Reparameterised Bell distribution

Probability mass function, distribution function, quantile function, and random generation for the Bell distribution reparameterised in terms of its mean.

Usage

dbell2(x, mu, log = FALSE)

pbell2(q, mu, lower.tail = TRUE, log.p = FALSE)

qbell2(p, mu, lower.tail = TRUE, log.p = FALSE)

rbell2(n, mu)

Arguments

x, q: integer vector of counts
mu: vector of positive means
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.

Value

dbell2 gives the probability mass function, pbell2 gives the distribution function, qbell2 gives the quantile function, and rbell2 generates random deviates.

Details

This implementation of dbell2 and pbell2 allows for automatic differentiation with RTMB with respect to mu.

The Bell distribution has mean $\mu = \theta e^{\theta}$, which is a bijection from $\theta > 0$ to $\mu > 0$ and is inverted by the principal branch of the Lambert W function, $$\theta = W(\mu).$$ Every positive mean therefore corresponds to exactly one $\theta$. All four functions simply apply this transformation and hand over to their bell counterparts.

In this parameterisation the variance is $\mu (1 + W(\mu))$, so the distribution is always overdispersed relative to the Poisson distribution, but the degree of overdispersion is determined by the mean rather than by a free parameter.

lambertW is AD-compatible to arbitrary order, so mu may be a parameter of a model fitted by Laplace approximation.

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 <- rbell2(1, 3)
d <- dbell2(x, 3)
p <- pbell2(x, 3)
q <- qbell2(p, 3)

# the two parameterisations agree
all.equal(dbell2(0:5, 3), dbell(0:5, lambertW(3)))
#> [1] TRUE