Lomax distribution

Lomax

Probability density function
Cumulative distribution function
Parameters	$\alpha >0$ shape (real) $\lambda >0$ scale (real)
Support	$x\geq 0$
PDF	${\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)}$
CDF	$1-\left[{1+{x \over \lambda }}\right]^{-\alpha }$
Quantile	$\lambda \left[\left(1-p\right)^{-{\frac {1}{\alpha }}}-1\right]$
Mean	${\lambda \over {\alpha -1}}{\text{ for }}\alpha >1$; undefined otherwise
Median	$\lambda \left({\sqrt[{\alpha }]{2}}-1\right)$
Mode	0
Variance	${\begin{cases}{{\lambda ^{2}\alpha } \over {(\alpha -1)^{2}(\alpha -2)}}&\alpha >2\\\infty &1<\alpha \leq 2\\{\text{Undefined}}&{\text{otherwise}}\end{cases}}$
Skewness	${\frac {2(1+\alpha )}{\alpha -3}}\,{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3\,$
Excess kurtosis	${\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4\,$
Entropy	$1+{\frac {1}{\alpha }}-\log {\frac {\alpha }{\beta }}$
MGF	$\alpha e^{-\lambda t}(-\lambda t)^{\alpha }\Gamma (-\alpha ,-\lambda t)\,$
CF	$\alpha e^{-i\lambda t}(-i\lambda t)^{\alpha }\Gamma (-\alpha ,-i\lambda t)\,$

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

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:

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

Non-central moments

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

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

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:

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

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

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

Relation to the 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 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 ${\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )$.

Relation to the F distribution

The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density $f(x)={\frac {1}{(1+x)^{2}}}$, 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:

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

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 (scale-) 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). Since the rate parameter may equivalently be reparameterized to a scale parameter, the Lomax distribution constitutes a scale mixture of exponentials (with the exponential scale parameter following an inverse-gamma distribution).

See also

• power law
• compound probability distribution
• hyperexponential distribution (finite mixture of exponentials)
• normal-exponential-gamma distribution (a normal scale mixture with Lomax mixing distribution)