Create a zero-inflated negative binomial distribution

Zero-inflated negative binomial distributions are frequently used to model counts with overdispersion and many zero observations.

Usage

ZINegativeBinomial(mu, theta, pi)

Arguments

mu

Location parameter of the negative binomial component of the distribution. Can be any positive number.

theta

Overdispersion parameter of the negative binomial component of the distribution. Can be any positive number.

pi

Zero-inflation probability, can be any value in [0, 1].

Value

A ZINegativeBinomial object.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let \(X\) be a zero-inflated negative binomial random variable with parameters mu = \(\mu\) and theta = \(\theta\).

Support: \(\{0, 1, 2, 3, ...\}\)

Mean: \((1 - \pi) \cdot \mu\)

Variance: \((1 - \pi) \cdot \mu \cdot (1 + (\pi + 1/\theta) \cdot \mu)\)

Probability mass function (p.m.f.):

$$ P(X = k) = \pi \cdot I_{0}(k) + (1 - \pi) \cdot f(k; \mu, \theta) $$

where \(I_{0}(k)\) is the indicator function for zero and \(f(k; \mu, \theta)\) is the p.m.f. of the NegativeBinomial distribution.

Cumulative distribution function (c.d.f.):

$$ P(X \le k) = \pi + (1 - \pi) \cdot F(k; \mu, \theta) $$

where \(F(k; \mu, \theta)\) is the c.d.f. of the NegativeBinomial distribution.

Moment generating function (m.g.f.):

Omitted for now.

See also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X
#> [1] "ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)"

## standard functions
pdf(X, 0:8)
#> [1] 0.46428571 0.15306122 0.10932945 0.07809246 0.05578033 0.03984309 0.02845935
#> [8] 0.02032811 0.01452008
cdf(X, 0:8)
#> [1] 0.4642857 0.6173469 0.7266764 0.8047688 0.8605492 0.9003923 0.9288516
#> [8] 0.9491797 0.9636998
quantile(X, seq(0, 1, by = 0.25))
#> [1]   0   0   1   3 Inf

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))
#> [1] 3

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)


## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)