INVERSE GAUSSIAN 3P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.InverseGaussian3P({"mu": *, "lambda": *, "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

$$X\sim {\mathrm{InverseGaussian}}_{3\mathrm{P}}(\mu ,\lambda ,\text{Loc})$$

Distribution Domain

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

Parameters Domain and Constraints

$$\mu \in {\mathbb{R}}^{+},\lambda \in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\mathrm{\Phi }\left(\sqrt{\frac{\lambda }{x-\text{Loc}}}(\frac{x-\text{Loc}}{\mu }-1)\right)+\exp \left(\frac{2\lambda }{\mu }\right)\mathrm{\Phi }(-\sqrt{\frac{\lambda }{x-\text{Loc}}}(\frac{x-\text{Loc}}{\mu }+1))$$

Probability Density Function

$$f_X\left(x\right)=\sqrt{\frac{\lambda }{2\pi (x-\text{Loc}{)}^3}}\exp [-\frac{\lambda (x-\mu -\text{Loc}{)}^2}{2{\mu }^2(x-\text{Loc})}]$$

Percent Point Function / Sample

$${\text{Sample}}_X=\{\begin{array}{c}\text{Loc}+x_0\text{if}{u}_2⩽\frac{\mu }{\mu +x_0}\\ \text{Loc}+\frac{{\mu }^2}{x_0}\text{if}{u}_2⩾\frac{\mu }{\mu +x_0}\end{array}$$

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=\frac{{\mu }^3}{\lambda }$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=3{\left(\frac{\mu }{\lambda }\right)}^{1/2}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=3+\frac{15\mu }{\lambda }$$

Parametric Median

$$\mathrm{Median}(X)=F_X^{-1}\left(\frac{1}{2}\right)$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\mu [{(1+\frac{9{\mu }^2}{4{\lambda }^2})}^{\frac{1}{2}}-\frac{3\mu }{2\lambda }]$$

Additional Information and Definitions

• $$\begin{gathered} \text{Computing an analytic expression for the inverse of the cumulative distribution function is not} \\ \text{feasible. Nonetheless, it is possible to generate a random sample from the distribution.} \end{gathered}$$
• $\text{Loc}:\text{Location parameter}$
• $\mathrm{\Phi }\left(x\right):\text{CDF normal standard distribution}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$
• $x_0=\mu +\frac{{\mu }^2[{\mathrm{\Phi }}^{-1}\left(u_1\right){]}^2}{2\lambda }-\frac{\mu }{2\lambda }\sqrt{4\mu \lambda [{\mathrm{\Phi }}^{-1}\left(u_1\right){]}^2+{\mu }^2([{\mathrm{\Phi }}^{-1}\left(u_1\right){]}^2{)}^2}$
• $u_1:\text{Uniform[0,1] random varible}$
• $u_2:\text{Uniform[0,1] random varible}$

Spreadsheet Documents

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