bell: Bell distribution

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

Bell distribution

Description

Probability mass function, distribution function, quantile function, and random generation for the Bell distribution.

Usage

dbell(x, theta, log = FALSE)

pbell(q, theta, lower.tail = TRUE, log.p = FALSE)

qbell(p, theta, lower.tail = TRUE, log.p = FALSE)

rbell(n, theta)

Arguments

x, q	integer vector of counts
theta	vector of positive Bell parameters
log, log.p	logical; if TRUE, probabilities/ densities p are returned as \log(p).
lower.tail	logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].
p	vector of probabilities
n	number of random values to return.

Details

This implementation of dbell and pbell allows for automatic differentiation with RTMB with respect to theta.

The Bell distribution (Castellares et al. 2018) is a one-parameter distribution for overdispersed counts with probability mass function

P(X = x;\, \theta) = \frac{e^{1 - e^{\theta}}\, \theta^{x}\, B_x}{x!}, \quad x = 0, 1, 2, \ldots,

for \theta > 0, where B_x is the x-th Bell number, i.e. the number of ways a set of x elements can be partitioned into non-empty subsets.

Its mean and variance are

E(X) = \theta e^{\theta}, \qquad \mathrm{Var}(X) = \theta e^{\theta} (1 + \theta),

so the dispersion index is \mathrm{Var}(X) / E(X) = 1 + \theta > 1 and the distribution is always overdispersed. Note that, unlike the negative binomial or generalised Poisson distributions, the Bell distribution has no separate dispersion parameter: the amount of overdispersion is tied to the mean. As \theta \to 0 it approaches the Poisson distribution.

The distribution arises as a compound Poisson sum

X = \sum_{i=1}^{N} Y_i, \qquad N \sim \mathrm{Pois}(e^{\theta} - 1), \quad Y_i \sim \mathrm{ztPois}(\theta),

with the Y_i independent of N, and rbell uses exactly this representation. It is therefore infinitely divisible, and a member of the one-parameter exponential family with natural parameter \log\theta and sufficient statistic x.

The Bell numbers grow faster than any exponential and overflow double precision at x = 219, so \log B_x is used throughout rather than B_x. As x is data, \log B_x is constant with respect to the parameters and is cached across calls.

Neither the distribution function nor the quantile function has a closed form; both are obtained by summing the probability mass function, with qbell choosing its summation range automatically from the mean and variance.

See bell2 for the parameterisation by the mean.

Value

dbell gives the probability mass function, pbell gives the distribution function, qbell gives the quantile function, and rbell generates random deviates.

References

Castellares, F., Ferrari, S. L. P., and Lemonte, A. J. (2018). On the Bell distribution and its associated regression model for count data. Applied Mathematical Modelling 56, 172-185. doi:10.1016/j.apm.2017.12.014

Examples

set.seed(123)
x <- rbell(1, 1)
d <- dbell(x, 1)
p <- pbell(x, 1)
q <- qbell(p, 1)

# mean and variance
theta <- 0.8
xs <- 0:100
sum(xs * dbell(xs, theta)) # theta * exp(theta)

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