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Beta Prime 4P distribution

What it represents

Beta Prime 4P adds location and scale to the Beta-prime distribution. It retains the interpretation of a positive ratio while allowing the observed origin to shift and the measurement unit to vary.

Historical clue

The “4P” label is a software convention indicating four parameters, not an independent historical name. The underlying family is beta type II or Pearson type VI.

Relationships that clarify its use

It is related to the two-parameter Beta prime by an affine transformation. It also connects with F and with Burr/Dagum models; the exact equivalence depends on parameterization and scale factors.

Data examples

  • positive ratios with a measurement threshold
  • income or size models with a specific origin and unit

Modelling warning

A loc estimate near the sample minimum is unstable; check it against censoring, measurement precision, and fit sensitivity.

Odds with a different zero and unit

Beta Prime 4P applies location and scale to a standard Beta Prime ratio. This freedom can represent a ratio measured from a nonzero floor or expressed in different units. The transformation preserves the tail shape of the excess.

What it does not automatically preserve is the odds or Gamma-ratio story. Adding location to a ratio changes its algebraic interpretation, and an arbitrary scale should correspond to the units of the problem. Near the endpoint, loc competes with the parameter governing initial mass; in the tail, shape determines which moments exist. Sensitivity analysis against the unshifted version helps separate physical need from statistical flexibility.

Decision guide

A good candidate when: Beta Prime flexibility is needed but support begins at a nonzero physical threshold and requires its own scale.

Compare it with: the unshifted Beta Prime before freeing loc. The extra parameter should represent an interpretable boundary; otherwise it may merely chase the sample minimum.

References

Beta Prime 4P Distribution: equations and calculator

Distribution defintion

X∼BetaPrime4P(α,β,Loc,Sc)X\sim\mathrm{BetaPrime}_{\mathrm{4P}}\left(\alpha,\beta,\text{Loc},\text{Sc}\right)

Distribution domain

x∈[Loc,∞)x\in [\text{Loc},\infty)

Parameters domain and parameters constraints

α∈R+,β∈R+,Loc∈R,Sc∈R+\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=I(z(x)1+z(x),α,β)F_{X}\left(x\right)=I\left(\frac{z(x)}{1+z(x)},\alpha,\beta\right)

Probability density function

fX(x)=z(x)α−1(1+z(x))−α−βSc×Beta(α,β)f_{X}\left(x\right)=\frac{z(x)^{\alpha-1} (1+z(x))^{-\alpha -\beta}}{\text{Sc}\times \text{Beta}(\alpha,\beta)}

Percent point function/Sample

FX−1(u)=Loc+ScI−1(u,α,β)1−I−1(u,α,β)F^{-1}_{X}\left(u\right)=\text{Loc}+\text{Sc}\frac{I^{-1}\left(u,\alpha,\beta\right)}{1-I^{-1}\left(u,\alpha,\beta\right)}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=Γ(k+α)Γ(β−k)Γ(α)Γ(β)if β>k\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{\Gamma\left(k+\alpha\right)\Gamma\left(\beta-k\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)} \quad \text{if }\beta>k

Parametric mean

Mean(X)=Loc+Scμ~1′=Loc+Scαβ−1if β>1\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\tilde{\mu}'_{1}=\text{Loc}+\text{Sc}\frac{\alpha}{\beta-1} \quad \text{if }\beta>1

Parametric variance

Variance(X)=Sc2(μ~2′−μ~1′2)=Sc2α(α+β−1)(β−2)(β−1)2if β>2\mathrm{Variance}(X)=\text{Sc}^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\text{Sc}^{2}\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} \quad \text{if }\beta>2

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2(2α+β−1)β−3β−2α(α+β−1)if β>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(2\alpha+\beta-1)}{\beta-3}\sqrt{\frac{\beta-2}{\alpha(\alpha+\beta-1)}} \quad \text{if }\beta>3

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2if β>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}} \quad \text{if }\beta>4

Parametric median

Median(X)=Loc+ScI−1(12,α,β)1−I−1(12,α,β)\mathrm{Median}(X)=\text{Loc}+\text{Sc}\frac{I^{-1}\left(\frac{1}{2},\alpha,\beta\right)}{1-I^{-1}\left(\frac{1}{2},\alpha,\beta\right)}

Parametric mode

Mode(X)=Loc+Scα−1β+1\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{\alpha-1}{\beta+1}

Additional information and definitions

X~∼BetaPrime(α,β)\tilde{X}\sim \mathrm{BetaPrime}\left( \alpha,\beta \right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}