PLAYGROUND

DISTRIBUTIONS / CONTINUOUS / F 4P

F 4P distribution

Why it exists

The four-parameter F distribution is a shifted and rescaled F. It can describe a positive ratio measured from a non-zero threshold, but it no longer automatically has the interpretation of a variance ratio.

Alternative names

It may also be called four-parameter F or shifted F. “Noncentral F” is a different extension, not a synonym.

Constructions and limits

With loc=0 and scale=1 it becomes F. It is not equivalent to a noncentral F: noncentrality changes the numerator mechanism, whereas this form transforms the observed axis.

Applications

It can be useful for an instrument with a non-zero zero-point and a specific unit. For an inferential F statistic, freeing loc or scale needs a justification outside the standard theory.

A free loc can remove the ratio interpretation and make comparison with a two-parameter F misleading.

The F shape separated from the F test

F 4P shifts and rescales a standard F variable. The result keeps two shape controls and a right tail, but it is no longer directly the ratio of two Chi Square variables divided by their degrees of freedom.

This version can describe a positive quantity measured from a nonzero threshold. It must not be used to read ANOVA or regression p values: classical F tables assume zero location, unit scale, and degrees fixed by design. Location, scale, and both degrees may compete during fitting. If the goal is description rather than classical inference, communicate observable quantiles and call the degrees shape parameters to avoid implying a nonexistent sample size.

Decision guide

A good candidate when: an F shape appears on a translated and rescaled measurement scale with interpretable bounds.

Compare it with: standard F before freeing location and scale. In classical inference, shifting F breaks the usual interpretation of its degrees of freedom.

References

F 4P Distribution: equations and calculator

Distribution defintion

X∼F4P(df1,df2,Loc,Sc)X\sim\mathrm{F_{4P}}\left(\text{df}_{1},\text{df}_{2},\text{Loc},\text{Sc}\right)

Distribution domain

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

Parameters domain and parameters constraints

df1∈R+,df2∈R+,Loc∈R,Sc∈R+\text{df}_{1}\in\mathbb{R}^{+},\text{df}_{2}\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=Idf1z(x)/(df1z(x)+df2)(df12,df22)F_{X}\left(x\right)=I_{\text{df}_{1} z(x)/(\text{df}_{1} z(x)+\text{df}_{2})}\left (\tfrac{\text{df}_{1}}{2},\tfrac{\text{df}_{2}}{2}\right)

Probability density function

fX(x)=1Sc×(df1z(x))df1df2df2(df1z(x)+df2)df1+df2z(x) Beta(df12,df22)f_{X}\left(x\right)=\frac{1}{\text{Sc}}\times \frac{\sqrt{\frac{(\text{df}_{1} z(x))^{\text{df}_{1}} \text{df}_{2}^{\text{df}_{2}}}{(\text{df}_{1} z(x)+\text{df}_{2})^{\text{df}_{1}+\text{df}_{2}}}}}{z(x)\,\text{Beta}\left(\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}

Percent point function/Sample

FX−1(u)=Loc+Scdf2×I−1(u,df12,df22)df1×(1−I−1(u,df12,df22))F^{-1}_{X}\left(u\right)=\text{Loc}+\text{Sc}\frac{\text{df}_{2}\times I^{-1}\left(u,\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}{\text{df}_{1}\times \left(1-I^{-1}\left(u,\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)\right)}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=Γ(df12+k)Γ(df12)Γ(df22−k)Γ(df22)(df2df1)kif df2>2k\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{\Gamma\left(\tfrac{\text{df}_{1}}{2}+k\right) }{\Gamma\left(\tfrac{\text{df}_{1}}{2}\right)}\frac{\Gamma\left(\tfrac{\text{df}_{2}}{2}-k\right) }{\Gamma\left(\tfrac{\text{df}_{2}}{2}\right) }\left(\frac{\text{df}_{2}}{\text{df}_{1}}\right)^k \quad \text{if }\text{df}_{2} > 2k

Parametric mean

Mean(X)=Loc+Scμ~1′=Loc+Scdf2df2−2if df2>2\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\tilde{\mu}'_{1}=\text{Loc}+\text{Sc}\frac{\text{df}_{2}}{\text{df}_{2}-2} \quad \text{if }\text{df}_{2} > 2

Parametric variance

Variance(X)=Sc2(μ~2′−μ~1′2)=Sc22 df22 (df1+df2−2)df1(df2−2)2(df2−4)if df2>4\mathrm{Variance}(X)=\text{Sc}^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\text{Sc}^{2}\frac{2\,\text{df}_{2}^2\,(\text{df}_{1}+\text{df}_{2}-2)}{\text{df}_{1} (\text{df}_{2}-2)^2 (\text{df}_{2}-4)} \quad \text{if }\text{df}_{2} > 4

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=(2df1+df2−2)8(df2−4)(df2−6)df1(df1+df2−2)if df2>6\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 \text{df}_{1}+\text{df}_{2}-2) \sqrt{8 (\text{df}_{2}-4)}}{(\text{df}_{2}-6) \sqrt{\text{df}_{1} (\text{df}_{1}+\text{df}_{2} -2)}}\quad \text{if }\text{df}_{2} > 6

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3(8+(df2−6)×Skewness(X)2)2df2−16+3if df2>8\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{3\left(8+\left(\text{df}_{2}-6\right)\times \mathrm{Skewness}(X)^{2}\right)}{2\text{df}_{2}-16}+3\quad \text{if }\text{df}_{2} > 8

Parametric median

Median(X)=Loc+Scdf2×I−1(12,df12,df22)df1×(1−I−1(12,df12,df22))\mathrm{Median}(X)=\text{Loc}+\text{Sc}\frac{\text{df}_{2}\times I^{-1}\left(\frac{1}{2},\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}{\text{df}_{1}\times \left(1-I^{-1}\left(\frac{1}{2},\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)\right)}

Parametric mode

Mode(X)=Loc+Scdf2(df1−2)df1(df2+2)if df1>2\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{\text{df}_{2}\left(\text{df}_{1}-2\right)}{\text{df}_{1}\left(\text{df}_{2}+2\right)} \quad \text{if }\text{df}_{1} > 2

Additional information and definitions

X~∼F(df1,df2)\tilde{X}\sim\mathrm{F}\left(\text{df}_{1},\text{df}_{2}\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}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}