Reparameterised hurdle negative binomial distribution

Probability mass function, distribution function, and random generation for the hurdle (zero-altered) negative binomial distribution, parameterised by the mean of the untruncated negative binomial.

Usage

dhnbinom2(x, mu, size, zeroprob = 0.5, log = FALSE)

phnbinom2(q, mu, size, zeroprob = 0.5, lower.tail = TRUE, log.p = FALSE)

rhnbinom2(n, mu, size, zeroprob = 0.5)

Arguments

x, q: integer vector of counts
mu: mean of the untruncated negative binomial, must be strictly positive
size: dispersion parameter, must be strictly positive
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

dhnbinom2 gives the probability mass function, phnbinom2 gives the distribution function, and rhnbinom2 generates random deviates.

Details

This implementation allows for automatic differentiation with RTMB.

This is hnbinom with the success probability replaced by $$\pi = \frac{\mathrm{size}}{\mathrm{size} + \mu},$$ so that $\mu$ is the mean of the untruncated negative binomial, whose variance is $\mu + \mu^2/\mathrm{size}$. Note that $\mu$ is not the mean of the hurdle distribution itself, which also depends on zeroprob.

As for all hurdle distributions, zeroprob is exactly the probability of observing a zero and may be larger or smaller than the negative binomial would give on its own.

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 <- rhnbinom2(5, mu = 3, size = 2, zeroprob = 0.3)
d <- dhnbinom2(x, mu = 3, size = 2, zeroprob = 0.3)
p <- phnbinom2(x, mu = 3, size = 2, zeroprob = 0.3)