dcombinom gives the density; pcombinom gives the cumulative
distribution function.
Arguments
x
vector of quantiles (non-negative integers \(\le\) size).
size
vector of numbers of trials.
mu
vector of means (of the count, i.e. \(0 < \mu <\)
size).
nu
vector of dispersion parameters; \(\nu = 1\) recovers the
binomial distribution, \(\nu > 1\) gives under-dispersion,
\(0 < \nu < 1\) over-dispersion, and \(\nu < 0\) super-dispersion
(see allow_negative_nu in combinomial).
log
logical; if TRUE the log-density is returned.
q
vector of quantiles.
lower.tail
logical; if TRUE (default), probabilities are
\(P(X \le q)\).
log.p
logical; if TRUE probabilities are returned on the
log scale.
Details
The CMB distribution (Shmueli et al. 2005; Kadane 2016) has probability
mass function
$$P(X = x) = \binom{n}{x}^{\nu} p^{x} (1-p)^{n-x} / Z(p, \nu, n)$$
for \(x = 0, 1, \ldots, n\), where \(Z\) normalizes over the support.
Following the mean parameterization used by the combinomial
family (cf. Huang 2017), mu is the mean count; the natural
parameter \(p\) is recovered internally by solving
\(E[X \mid n, p, \nu] = \mu\).
References
Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., Boatwright, P. (2005).
A useful distribution for fitting discrete data: revival of the
Conway-Maxwell-Poisson distribution.
Journal of the Royal Statistical Society C54(1):127--142.
Kadane, J. B. (2016). Sums of possibly associated Bernoulli variables:
the Conway-Maxwell-Binomial distribution.
Bayesian Analysis11(2):403--420.
See Also
combinomial for the CMB family in glmmTMB;
Distributions for other standard distributions, including
dbinom for the binomial distribution.