Zero-truncated beta-binomial distribution

Probability mass function, distribution function and random generation for the zero-truncated beta-binomial distribution.

Usage

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

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

rztbetabinom(n, size, shape1, shape2)

Arguments

x

integer vector of counts

size

number of trials (zero or more)

shape1, shape2

positive shape parameters of the mixing beta distribution

log

logical; return log-density if TRUE

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.

Value

dztbetabinom gives the probability mass function, pztbetabinom gives the distribution function, and rztbetabinom generates random deviates.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, ..., n). Any zero-truncated distribution is defined as $$P(X=x | X>0) = P(X=x) / (1 - P(X=0)),$$ where \(P(X=x)\) is the probability mass function of the corresponding untruncated distribution.

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.

See also

betabinom, zibetabinom, hbetabinom, ztbinom

Examples

set.seed(123)
x <- rztbetabinom(5, size = 10, shape1 = 2, shape2 = 3)
d <- dztbetabinom(x, size = 10, shape1 = 2, shape2 = 3)
p <- pztbetabinom(x, size = 10, shape1 = 2, shape2 = 3)