EXPONENTIATED KUMARASWAMY DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.continuous.ExponentiatedKumaraswamy({"alpha": *, "beta": *, "lambda": *, "min": *, "max": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

python

## 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 ExponentiatedKumaraswamy (α, β, λ, min, max)

Distribution Domain

x \in (min, max)

Parameters Domain and Constraints

α \in R^{+}, β \in R^{+}, λ \in R^{+}, min \in R, max \in R

Cumulative Distribution Function

F_{X} (x) = {[1 - {(1 - z (x)^{α})}^{β}]}^{λ}

Probability Density Function

f_{X} (x) = \frac{α β λ}{max - min} z (x)^{α - 1} {(1 - z (x)^{α})}^{β - 1} {[1 - {(1 - z (x)^{α})}^{β}]}^{λ - 1}

Percent Point Function / Sample

F_{X}^{- 1} (u) = min + (max - min) {[1 - {(1 - u^{1 / λ})}^{1 / β}]}^{1 / α}

Parametric Centered Moments

{\tilde{μ}}_{k}^{'} = E [{\tilde{X}}^{k}] = λ \sum_{j = 0}^{\infty} (- 1)^{j} (\binom{k / α}{j}) Beta (\frac{j}{β} + 1, λ)

Parametric Mean

Mean (X) = min + (max - min) \times {\tilde{μ}}_{1}^{'}

Parametric Variance

Variance (X) = {(max - min)}^{2} ({\tilde{μ}}_{2}^{'} - {\tilde{μ}}_{1}^{' 2})

Parametric Skewness

Skewness (X) = \frac{{\tilde{μ}}_{3}^{'} - 3 {\tilde{μ}}_{2}^{'} {\tilde{μ}}_{1}^{'} + 2 {\tilde{μ}}_{1}^{' 3}}{({\tilde{μ}}_{2}^{'} - {\tilde{μ}}_{1}^{' 2})^{1.5}}

Parametric Kurtosis

Kurtosis (X) = \frac{{\tilde{μ}}_{4}^{'} - 4 {\tilde{μ}}_{1}^{'} {\tilde{μ}}_{3}^{'} + 6 {\tilde{μ}}_{1}^{' 2} {\tilde{μ}}_{2}^{'} - 3 {\tilde{μ}}_{1}^{' 4}}{({\tilde{μ}}_{2}^{'} - {\tilde{μ}}_{1}^{' 2})^{2}}

Parametric Median

Median (X) = F_{X}^{- 1} (0.5)

Parametric Mode

Mode (X) = \arg max_{x \in (min, max)} f_{X} (x)

Additional Information and Definitions

$\tilde{X} \sim ExponentiatedKumaraswamy (α, β, λ, 0, 1)$
$z (x) = (x - min) / (max - min)$
$u : Uniform[0,1] random varible$
$Beta (x, y) : Beta function$
$(\binom{ν}{j}) : Generalized binomial coefficient$

Spreadsheet Documents

EXPONENTIATED KUMARASWAMY DISTRIBUTION ​

Phitter implementation ​

Equations ​

EXPONENTIATED KUMARASWAMY DISTRIBUTION

Phitter implementation

Equations