Probability Mass Function of the Hurdle Negative Binomial Distribution

Computes the probability (or log-probability) of observing a count $y$ under a hurdle model with a Bernoulli process for zeros and a truncated Negative Binomial distribution for positive counts ($y > 0$).

Usage

dhurdle.nb(y, mu, size, pi, log = FALSE)

Arguments

y: Numeric vector of observed count values.
mu: Mean parameter ($\mu$) of the Negative Binomial component.
size: Dispersion (shape) parameter ($r$) of the Negative Binomial component.
pi: Probability of observing a structural zero (from the hurdle component).
log: Logical; if TRUE, probabilities $p$ are returned on the log scale.

Value

A numeric vector of:

Probabilities $P(Y = y)$ if log = FALSE.
Log-probabilities $\log P(Y = y)$ if log = TRUE.

Details

The hurdle Negative Binomial (HNB) distribution models counts $Y$ as:

$$ P(Y = 0) = \pi, $$ $$ P(Y = y) = (1 - \pi) \frac{f_{NB}(y)}{1 - f_{NB}(0)}, \quad \text{for } y > 0, $$

where $f_{NB}(y)$ is the Negative Binomial probability mass function with parameters $\mu$ and $r$ (dispersion).

Internally, computations are performed on the log scale for numerical stability. The function handles vectorized inputs and returns a numeric vector of the same length as y.

Examples

# Example parameters
mu <- 2
size <- 1
pi <- 0.3

# Compute hurdle NB probabilities for counts 0–5
dhurdle.nb(0:5, mu, size, pi)
#> [1] 0.30000000 0.23333333 0.15555556 0.10370370 0.06913580 0.04609053

# Log-scale probabilities
dhurdle.nb(0:5, mu, size, pi, log = TRUE)
#> [1] -1.203973 -1.455287 -1.860752 -2.266217 -2.671683 -3.077148

# Compare structural zero vs positive counts
dhurdle.nb(0, mu, size, pi)      # P(Y = 0)
#> [1] 0.3
dhurdle.nb(1, mu, size, pi)      # P(Y = 1)
#> [1] 0.2333333