bnbinom: Beta-negative binomial distribution

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Beta-negative binomial distribution

Description

Probability mass function, distribution function and random generation for the beta-negative binomial distribution.

Usage

dbnbinom(x, size, shape1, shape2, log = FALSE)

pbnbinom(q, size, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

rbnbinom(n, size, shape1, shape2)

Arguments

x	vector of non-negative counts.
size	positive number of successes (need not be an integer).
shape1	positive shape parameter 1 of the beta prior.
shape2	positive shape parameter 2 of the beta prior.
log	logical; if TRUE, probabilities are returned on the log scale.
q	vector of quantiles.
lower.tail	logical; if TRUE (default), probabilities are P[X \le q], otherwise P[X > q].
log.p	logical; if TRUE, probabilities are returned on the log scale.
n	number of random values to return (for rbnbinom).

Details

dbnbinom allows for automatic differentiation with RTMB.

The beta-negative binomial arises by giving the success probability of a negative binomial a beta prior, in the same way that the beta-binomial does for the binomial:

P(X = k;\, r, a, b) = \frac{\Gamma(k + r)}{k!\, \Gamma(r)} \frac{B(a + r,\, b + k)}{B(a,\, b)}, \quad k = 0, 1, 2, \ldots

The extra beta layer gives a much heavier tail than the negative binomial: the mean rb / (a - 1) exists only for a > 1 and the variance

\frac{r b (r + a - 1)(b + a - 1)}{(a - 2)(a - 1)^2}

only for a > 2. As a \to \infty with b/(a+b) held fixed the distribution collapses to the negative binomial.

Note that the three parameters are only weakly identified: the likelihood has a long ridge along which size and shape2 trade off against each other, so fitting all three at once needs either a lot of data or a restriction. The mean parameterisation is usually the more stable one to estimate in.

The distribution function has no closed form and is computed by summing the probability mass function over 0, \ldots, q, so its cost grows with the largest q. It is AD-compatible in the parameters, while q must be numeric data. This is also what one-step-ahead (OSA) residuals via RTMB::oneStepPredict need, so these are supported, e.g. with method = "cdf" and discrete = TRUE.

Value

dbnbinom gives the probability mass function, pbnbinom gives the distribution function, and rbnbinom generates random deviates.

References

Johnson, N. L., Kemp, A. W. and Kotz, S. (2005) Univariate Discrete Distributions, 3rd edition, Wiley, doi:10.1002/0471715816.

See Also

bnbinom2, betabinom, nbinom2

Examples

set.seed(123)
x <- rbnbinom(5, size = 3, shape1 = 4, shape2 = 2)
d <- dbnbinom(x, size = 3, shape1 = 4, shape2 = 2)
p <- pbnbinom(x, size = 3, shape1 = 4, shape2 = 2)

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.