Lomax distribution

Lomax
Parameters	$\lambda >0$ scale (real) $\alpha >0$ shape (real)
Support	$x\geq 0$
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	${{\lambda ^{2}\alpha } \over {(\alpha -1)^{2}(\alpha -2)}}{\text{ for }}\alpha >2$ $\infty {\text{ for }}1<\alpha \leq 2$ 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\}

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 )}}

Related distributions

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)}.

Gamma-exponential mixture

The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution. If λ ~ Gamma(scale=k, shape=θ) and X ~ Exponential(rate=λ) then the marginal distribution of X is Lomax(scale=1/k, shape=θ).

References

↑ 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)
↑ 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.
↑ 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 .

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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