Beta-binomial distribution

Density, distribution function and random generation for the beta-binomial distribution.

Usage

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

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

rbetabinom(n, size, shape1, shape2)

Arguments

x

vector of non-negative counts.

size

vector of total counts (number of trials). Needs to be >= x.

shape1

positive shape parameter 1 of the Beta prior.

shape2

positive shape parameter 2 of the Beta prior.

log

logical; if TRUE, densities 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 rbetabinom).

Value

dbetabinom gives the density, pbetabinom gives the distribution function, and rbetabinom generates random samples.

Details

This implementation of dbetabinom allows for automatic differentiation with RTMB.

$$P(X = k;\, n, a, b) = \binom{n}{k} \frac{B(k+a,\, n-k+b)}{B(a,\, b)}, \quad k = 0, 1, \ldots, n.$$

The distribution function has no closed form and is computed by summing the probability mass function over \(0, \ldots, q\). It is AD-compatible in the parameters, while q and size 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.

Examples

set.seed(123)
x <- rbetabinom(1, 10, 2, 5)
d <- dbetabinom(x, 10, 2, 5)
p <- pbetabinom(x, 10, 2, 5)