F 4P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.F4P({"df1": *, "df2": *, "loc": *, "scale": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim {\mathrm{F}}_{4\mathrm{P}}({\text{df}}_1,{\text{df}}_2,\text{Loc},\text{Sc})$$

Distribution Domain

$$x\in [\text{Loc},\mathrm{\infty })$$

Parameters Domain and Constraints

$${\text{df}}_1\in {\mathbb{R}}^{+},{\text{df}}_2\in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R},\text{Sc}\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=I_{{\text{df}}_{1}z(x)/({\text{df}}_{1}z(x)+{\text{df}}_2)}(\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2})$$

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\text{Sc}}\times \frac{\sqrt{\frac{({\text{df}}_{1}z(x){)}^{{\text{df}}_1}{\text{df}}_2^{{\text{df}}_2}}{({\text{df}}_{1}z(x)+{\text{df}}_2{)}^{{\text{df}}_1+{\text{df}}_2}}}}{z(x)\text{Beta}(\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2})}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\text{Sc}\frac{{\text{df}}_2\times I^{-1}(u,\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2})}{{\text{df}}_1\times (1-I^{-1}(u,\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2}))}$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\frac{\mathrm{\Gamma }(\frac{{\text{df}}_1}{2}+k)}{\mathrm{\Gamma }\left(\frac{{\text{df}}_1}{2}\right)}\frac{\mathrm{\Gamma }(\frac{{\text{df}}_2}{2}-k)}{\mathrm{\Gamma }\left(\frac{{\text{df}}_2}{2}\right)}{\left(\frac{{\text{df}}_2}{{\text{df}}_1}\right)}^k\text{if }{\text{df}}_2>2k$$

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+\text{Sc}{\tilde{\mu }}_1^{\prime }=\text{Loc}+\text{Sc}\frac{{\text{df}}_2}{{\text{df}}_2-2}\text{if }{\text{df}}_2>2$$

Parametric Variance

$$\mathrm{Variance}(X)={\text{Sc}}^2({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})={\text{Sc}}^2\frac{2{\text{df}}_2^2({\text{df}}_1+{\text{df}}_2-2)}{{\text{df}}_1({\text{df}}_2-2{)}^2({\text{df}}_2-4)}\text{if }{\text{df}}_2>4$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}=\frac{(2{\text{df}}_1+{\text{df}}_2-2)\sqrt{8({\text{df}}_2-4)}}{({\text{df}}_2-6)\sqrt{{\text{df}}_1({\text{df}}_1+{\text{df}}_2-2)}}\text{if }{\text{df}}_2>6$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\tilde{\mu }}_4^{\prime }-4{\tilde{\mu }}_1^{\prime }{\tilde{\mu }}_3^{\prime }+6{\tilde{\mu }}_1^{\prime 2}{\tilde{\mu }}_2^{\prime }-3{\tilde{\mu }}_1^{\prime 4}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^2}=\frac{3(8+({\text{df}}_2-6)\times \mathrm{Skewness}(X{)}^2)}{2{\text{df}}_2-16}+3\text{if }{\text{df}}_2>8$$

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\text{Sc}\frac{{\text{df}}_2\times I^{-1}(\frac{1}{2},\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2})}{{\text{df}}_1\times (1-I^{-1}(\frac{1}{2},\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2}))}$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{{\text{df}}_2({\text{df}}_1-2)}{{\text{df}}_1({\text{df}}_2+2)}\text{if }{\text{df}}_1>2$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{F}({\text{df}}_1,{\text{df}}_2)$
• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z\left(x\right)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• $I(x,a,b):\text{Regularized incomplete beta function}$
• $I^{-1}(x,a,b):\text{Inverse of regularized incomplete beta function}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document