LOGLOGISTIC 3P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.LogLogistic3P({"loc": *, "alpha": *, "beta": *})

💡 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

$$X\sim {\mathrm{LogLogistic}}_{3\mathrm{P}}(\text{Loc},\alpha ,\beta )$$

Distribution Domain

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

Parameters Domain and Constraints

$$\text{Loc}\in \mathbb{R},\alpha \in {\mathbb{R}}^{+},\beta \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\frac{1}{1+((x-\text{Loc})/\alpha {)}^{-\beta }}$$

Probability Density Function

$$f_X\left(x\right)=\frac{(\beta /\alpha )((x-\text{Loc})/\alpha {)}^{\beta -1}}{{(1+((x-\text{Loc})/\alpha {)}^{\beta })}^2}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\alpha {\left(\frac{u}{1-u}\right)}^{1/\beta }$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx={\alpha }^k\text{Beta}(1-k/\beta ,1+k/\beta )={\alpha }^k\frac{k\pi /\beta }{\sin (k\pi /\beta )}$$

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+{\tilde{\mu }}_1^{\prime }$$

Parametric Variance

$$\mathrm{Variance}(X)={\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\tilde{\mu }}_4^{\prime }-4{\tilde{\mu }}_1^{\prime }{\tilde{\mu }}_3^{\prime }+6{\tilde{\mu }}_1^{\prime 2}{\tilde{\mu }}_2^{\prime }-3{\tilde{\mu }}_1^{\prime 4}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^2}$$

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\alpha$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\alpha {\left(\frac{\beta -1}{\beta +1}\right)}^{1/\beta }$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{LogLogistic}(\alpha ,\beta )$
• $\text{Loc}:\text{Location parameter}$
• $\alpha :\text{Scale parameter}$
• $u:\text{Uniform[0,1] random varible}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

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