The function computes the probability mass function of the flexible beta-binomial distribution.
dFBB(x, size, mu, theta = NULL, phi = NULL, p, w)A vector with the same length as x.
a vector of quantiles.
the total number of trials.
the mean parameter. It must lie in (0, 1).
the overdispersion parameter. It must lie in (0, 1).
the precision parameter, an alternative way to specify the overdispersion parameter theta. It must be a real positive value.
the mixing weight. It must lie in (0, 1).
the normalized distance among component means. It must lie in (0, 1).
The FBB distribution is a special mixture of two beta-binomial distributions with probability mass function $$f_{FBB}(x;\mu,\phi,p,w) = p BB(x;\lambda_1,\phi)+(1-p)BB(x;\lambda_2,\phi),$$ for \(x \in \lbrace 0, 1, \dots, n \rbrace\), where \(BB(x;\cdot,\cdot)\) is the beta-binomial distribution with a mean-precision parameterization. Moreover, \(\phi=(1-\theta)/\theta>0\) is a precision parameter, \(0<p<1\) is the mixing weight, \(0<\mu=p\lambda_1+(1-p)\lambda_2<1\) is the overall mean, \(0<w<1\) is the normalized distance between component means, and \(\lambda_1=\mu+(1-p)w\) and \(\lambda_2=\mu-pw\) are the scaled means of the first and second component of the mixture, respectively.
Ascari, R., Migliorati, S. (2021). A new regression model for overdispersed binomial data accounting for outliers and an excess of zeros. Statistics in Medicine, 40(17), 3895--3914. doi:10.1002/sim.9005
dFBB(x = c(5,7,8), size=10, mu = .3, phi = 20, p = .5, w = .5)
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