DAGUM 4P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Dagum4P({"a": *, "b": *, "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

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

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)={(1+{\left(\frac{x-\text{Loc}}{b}\right)}^{-a})}^{-p}$$

Probability Density Function

$$f_X\left(x\right)=\frac{ap}{x-\text{Loc}}\left(\frac{(\frac{x-\text{Loc}}{b}{)}^{ap}}{{((\frac{x-\text{Loc}}{b}{)}^a+1)}^{p+1}}\right)$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+b(u^{-1/p}-1{)}^{-1/a}$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=p{b}^k\cdot \text{Beta}(\frac{ap+k}{a},\frac{a-k}{a})$$

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

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

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+b{\left(\frac{ap-1}{a+1}\right)}^{\frac{1}{a}}$$

Additional Information and Definitions

• $\overline{X}\sim \mathrm{Dagum}(a,b,p)$
• $\text{Loc}:\text{Location parameter}$
• $b:\text{Scale parameter}$
• $u:\text{Uniform[0,1] random varible}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

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