MCDONALD DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.continuous.McDonald({"a": *, "b": *, "c": *, "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 McDonald (a, b, c, min, max)

Distribution Domain

x \in (min, max)

Parameters Domain and Constraints

a \in R^{+}, b \in R^{+}, c \in R^{+}, min \in R, max \in R

Cumulative Distribution Function

F_{X} (x) = I_{z (x)^{c}} (a, b)

Probability Density Function

f_{X} (x) = \frac{c}{(max - min) Beta (a, b)} z (x)^{a c - 1} {(1 - z (x)^{c})}^{b - 1}

Percent Point Function / Sample

F_{X}^{- 1} (u) = min + (max - min) {[I_{u}^{- 1} (a, b)]}^{1 / c}

Parametric Centered Moments

{\tilde{μ}}_{k}^{'} = E [{\tilde{X}}^{k}] = \frac{Beta (a + \frac{k}{c}, b)}{Beta (a, b)}

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) = min + (max - min) {[I_{0.5}^{- 1} (a, b)]}^{1 / c}

Parametric Mode

Mode (X) = min + (max - min) {(\frac{a c - 1}{a c + b c - c - 1})}^{1 / c}

Additional Information and Definitions

$\tilde{X} \sim McDonald (a, b, c, 0, 1)$
$If X^{c} \sim Beta (a, b) then X \sim McDonald (a, b, c)$
$z (x) = (x - min) / (max - min)$
$u : Uniform[0,1] random varible$
$I_{x} (a, b) : Regularized incomplete beta function$
$I_{u}^{- 1} (a, b) : Inverse of regularized incomplete beta function$
$Beta (a, b) : Beta function$
$Mode formula valid when a c > 1 and b > 1$

Spreadsheet Documents

MCDONALD DISTRIBUTION ​

Phitter implementation ​

Equations ​

MCDONALD DISTRIBUTION

Phitter implementation

Equations