LOGGAMMA DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.LogGamma({"c": *, "mu": *, "sigma": *})

💡 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

$$X\sim \mathrm{LogGamma}(c,\mu ,\sigma )$$

Distribution Domain

$$x\in (0,\mathrm{\infty })$$

Parameters Domain and Constraints

$$c\in {\mathbb{R}}^{+},\mu \in \mathbb{R},\sigma \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\frac{\gamma (c,e^x)}{\mathrm{\Gamma }\left(c\right)}=\text{P}(c,e^{z(x)})$$

Probability Density Function

$$f_X\left(x\right)=\frac{\exp (cz(x)-e^{z(x)})}{\sigma \mathrm{\Gamma }\left(c\right)}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\mu +\sigma \ln \left({\text{P}}^{-1}(u,c)\right)$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\mu +\sigma {\psi }_0$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}={\alpha }^2{\psi }_1\left(c\right)$$

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\mu +\sigma \ln \left({\text{P}}^{-1}(1/2,c)\right)$$

Parametric Mode

$$\mathrm{Mode}(X)=\mu +\sigma \ln (c)$$

Additional Information and Definitions

• $\mu :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $z\left(x\right)=(x-\mu )/\sigma$
• $u:\text{Uniform[0,1] random varible}$
• $\text{P}(a,x)=\frac{\gamma (a,x)}{\mathrm{\Gamma }(a)}:\text{Regularized lower incomplete gamma function}$
• ${\text{P}}^{-1}(a,u):\text{Inverse of regularized lower incomplete gamma function}$
• $\gamma (a,x):\text{Lower incomplete gamma function}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$
• ${\psi }_0\left(x\right):\text{Digamma function}$
• ${\psi }_n\left(x\right):\text{Polygamma function of order }n\in \mathbb{N}$

Spreadsheet Documents

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