PARETO SECOND KIND DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.ParetoSecondKind({"xm": *, "alpha": *, "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

$$X\sim \mathrm{ParetoSecondKind}(x_{\mathrm{m}},\alpha ,\text{Loc})$$

Distribution Domain

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

Parameters Domain and Constraints

$$x_{\mathrm{m}}\in {\mathbb{R}}^{+},\alpha \in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R}$$

Cumulative Distribution Function

$$F_X\left(x\right)=1-{\left[1+\frac{x-\text{Loc}}{x_{\mathrm{m}}}\right]}^{-\alpha }$$

Probability Density Function

$$f_X\left(x\right)=\frac{\alpha }{x_{\mathrm{m}}}{\left[1+\frac{x-\text{Loc}}{x_{\mathrm{m}}}\right]}^{-(\alpha +1)}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+x_{\mathrm{m}}[{(1-p)}^{-\frac{1}{\alpha }}-1]$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\frac{x_{\mathrm{m}}^k\mathrm{\Gamma }(\alpha -k)\mathrm{\Gamma }(1+k)}{\mathrm{\Gamma }(\alpha )}$$

Parametric Mean

$$\mathrm{Mean}(X)={\tilde{\mu }}_1^{\prime }=\frac{x_{\mathrm{m}}}{\alpha -1}\text{if }\alpha >1$$

Parametric Variance

$$\mathrm{Variance}(X)={\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}=\frac{x_{\mathrm{m}}^2\alpha }{(\alpha -1{)}^2(\alpha -2)}\text{if }\alpha >2$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}=\frac{2(1+\alpha )}{\alpha -3}\sqrt{\frac{\alpha -2}{\alpha }}\text{if }\alpha >3$$

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=x_{\mathrm{m}}(\sqrt[\alpha ]{2}-1)$$

Parametric Mode

$$\mathrm{Mode}(X)=0$$

Additional Information and Definitions

• $X\sim \mathrm{ParetoSecondKind}(x_{\mathrm{m}},\alpha ,0)$
• $x_{\mathrm{m}}:\text{Scale parameter}$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$

Spreadsheet Documents

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