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

Characterization

Probability density function

The probability density function (pdf) for the Lomax distribution is given by

${\displaystyle p(x)={\frac {\alpha }{\lambda }}\left(1+{\frac {x}{\lambda }}\right)^{-(\alpha +1)},\qquad x\geq 0,}$

with shape parameter ${\displaystyle \alpha >0}$ and scale parameter ${\displaystyle \lambda >0}$. The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:

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

Non-central moments

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

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

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:

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

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

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

Relation to the generalized Pareto distribution

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

${\displaystyle \mu =0,~\xi ={1 \over \alpha },~\sigma ={\lambda \over \alpha }.}$

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

Relation to the F distribution

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

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

Relation to the 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.

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)