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]

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

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

with shape parameter ${\displaystyle \alpha >0}${\displaystyle \alpha >0} and scale parameter ${\displaystyle \lambda >0}${\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:

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

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

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

The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:

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

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

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

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

The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then ${\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}${\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}.

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

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:

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

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

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).

• Power law
• Compound probability distribution
• Hyperexponential distribution (finite mixture of exponentials)
• Normal-exponential-gamma distribution (a normal scale mixture with Lomax mixing distribution)

• Continuous distributions
• Compound probability distributions
• Probability distributions with non-finite variance

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