Lomax distribution

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
Cumulative distribution function
Parameters

shape (real)

scale (real)
Support
PDF
CDF
Mean
Otherwise undefined
Median
Mode 0
Variance

Otherwise undefined
Skewness
Ex. kurtosis

Characterization

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:

.

Non-central moments

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:

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

Relation to the generalized Pareto distribution

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

Relation to the beta prime distribution

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 F distribution

The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density , the same distribution as an F(2,2) distribution. This is the distribution of the ratio of two independent and identically distributed random variables with exponential distributions.

Relation to the 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:

Relation to the (log-) logistic distribution

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 connection

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 also

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). "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.
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