Conway-Maxwell-binomial distribution — combinom • RTMBdist

Probability mass function, distribution function, quantile function, and random generation for the Conway-Maxwell-binomial (CMB) distribution.

Usage

dcombinom(x, size, prob, nu = 1, log = FALSE)

pcombinom(q, size, prob, nu = 1, lower.tail = TRUE, log.p = FALSE)

qcombinom(p, size, prob, nu = 1, lower.tail = TRUE, log.p = FALSE)

rcombinom(n, size, prob, nu = 1)

Arguments

x, q: integer vector of counts in $\{0, 1, \ldots, $size$\}$
size: vector of numbers of trials (non-negative integers)
prob: vector of success probabilities in $(0, 1)$
nu: vector of dispersion parameters; nu = 1 gives the binomial distribution, nu > 1 under-dispersion and nu < 1 over-dispersion. May be negative.
log, log.p: logical; if TRUE, probabilities/densities are returned as $\log(p)$.
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise $P[X > x]$.
p: vector of probabilities
n: number of random values to return

Value

dcombinom gives the probability mass function, pcombinom gives the distribution function, qcombinom gives the quantile function, and rcombinom generates random deviates.

Details

This implementation of dcombinom and pcombinom allows for automatic differentiation with RTMB, including differentiation with respect to x, such that one-step-ahead (OSA) residuals are supported.

The CMB distribution generalises the binomial distribution by an additional dispersion parameter $\nu$ in the same way that the Conway-Maxwell-Poisson distribution generalises the Poisson distribution. Its probability mass function is

$$P(X = x;\, n, p, \nu) = \frac{1}{Z(n, p, \nu)} \binom{n}{x}^{\nu} p^x (1-p)^{n-x}, \quad x = 0, 1, \ldots, n,$$

with normalising constant

$$Z(n, p, \nu) = \sum_{k=0}^{n} \binom{n}{k}^{\nu} p^k (1-p)^{n-k}.$$

For $\nu = 1$ this reduces to the binomial distribution. Values $\nu > 1$ give under-dispersion and $\nu < 1$ over-dispersion relative to a binomial distribution with the same mean. As the support is finite, $Z$ converges for every real $\nu$, so $\nu$ is not restricted to be positive; negative values yield strongly over-dispersed, U-shaped distributions.

The distribution arises as the sum of $n$ exchangeable, possibly associated Bernoulli variables, where $\nu$ controls the association. Note that prob is not the mean divided by size unless $\nu = 1$; the mean has no closed form for general $\nu$ and must be obtained by summation over the support.

References

Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., and Boatwright, P. (2005). A useful distribution for fitting discrete data: revival of the Conway-Maxwell-Poisson distribution. Journal of the Royal Statistical Society: Series C 54(1), 127-142.

Kadane, J. B. (2016). Sums of possibly associated Bernoulli variables: the Conway-Maxwell-binomial distribution. Bayesian Analysis 11(2), 403-420.

https://en.wikipedia.org/wiki/Conway-Maxwell-binomial_distribution

Examples

set.seed(123)
x <- rcombinom(1, size = 10, prob = 0.4, nu = 1.5)
d <- dcombinom(x, 10, 0.4, 1.5)
p <- pcombinom(x, 10, 0.4, 1.5)
q <- qcombinom(p, 10, 0.4, 1.5)

# nu = 1 recovers the binomial distribution
all.equal(dcombinom(0:10, 10, 0.3, 1), dbinom(0:10, 10, 0.3))
#> [1] TRUE