Zero-Inflated Negative Binomial Distribution#

Univariate, Discrete, Non-Negative, Zero-inflated

The Zero-Inflated Negative Binomial (ZINB) distribution is a discrete probability distribution used to model count data characterized by an excess of zeros. It combines two components: a NegativeBinomial component, which describe count values and a zero-inflation component, which accounts for the extra zeros.

Key properties and parameters#

Support	\(x \in \mathbb{N}_0\)
Mean	\(\psi\mu\)
Variance	\(\psi \left(\frac{{\mu^2}}{{\alpha}}\right) + \psi \mu + \psi \mu^2 - \psi^2 \mu^2\)

Parameters:

\(\psi\) : (float) Expected proportion of Negative Binomial variates, \(0 \leq \psi \leq 1\).
\(\mu\) : (float) Poisson distribution mean parameter, \(\mu > 0\).
\(\alpha\) : (float) Gamma distribution shape parameter, \(\alpha > 0\).
\(n\) : (int) Number of target success trials, \(n \geq 0\).
\(p\) : (float) Probability of success in each trial, \(0 \leq p \leq 1\).

Alternative Parameterization:

The ZINB distribution can be parametrized either in terms of \(\psi\), \(\mu\) and \(\alpha\) or in terms of \(\psi\), \(n\) and \(p\). The link between the two parameterizations is given by:

\[\begin{split} \begin{align*} \mu = \frac{n(1-p)}{p} \\ \alpha = n \end{align*} \end{split}\]

Probability Mass Function (PMF)#

\[\begin{split} f(x \mid \psi, \mu, \alpha) = \left\{ \begin{array}{l} (1-\psi) + \psi \left ( \frac{\alpha}{\alpha+\mu} \right) ^\alpha, \text{if } x = 0 \\ \psi \frac{\Gamma(x+\alpha)}{x! \Gamma(\alpha)} \left ( \frac{\alpha}{\mu+\alpha} \right)^\alpha \left( \frac{\mu}{\mu+\alpha} \right)^x, \text{if } x=1,2,3,\ldots \end{array} \right. \end{split}\]

Parameters \(\psi\), \(\mu\) and \(\alpha\)

../../_images/zeroinflatednegativebinomial_0_2.png

Parameters \(\psi\), \(n\) and \(p\)

../../_images/zeroinflatednegativebinomial_1_0.png

Cumulative Distribution Function (CDF)#

The CDF of the Zero-Inflated Negative Binomial distribution is given by:

\[ F(x \mid \psi, \mu, \alpha) = \sum_{i=0}^{x} f(i \mid \psi, \mu, \alpha) \]