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Bradford distribution

Quick view

The Bradford distribution is bounded and has a density that falls smoothly from the lower endpoint. Its shape parameter controls how unequal the concentration is across the interval.

If you are coming from another distribution

For a small shape value it approaches a uniform distribution on the interval. With finite support and moderate skewness it can compete with Beta, triangular, or trapezoidal models, but its logarithmic mechanism is different.

History and terminology

The model was introduced by S. C. Bradford in the context of how scientific articles are scattered across journals. Bradford’s law in bibliometrics describes productivity zones, while this distribution is a continuous representation associated with that phenomenon.

A familiar situation

It can describe a quantity spread across a known range when multiplicative or relative position makes one region more populated. The original bibliometric application does not imply that every citation count is a continuous variable.

Fitting with care

If min and max are limits of an ordinal or discretized count, a continuous distribution may be an approximation rather than the literal data-generating model.

From journal shelves to a bounded law

In 1934 S. C. Bradford observed that articles on a subject were scattered unevenly among journals: a small core produced much, while a long periphery contributed little. Leimkuhler, and later Morse with Leimkuhler, expressed that bibliometric law through an exact distribution with a logarithmic relationship.

In its bounded continuous form, c tilts mass toward one endpoint, and Uniform appears as c approaches zero. The information-science genealogy supplies a concrete application, but the curve can also be used phenomenologically on other intervals. In that setting, explain why reciprocal-linear decay represents the mechanism instead of selecting it merely for its uncommon name.

Decision guide

A good candidate when: the data are bounded and show a simple monotone tilt toward one endpoint with no interior mode.

Compare it with: Triangular, Power Function, and Beta. If the two endpoints behave differently or an interior mode appears, Bradford’s single shape parameter is too restrictive.

References

  • SciPy reference: scipy.stats.bradford — definition and parameterization

  • Morse, P. M. & Leimkuhler, F. F. (1979). Exact solution for the Bradford distribution and its use in modeling informational data. Operations Research, 27(1), 187–198.

  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.

  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.

  • Bradford, S. C. (1934). Sources of information on specific subjects. Engineering, 137, 85–86.

Bradford Distribution: equations and calculator

Distribution defintion

X∼Bradford(c,min,max)X\sim\mathrm{Bradford}\left(c,\text{min},\text{max}\right)

Distribution domain

x∈(min,max)x\in\left(\text{min},\text{max}\right)

Parameters domain and parameters constraints

c∈R+,min∈R,max∈R,min<maxc\in\mathbb{R}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R},\text{min} < \text{max}

Cumulative distribution function

FX(x)=ln⁡(1+c⋅z(x))kF_{X}\left(x\right)=\frac{\ln\left(1+c\cdot z(x)\right)}{k}

Probability density function

fX(x)=ck(1+c⋅z(x))(max−min)f_{X}\left(x\right)=\frac{c}{k\left(1+c\cdot z(x)\right)\left(\text{max}-\text{min}\right)}

Percent point function/Sample

FX−1(u)=min+(max−min)×(1+c)u−1cF^{-1}_{X}\left(u\right)=\text{min}+(\text{max}-\text{min})\times \frac{\left(1+c\right)^{u}-1}{c}

Non-central parametric moments

μ~k′=E[X~k]=∫01xkfX~(x)dx\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{1}x^{k}f_{\tilde{X}}\left(x\right)dx

Parametric mean

Mean(X)=min+(max−min)⋅μ~1′=min+(max−min)⋅c−kck\mathrm{Mean}(X)=\text{min}+\left(\text{max}-\text{min}\right)\cdot\tilde{\mu}'_{1}=\text{min}+\left(\text{max}-\text{min}\right)\cdot\frac{c-k}{ck}

Parametric variance

Variance(X)=(max−min)2⋅(μ~2′−μ~1′2)=(max−min)2⋅(c+2)k−2c2ck2\mathrm{Variance}(X)=\left(\text{max}-\text{min}\right)^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\left(\text{max}-\text{min}\right)^{2}\cdot\frac{\left(c+2\right)k-2c}{2ck^{2}}

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2(12c2−9kc(c+2)+2k2(c(c+3)+3))c(c(k−2)+2k)(3c(k−2)+6k)\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{\sqrt{2}\left(12c^{2}-9kc\left(c+2\right)+2k^{2}\left(c\left(c+3\right)+3\right)\right)}{\sqrt{c\left(c\left(k-2\right)+2k\right)}\left(3c\left(k-2\right)+6k\right)}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3+c3(k−3)(k(3k−16)+24)+12kc2(k−4)(k−3)+6ck2(3k−14)+12k33c(c(k−2)+2k)2\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}}=3+\frac{c^{3}\left(k-3\right)\left(k\left(3k-16\right)+24\right)+12kc^{2}\left(k-4\right)\left(k-3\right)+6ck^{2}\left(3k-14\right)+12k^{3}}{3c\left(c\left(k-2\right)+2k\right)^{2}}

Parametric median

Median(X)=min+(max−min)⋅(1+c)frac12−1c\mathrm{Median}(X)=\text{min}+(\text{max}-\text{min})\cdot\frac{\left(1+c\right)^{frac{1}{2}}-1}{c}

Parametric mode

Mode(X)=min\mathrm{Mode}(X)=\text{min}

Additional information and definitions

X~∼Bradford(c,0,1)\tilde{X}\sim\mathrm{Bradford}\left(c,0,1\right)
k=ln⁡(1+c)k=\ln(1+c)
z(x)=(x−min)/(max−min)z\left(x\right)=\left(x-\text{min}\right)/\left(\text{max}-\text{min}\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}