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ⓘ Lomax distribution. The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business ..




Lomax distribution
                                     

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

                                     

1. Characterization

Probability density function

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

p x = α λ } exists only if the shape parameter α {\displaystyle \alpha } strictly exceeds ν {\displaystyle \nu }, when the moment has the value E X ν = λ ν Γ α − ν Γ 1 + ν Γ α {\displaystyle EX^{\nu }={\frac {\lambda ^{\nu }\Gamma \alpha -\nu\Gamma 1+\nu}{\Gamma \alpha}}}
                                     

2.1. 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:

If Y ∼ Pareto x m = λ, α, then Y − x m ∼ Lomax α, λ. {\displaystyle {\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:

If X ∼ Lomax α, λ then X ∼ PII x m = λ, α, μ = 0. {\displaystyle {\text{If }}X\sim {\mbox{Lomax}}\alpha,\lambda{\text{ then }}X\sim {\text{PII}}x_{m}=\lambda,\alpha,\mu =0.}
                                     

2.2. Related distributions Relation to the generalized Pareto distribution

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

μ = 0, ξ = 1 α, σ = λ α. {\displaystyle \mu =0,~\xi ={1 \over \alpha },~\sigma ={\lambda \over \alpha }.}
                                     

2.3. Related distributions 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 X λ ∼ β ′ 1, α {\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }1,\alpha}.

                                     

2.4. Related distributions Relation to the F distribution

The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density f x = 1 + x 2 {\displaystyle fx={\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.

                                     

2.5. Related 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:

α = 2 − q − 1, λ = 1 λ q − 1. {\displaystyle \alpha =,~\lambda ={1 \over \lambda _{q}q-1}.}
                                     

2.6. Related distributions Relation to the log- logistic distribution

The logarithm of a Lomaxshape=1.0, scale=λ-distributed variable follows a logistic distribution with location logλ and scale 1.0. This implies that a Lomaxshape=1.0, scale=λ-distribution equals a log-logistic distribution with shape β=1.0 and scale α=logλ.

                                     

2.7. Related distributions 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 ~ Gammashape=k, scale=θ and X |λ ~ Exponentialrate=λ then the marginal distribution of X |k,θ is Lomaxshape=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.

                                     
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