BRADFORD DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.continuous.Bradford({"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 Bradford (c, min, max)

Distribution Domain

x \in (min, max)

Parameters Domain and Constraints

c \in R^{+}, min \in R, max \in R, min < max

Cumulative Distribution Function

F_{X} (x) = \frac{\ln (1 + c \cdot z (x))}{k}

Probability Density Function

f_{X} (x) = \frac{c}{k (1 + c \cdot z (x)) (max - min)}

Percent Point Function / Sample

F_{X}^{- 1} (u) = min + (max - min) \times \frac{{(1 + c)}^{u} - 1}{c}

Parametric Centered Moments

{\tilde{μ}}_{k}^{'} = E [{\tilde{X}}^{k}] = \int_{0}^{1} x^{k} f_{\tilde{X}} (x) d x

Parametric Mean

Mean (X) = min + (max - min) \cdot {\tilde{μ}}_{1}^{'} = min + (max - min) \cdot \frac{c - k}{c k}

Parametric Variance

Variance (X) = {(max - min)}^{2} \cdot ({\tilde{μ}}_{2}^{'} - {\tilde{μ}}_{1}^{' 2}) = {(max - min)}^{2} \cdot \frac{(c + 2) k - 2 c}{2 c k^{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}} = \frac{\sqrt{2} (12 c^{2} - 9 k c (c + 2) + 2 k^{2} (c (c + 3) + 3))}{\sqrt{c (c (k - 2) + 2 k)} (3 c (k - 2) + 6 k)}

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}} = 3 + \frac{c^{3} (k - 3) (k (3 k - 16) + 24) + 12 k c^{2} (k - 4) (k - 3) + 6 c k^{2} (3 k - 14) + 12 k^{3}}{3 c {(c (k - 2) + 2 k)}^{2}}

Parametric Median

Median (X) = min + (max - min) \cdot \frac{{(1 + c)}^{f r a c 12} - 1}{c}

Parametric Mode

Mode (X) = min

Additional Information and Definitions

$\tilde{X} \sim Bradford (c, 0, 1)$
$k = \ln (1 + c)$
$z (x) = (x - min) / (max - min)$
$u : Uniform[0,1] random varible$

Spreadsheet Documents

BRADFORD DISTRIBUTION ​

Phitter implementation ​

Equations ​

BRADFORD DISTRIBUTION

Phitter implementation

Equations