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]
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Mean | ; undefined otherwise | ||
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Mode | 0 | ||
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The probability density function (pdf) for the Lomax distribution is given by
with shape parameter and scale parameter . The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:
The th non-central moment exists only if the shape parameter strictly exceeds , when the moment has the value
The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:
The Lomax distribution is a Pareto Type II distribution with xm=λ and μ=0:[5]
The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:
The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then .
The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density , 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:
The logarithm of a Lomax(shape = 1.0, scale = λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0. This implies that a Lomax(shape = 1.0, scale = λ)-distribution equals a log-logistic distribution with shape β = 1.0 and scale α = log(λ).
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).