GENERALIZED GAMMA 4P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.GeneralizedGamma4P({"a": *, "d": *, "p": *, "loc": *})

💡 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

$$X\sim {\mathrm{GeneralizedGamma}}_{4\mathrm{P}}(a,d,p,\text{Loc})$$

Distribution Domain

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

Parameters Domain and Constraints

$$a\in {\mathbb{R}}^{+},d\in {\mathbb{R}}^{+},p\in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\text{P}(d/p,((x-\text{Loc})/a{)}^p)=\frac{\gamma (d/p,((x-\text{Loc})/a{)}^p)}{\mathrm{\Gamma }(d/p)}$$

Probability Density Function

$$f_X\left(x\right)=\frac{p/a^d}{\mathrm{\Gamma }(d/p)}(x-\text{Loc}{)}^{d-1}{e}^{-((x-\text{Loc})/a{)}^p}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+a{\text{P}}^{-1}{(\frac{d}{p},u)}^{\frac{1}{p}}$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=a^k\frac{\mathrm{\Gamma }(\frac{d+k}{p})}{\mathrm{\Gamma }(\frac{d}{p})}$$

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

$$\mathrm{Median}(X)=\text{Loc}+a{\text{P}}^{-1}{(\frac{d}{p},\frac{1}{2})}^{\frac{1}{p}}$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+a{\left(\frac{d-1}{p}\right)}^{\frac{1}{p}}\text{if }d>1$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{GeneralizedGamma}(a,d,p)$
• $\text{Loc}:\text{Location parameter}$
• $a:\text{Scale parameter}$
• $u:\text{Uniform[0,1] random varible}$
• $\text{P}(a,x)=\frac{\gamma (a,x)}{\mathrm{\Gamma }(a)}:\text{Regularized lower incomplete gamma function}$
• ${\text{P}}^{-1}(a,u):\text{Inverse of regularized lower incomplete gamma function}$
• $\gamma (a,x):\text{Lower incomplete gamma function}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$

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