JOHNSON SB DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.JohnsonSB({"xi": *, "lambda": *, "gamma": *, "delta": *})

💡 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

$$X\sim \mathrm{JohnsonSB}(\xi ,\lambda ,\gamma ,\delta )$$

Distribution Domain

$$x\in (\xi ,\xi +\lambda )$$

Parameters Domain and Constraints

$$\xi \in \mathbb{R},\lambda \in {\mathbb{R}}^{+},\gamma \in \mathbb{R},\delta \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\mathrm{\Phi }(\gamma +\delta \ln \frac{z(x)}{1-z(x)})$$

Probability Density Function

$$f_X\left(x\right)=\frac{\delta }{\lambda \sqrt{2\pi }z(1-z(x))}\exp [-\frac{1}{2}{(\gamma +\delta \ln \frac{z(x)}{1-z(x)})}^2]$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\frac{\lambda \exp \left(\frac{{\mathrm{\Phi }}^{-1}(u)-\gamma }{\delta }\right)}{1+\exp \left(\frac{{\mathrm{\Phi }}^{-1}(u)-\gamma }{\delta }\right)}+\xi$$

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\frac{\lambda \exp \left(\frac{{\mathrm{\Phi }}^{-1}\left(1/2\right)-\gamma }{\delta }\right)}{1+\exp \left(\frac{{\mathrm{\Phi }}^{-1}\left(1/2\right)-\gamma }{\delta }\right)}+\xi$$

Parametric Mode

$$\mathrm{Mode}(X)=\arg \underset{x}{max}{f}_X\left(x\right)$$

Additional Information and Definitions

• $\xi :\text{Location parameter}$
• $\lambda :\text{Scale parameter}$
• $z\left(x\right)=(x-\xi )/\lambda$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Phi }\left(x\right):\text{CDF normal standard distribution}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$

Spreadsheet Documents

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