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Pareto Second Kind distribution

Profile

Pareto Type II generalizes the classical Pareto form to model mass near the minimum more flexibly. It keeps a power-law right tail and is known as Lomax when the threshold is set to zero.

Feature Reading
Support Support begins at loc and is unbounded to the right. xm and alpha set threshold scale and tail shape.
Shape It is Lomax under a common parameterization and closely related to Burr XII. It also connects with Generalized Pareto for exceedances, but parameters are not automatically interchangeable.

Origin and terminology

Pareto type classifications were consolidated in distribution compendia; Type II became common in reliability, insurance, and economics under several names.

In real models

  • claims and excess losses above a deductible
  • durations, sizes, and thresholded heavy-tailed quantities

Comparisons

Check whether loc represents a physical minimum, deductible, or only a parameterization; those readings imply different decisions.

A Pareto tail without a jump at the threshold

The Lomax distribution shifts the Pareto idea so that excess begins at zero and density is positive from the origin. It is also called Pareto type II. Lomax used it in 1954 for business failure data, where a power tail coexisted with many small observations.

It can be constructed by mixing an Exponential variable over a Gamma rate, a story that interprets the tail as heterogeneity across units. This mixture connects it with survival and risk models. Unlike Pareto I, it does not require density to begin abruptly at a positive minimum. Names vary among texts and some parameterizations add location, so support and definition must be checked before comparing tail indices.

Decision guide

A good candidate when: a positive excess over an origin is modelled and a power tail is expected without a separate multiplicative minimum.

Compare it with: Pareto I, Generalized Pareto, and Loglogistic. Clarify whether data are absolute levels or excesses because that changes the meaning of scale.

References

  • SciPy reference: scipy.stats.lomax — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Lomax, K. S. (1954). Business failures: Another example of the analysis of failure data. Journal of the American Statistical Association, 49(268), 847–852.
  • Clauset, A., Shalizi, C. R. & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661–703.

Pareto Second Kind Distribution: equations and calculator

Distribution defintion

X∼ParetoSecondKind(xm,α,Loc)X\sim\mathrm{ParetoSecondKind}\left(x_\mathrm{m},\alpha,\text{Loc}\right)

Distribution domain

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Parameters domain and parameters constraints

xm∈R+,α∈R+,Loc∈Rx_\mathrm{m}\in\mathbb{R}^{+},\alpha\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Cumulative distribution function

FX(x)=1−[1+x−Locxm]−αF_{X}\left(x\right)=1-\left[{1+\frac{x-\text{Loc}}{x_\mathrm{m}}}\right]^{-\alpha}

Probability density function

fX(x)=αxm[1+x−Locxm]−(α+1)f_{X}\left(x\right)=\frac{\alpha}{x_\mathrm{m}} \left[{1+\frac{x-\text{Loc}}{x_\mathrm{m}}}\right]^{-(\alpha+1)}

Percent point function/Sample

FX−1(u)=Loc+xm[(1−p)−1α−1]F^{-1}_{X}\left(u\right)=\text{Loc}+x_\mathrm{m} \left[\left(1-p\right)^{-\frac{1}{\alpha}} -1\right]

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=xmkΓ(α−k)Γ(1+k)Γ(α)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{x_\mathrm{m}^k \Gamma(\alpha-k)\Gamma(1+k)}{\Gamma(\alpha)}

Parametric mean

Mean(X)=μ~1′=xmα−1if α>1\mathrm{Mean}(X)=\tilde{\mu}'_{1}=\frac{x_\mathrm{m}}{{\alpha -1}} \quad \text{if }\alpha>1

Parametric variance

Variance(X)=μ~2′−μ~1′2=xm2α(α−1)2(α−2)if α>2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}=\frac{x_\mathrm{m}^2 \alpha}{(\alpha-1)^2(\alpha-2)} \quad \text{if }\alpha>2

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2(1+α)α−3 α−2αif α>3\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}=\frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}} \quad \text{if }\alpha>3

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=6(α3+α2−6α−2)α(α−3)(α−4)if α>4\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}=\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)} \quad \text{if }\alpha>4

Parametric median

Median(X)=xm(2α−1)\mathrm{Median}(X)=x_\mathrm{m}\left(\sqrt[\alpha]{2}-1\right)

Parametric mode

Mode(X)=0\mathrm{Mode}(X)=0

Additional information and definitions

X∼ParetoSecondKind(xm,α,0)X\sim\mathrm{ParetoSecondKind}\left(x_\mathrm{m},\alpha,0\right)
xm:Scale parameterx_\mathrm{m}:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}