Create a hurdle negative binomial distribution

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

Usage

HurdleNegativeBinomial(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-hurdle probability, can be any value in [0, 1].

Value

A HurdleNegativeBinomial 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 hurdle negative binomial random variable with parameters mu = \(\mu\) and theta = \(\theta\).

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

Mean: $$ \mu \cdot \frac{\pi}{1 - F(0; \mu, \theta)} $$

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

Variance: $$ m \cdot \left(1 + \frac{\mu}{\theta} + \mu - m \right) $$

where \(m\) is the mean above.

Probability mass function (p.m.f.): \(P(X = 0) = 1 - \pi\) and for \(k > 0\)

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

where \(f(k; \mu, \theta)\) is the p.m.f. of the NegativeBinomial distribution.

Cumulative distribution function (c.d.f.): \(P(X \le 0) = 1 - \pi\) and for \(k > 0\)

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

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

Omitted for now.

See also

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

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X
#> [1] "HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)"

## standard functions
pdf(X, 0:8)
#> [1] 0.25000000 0.21428571 0.15306122 0.10932945 0.07809246 0.05578033 0.03984309
#> [8] 0.02845935 0.02032811
cdf(X, 0:8)
#> [1] 0.2500000 0.4642857 0.6173469 0.7266764 0.8047688 0.8605492 0.9003923
#> [8] 0.9288516 0.9491797
quantile(X, seq(0, 1, by = 0.25))
#> [1]   0   0   2   4 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)