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Burr 4P distribution

The idea in one sentence

Burr 4P extends the Burr family with a location parameter. Tail shape and scale remain Burr-like, while the lower threshold can be separated from zero on the observed measurement scale.

Support and interpretation

The support starts at loc and extends to infinity. A, B, and C control scale and shape; location changes the origin, not the heavy- or light-tailed nature of the family.

Its place in the family

It keeps Burr XII’s links with GB2, Beta prime, Dagum, and loglogistic models. Adding loc can suit lifetimes with a minimum warranty period or sizes with a physical minimum, but it removes the simple ratio-from-zero interpretation.

An applied reading

The four-parameter version can fit a heavy tail after a delay or threshold. In engineering, separating time in service from time exposed to failure is one possible interpretation; it must be supported by the measurement process.

Estimates of loc are often correlated with A; a maximum-likelihood value without a sensitivity profile can hide that weakness.

Moving the floor beneath a Burr tail

Burr 4P adds location to the type XII shape. The variable can be read as a threshold loc plus a positive Burr excess. This helps when claims, sizes, or durations only begin beyond a deductible or physical limit.

The new parameter does not change the tail index of the excess, but it changes which observations count as near the origin. That interaction can substantially alter both estimated shapes. A known contractual deductible may be fixed confidently; a threshold learned from the smallest observation is far less certain. If data were already truncated or censored by a reporting rule, shifting the density is not a substitute for a likelihood that represents the observation mechanism.

Decision guide

A good candidate when: Burr XII describes body and tail well and there is also a physically justified minimum time or size.

Compare it with: unshifted Burr and 3P Loglogistic. Check loc stability after removing the smallest observations; a boundary that moves substantially is usually weakly identified.

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.
  • Burr, I. W. (1942). Cumulative frequency functions. The Annals of Mathematical Statistics, 13(2), 215–232.
  • Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley.

Burr 4P Distribution: equations and calculator

Distribution defintion

X∼Burr4P(A,B,C,Loc)X\sim \mathrm{Burr_{4P}}\left( A,B,C,\text{Loc} \right)

Distribution domain

x∈[Loc,∞)x\in [\text{Loc},\infty)

Parameters domain and parameters constraints

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

Cumulative distribution function

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

Probability density function

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

Percent point function/Sample

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

Non-central parametric moments

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

Parametric mean

Mean(X)=Loc+μ~1′\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}

Parametric variance

Variance(X)=μ~2′−μ~1′2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5\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}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)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}}

Parametric median

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

Parametric mode

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

Additional information and definitions

X~∼Burr(A,B,C)\tilde{X}\sim \mathrm{Burr}\left( A,B,C \right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
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}