Lomax distribution

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

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

with shape parameter $\backslash alpha\; >\; 0$ and scale parameter $\backslash lambda\; >\; 0$. The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:

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

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

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

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

:$\backslash text\{If\; \}\; Y\; \backslash sim\; \backslash operatorname\{Pareto\}(x\_m\; =\; \backslash lambda,\; \backslash alpha),\; \backslash text\{\; then\; \}\; Y\; -\; x\_m\; \backslash sim\; \backslash operatorname\{Lomax\}(\backslash alpha,\backslash lambda).$

The Lomax distribution is a Pareto Type II distribution with x_m = λ and μ = 0:{{citation|title=Statistical Size Distributions in Economics and Actuarial Sciences|volume=470|series=Wiley Series in Probability and Statistics|first1=Christian|last1=Kleiber|first2=Samuel|last2=Kotz|publisher=John Wiley & Sons|year=2003|isbn=9780471457169|page=60|url=https://books.google.com/books?id=7wLGjyB128IC&pg=PA60}}.

:$\backslash text\{If\; \}\; X\; \backslash sim\; \backslash operatorname\{Lomax\}(\backslash alpha,\; \backslash lambda)\; \backslash text\{\; then\; \}\; X\; \backslash sim\; \backslash text\{P(II)\}\backslash left(x\_m\; =\; \backslash lambda,\; \backslash alpha,\; \backslash mu\; =\; 0\backslash right).$

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

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

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

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

:$\backslash alpha\; =\; \{\{2\; -\; q\}\; \backslash over\; \{q\; -\; 1\}\},\; ~\; \backslash lambda\; =\; \{1\; \backslash over\; \backslash lambda\_q(q\; -\; 1)\}\; .$

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)

{{ProbDistributions|continuous-semi-infinite}}

Category:Continuous distributions

Category:Compound probability distributions

Category:Probability distributions with non-finite variance