:{{cn|date=October 2012}} | |||
:<math> | |||
\text{If } X \sim \mbox{Lomax}(\lambda,\alpha) \text{ then } X \sim \text{P(II)}(x_m = \lambda, \alpha, \mu=0).</math> | |||
== Relation to generalized Pareto distribution == | |||
The Lomax distribution is a special case of the [[generalized Pareto distribution]]. Specifically: | |||
== | :<math> \mu = 0,~ \xi = {1 \over \alpha},~ \sigma = {\lambda \over \alpha} .</math> | ||
== | == 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: | |||
== | :<math> \alpha = { {2-q} \over {q-1}}, ~ \lambda = {1 \over \lambda_q (q-1)} .</math> | ||
== | == Non-central moments == | ||
The <math>\nu</math>th non-central moment <math>E[X^\nu]</math> exists only if the shape parameter <math>\alpha</math> strictly exceeds <math>\nu</math>, when the moment has the value | |||
:<math> E(X^\nu) = \frac{ \lambda^\nu \Gamma(\alpha-\nu)\Gamma(1+\nu)}{\Gamma(\alpha)}</math> | |||
}} | |||
==See also== | == See also == | ||
*[[ | *[[Power law]] | ||
==References== | |||
<references /> | |||
[[Category: | {{ProbDistributions|continuous-semi-infinite}} | ||
[[Category: | [[Category:Continuous distributions]] | ||
[[Category: | [[Category:Probability distributions with non-finite variance]] | ||
[[Category:Probability distributions]] | |||
Revision as of 15:00, 18 August 2014
Template:Probability distribution
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.[1][2] It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.[3]
Characterization
Probability density function
The probability density function (pdf) for the Lomax distribution is given by
![{\displaystyle p(x)={\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)},\qquad x\geq 0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe53751e161afafd689924287b9e24be2fcb8d2c)
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:
- .
Differential equation
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:
The Lomax distribution is a Pareto Type II distribution with xm=λ and μ=0:Template:Cn
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:
Non-central moments
The th non-central moment exists only if the shape parameter strictly exceeds , when the moment has the value
![{\displaystyle E[X^{\nu }]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d65f8be1ba81337fe605abe924088385e2cf49d5)
See also
References
- ↑ Lomax, K. S. (1954) "Business Failures; Another example of the analysis of failure data". Journal of the American Statistical Association, 49, 847–852. Template:Jstor
- ↑ 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
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