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Burr distribution

What it describes

The Burr distribution is a family of right-tailed models designed to describe positive data more flexibly than a single power law. Its parameters can change scale, near-origin concentration, and tail decay.

Its support is positive. Extreme behaviour depends on the shape combination, especially the tail parameter; some moments may not exist even when a finite sample has a finite mean.

History and names

I. W. Burr introduced a collection of frequency functions in 1942 for skewed data. The family is often organized as Burr type XII among the Burr distributions.

How it connects to other distributions

Burr XII is closely tied to the beta type II and GB2 families. Particular parameter choices produce forms related to loglogistic, Pareto, and Dagum models, so names vary across fields.

Where it appears

  • reliability, lifetimes, and failure risk
  • income, particle-size, and other heavy-tailed phenomena

Fitting cautions

Comparing parameters across software without checking scale, shape, and location conventions can produce incorrect conclusions.

The twelfth curve that eclipsed the other eleven

In 1942 Irving Burr presented a system of twelve cumulative functions. The law now usually called simply Burr is his type XII. Its combination of two shapes and a scale allows separate control of behaviour near zero and the power-law tail.

In economics it is also called Singh Maddala. Loglogistic is a special case, and reciprocal transformations connect it with Burr type III and Dagum under suitable parameterizations. This network of names demands bibliographic care. In actuarial applications the flexible tail is attractive, but two free shapes can trade off while describing the centre. Diagnostics should examine extreme probabilities rather than global fit criteria alone.

Decision guide

A good candidate when: positive data require flexible body shape together with a potentially heavy right tail, as in heterogeneous lifetimes or sizes.

Compare it with: Loglogistic, Pareto, and Dagum. Compare survival probabilities on a logarithmic scale: models that look alike centrally can diverge sharply in the tail.

References

  • SciPy reference: scipy.stats.burr12 — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.
  • Burr, I. W. (1942). Cumulative frequency functions. The Annals of Mathematical Statistics, 13(2), 215–232.

Burr Distribution: equations and calculator

Distribution defintion

X∼Burr(A,B,C)X\sim\mathrm{Burr}\left(A,B,C\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

A∈R+,B∈R,C∈R+A\in\mathbb{R}^{+},B\in\mathbb{R},C\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−[1+(xA)B]−CF_{X}\left(x\right)=1-\left[1+\left(\frac{x}{A}\right)^{B}\right]^{-C}

Probability density function

fX(x)=BCA(xA)B−1[1+(xA)B]−C−1f_{X}\left(x\right)=\frac{BC}{A}\left(\frac{x}{A}\right)^{B-1}\left[1+\left(\frac{x}{A}\right)^{B}\right]^{-C-1}

Percent point function/Sample

FX−1(u)=A[(1−u)−1c−1]1BF^{-1}_{X}\left(u\right)=A\left[(1-u)^{-\frac{1}{c}}-1\right]^{\frac{1}{B}}

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=AkC×Beta(BC−kB,B+KB)\mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k}f_{X}\left(x\right)dx=A^{k}C\times \text{Beta}\left(\frac{BC-k}{B},\frac{B+K}{B}\right)

Parametric mean

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=A[(12)−1c−1]1B\mathrm{Median}(X)=A\left[\left(\frac{1}{2}\right)^{-\frac{1}{c}}-1\right]^{\frac{1}{B}}

Parametric mode

Mode(X)=A(B−1BC+1)1B\mathrm{Mode}(X)=A\left(\frac{B-1}{BC+1}\right)^{\frac{1}{B}}

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}