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\title{The Beta-Binomial Distribution}
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Density, distribution function, and random
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generation for the beta-binomial distribution.
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dbetabin(x, size, prob, rho, log = FALSE)
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pbetabin(q, size, prob, rho, log.p = FALSE)
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rbetabin(n, size, prob, rho)
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dbetabin.ab(x, size, shape1, shape2, log = FALSE)
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pbetabin.ab(q, size, shape1, shape2, log.p = FALSE)
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rbetabin.ab(n, size, shape1, shape2)
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\item{x, q}{vector of quantiles.}
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% \item{p}{vector of probabilities.}
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\item{size}{number of trials.}
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\item{n}{number of observations.
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Must be a positive integer of length 1.}
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the probability of success \eqn{\mu}{mu}.
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Must be in the unit open interval \eqn{(0,1)}.
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the correlation parameter \eqn{\rho}{rho}.
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Must be in the unit open interval \eqn{(0,1)}.
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\item{shape1, shape2}{
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the two (positive) shape parameters of the standard
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beta distribution. They are called \code{a} and \code{b} in
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\code{\link[base:Special]{beta}} respectively.
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If \code{TRUE} then all probabilities \code{p} are given as \code{log(p)}.
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\code{dbetabin} and \code{dbetabin.ab} give the density,
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\code{pbetabin} and \code{pbetabin.ab} give the distribution function, and
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% \code{qbetabin} and \code{qbetabin.ab} gives the quantile function, and
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\code{rbetabin} and \code{rbetabin.ab} generate random deviates.
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The beta-binomial distribution is a binomial distribution whose
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probability of success is not a constant but it is generated from a
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beta distribution with parameters \code{shape1} and \code{shape2}.
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Note that the mean of this beta distribution is
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\code{mu = shape1/(shape1+shape2)}, which therefore is the
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mean or the probability of success.
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See \code{\link{betabinomial}} and \code{\link{betabin.ab}},
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the \pkg{VGAM} family functions for
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estimating the parameters, for the formula of the probability density
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function and other details.
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\code{pbetabin} and \code{pbetabin.ab} can be particularly slow.
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The functions here ending in \code{.ab} are called from those
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functions which don't.
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The simple transformations
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\eqn{\mu=\alpha / (\alpha + \beta)}{mu=alpha/(alpha+beta)} and
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\eqn{\rho=1/(1 + \alpha + \beta)}{rho=1/(1+alpha+beta)} are used,
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where \eqn{\alpha}{alpha} and \eqn{\beta}{beta} are the two
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\code{\link{betabinomial}},
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\code{\link{betabin.ab}}.
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N = 9; xx = 0:N; s1 = 2; s2 = 3
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dy = dbetabin.ab(xx, size = N, shape1 = s1, shape2 = s2)
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barplot(rbind(dy, dbinom(xx, size = N, prob = s1/(s1+s2))),
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beside = TRUE, col = c("blue","green"), las = 1,
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main = paste("Beta-binomial (size=",N,", shape1=",s1,
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", shape2=",s2,") (blue) vs\n",
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" Binomial(size=", N, ", prob=", s1/(s1+s2), ") (green)", sep = ""),
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names.arg = as.character(xx), cex.main = 0.8)
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sum(dy*xx) # Check expected values are equal
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sum(dbinom(xx, size = N, prob = s1/(s1+s2))*xx)
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cumsum(dy) - pbetabin.ab(xx, N, shape1 = s1, shape2 = s2)
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y = rbetabin.ab(n = 10000, size = N, shape1 = s1, shape2 = s2)
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barplot(rbind(dy, ty/sum(ty)),
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beside = TRUE, col = c("blue","red"), las = 1,
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main = paste("Beta-binomial (size=",N,", shape1=",s1,
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", shape2=",s2,") (blue) vs\n",
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" Random generated beta-binomial(size=", N, ", prob=", s1/(s1+s2),
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") (red)", sep = ""), cex.main = 0.8,
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names.arg = as.character(xx))
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\keyword{distribution}
added
removed
man/betabinUC.Rd
