Lomax distribution - Alchetron, The Free Social Encyclopedia


Parameters λ > 0 {\displaystyle \lambda >0} scale (real) α > 0 {\displaystyle \alpha >0} shape (real) Support x ≥ 0 {\displaystyle x\geq 0} PDF α λ [ 1 + x λ ] − ( α + 1 ) {\displaystyle {\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)}} CDF 1 − [ 1 + x λ ] − α {\displaystyle 1-\left[{1+{x \over \lambda }}\right]^{-\alpha }} Mean λ α − 1 for α > 1 {\displaystyle {\lambda \over {\alpha -1}}{\text{ for }}\alpha >1} Otherwise undefined Median λ ( 2 α − 1 ) {\displaystyle \lambda ({\sqrt[{\alpha }]{2}}-1)}

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.

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 ,

with shape parameter α > 0 and scale parameter λ > 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 .

Differential equation

The pdf of the Lomax distribution is a solution to the following differential equation:

{ ( λ + x ) p ′ ( x ) + ( α + 1 ) p ( x ) = 0 , p ( 0 ) = α λ }

Non-central moments

The ν th non-central moment E [ X ν ] exists only if the shape parameter α strictly exceeds ν , when the moment has the value

E ( X ν ) = λ ν Γ ( α − ν ) Γ ( 1 + ν ) Γ ( α )

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 ( λ , α ) .

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

If X ∼ Lomax ( λ , α ) then X ∼ P(II) ( x m = λ , α , μ = 0 ) .

Relation to generalized Pareto distribution

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

μ = 0 , ξ = 1 α , σ = λ α .

Relation to 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 q − 1 , λ = 1 λ q ( q − 1 ) .

Gamma-exponential mixture

The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution. If λ ~ Gamma(scale=k, shape=θ) and X ~ Exponential(rate=λ) then the marginal distribution of X is Lomax(scale=1/k, shape=θ).

References

Lomax distribution Wikipedia

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