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 ) = α λ ( 1 + x λ ) − ( α + 1 ) , x ≥ 0 , {\displaystyle p(x)={\frac {\alpha }{\lambda }}\left(1+{\frac {x}{\lambda }}\right)^{-(\alpha +1)},\qquad x\geq 0,}
with shape parameter α > 0 {\displaystyle \alpha >0} and scale parameter λ > 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:
p ( x ) = α λ α ( x + λ ) α + 1 . {\displaystyle p(x)={\frac {\alpha \lambda ^{\alpha }}{(x+\lambda )^{\alpha +1}}}.}
Non-central moments The ν {\displaystyle \nu } th non-central moment E [ X ν ] {\displaystyle E\left[X^{\nu }\right]} 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 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:
If Y ∼ Pareto ( x m = λ , α ) , then Y − x m ∼ Lomax ( α , λ ) . {\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 xm = λ and μ = 0:
If X ∼ Lomax ( α , λ ) then X ∼ P(II) ( x m = λ , α , μ = 0 ) . {\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:
μ = 0 , ξ = 1 α , σ = λ α . {\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 X λ ∼ β ′ ( 1 , α ) {\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )} .
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