Hurdle binomial distribution

Probability mass function, distribution function, and random generation for the hurdle (zero-altered) binomial distribution.

Usage

dhbinom(x, size, prob, zeroprob = 0.5, log = FALSE)

phbinom(q, size, prob, zeroprob = 0.5, lower.tail = TRUE, log.p = FALSE)

rhbinom(n, size, prob, zeroprob = 0.5)

Arguments

x, q: integer vector of counts
size: number of trials (zero or more)
prob: probability of success on each trial
zeroprob: probability of a zero, between 0 and 1
log, log.p: logical; return log-density if TRUE
lower.tail: logical; if TRUE, probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
n: number of random values to return.

Value

dhbinom gives the probability mass function, phbinom gives the distribution function, and rhbinom generates random deviates.

Details

This implementation allows for automatic differentiation with RTMB.

A hurdle distribution models the zeros and the positive counts as two separate processes: the probability of a zero is a free parameter, and the positive counts follow the corresponding zero-truncated distribution. Writing $p_0$ for zeroprob, $$P(X = 0) = p_0, \qquad P(X = x) = (1 - p_0)\,\frac{P_{\mathrm{Bin}}(x;\,n,\pi)}{1 - \pi_0}, \quad x = 1, \ldots, n.$$ where $\pi_0 = P_{\mathrm{Bin}}(0;\,n,\pi)$ is the probability of a zero under the ordinary binomial.

Unlike zero-inflation, which can only add zeros to those the binomial already produces, zeroprob here is exactly the probability of a zero and may be larger or smaller than $\pi_0$. The two coincide with the ordinary binomial when zeroprob equals $\pi_0$.

References

Mullahy, J. (1986) Specification and testing of some modified count data models. Journal of Econometrics, 33, 341-365.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

set.seed(123)
x <- rhbinom(5, size = 10, prob = 0.3, zeroprob = 0.4)
d <- dhbinom(x, size = 10, prob = 0.3, zeroprob = 0.4)
p <- phbinom(x, size = 10, prob = 0.3, zeroprob = 0.4)