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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]

Lomax
Probability density function
PDF of the Lomax distribution
Cumulative distribution function
Lomax distribution CDF plot
Parameters shape (real)
scale (real)
Support
PDF
CDF
Mean
Otherwise undefined
Median
Mode0
Variance

Otherwise undefined
Skewness
Ex. kurtosis

Contents

CharacterizationEdit

Probability density functionEdit

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:

 .

Non-central momentsEdit

The  th non-central moment   exists only if the shape parameter   strictly exceeds  , when the moment has the value

 

Related distributionsEdit

Relation to the Pareto distributionEdit

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:[5]

 

Relation to the generalized Pareto distributionEdit

The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:

 

Relation to the beta prime distributionEdit

The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then  .

Relation to the q-exponential distributionEdit

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:

 

Relation to the (log-) logistic distributionEdit

The logarithm of a Lomax(shape=1.0, scale=λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0. This implies that a Lomax(shape=1.0, scale=λ)-distribution equals a log-logistic distribution with shape β=1.0 and scale α=log(λ).

Gamma-exponential mixture connectionEdit

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).


See alsoEdit

ReferencesEdit

  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. 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. ^ Van Hauwermeiren M and Vose D (2009). A Compendium of Distributions [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com.
  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.