The function computes the probability density function of the flexible beta distribution. It can also compute the probability density function of the augmented flexible beta distribution by assigning positive probabilities to zero and/or one values and a (continuous) flexible beta density to the interval (0,1).
dFB(x, mu, phi, p, w, q0 = NULL, q1 = NULL)A vector with the same length as x.
a vector of quantiles.
the mean parameter. It must lie in (0, 1).
the precision parameter. 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 probability of augmentation in zero. It must lie in (0, 1). In case of no augmentation, it is NULL (default).
the probability of augmentation in one. It must lie in (0, 1). In case of no augmentation, it is NULL (default).
The FB distribution is a special mixture of two beta distributions with probability density function $$f_{FB}(x;\mu,\phi,p,w)=p f_B(x;\lambda_1,\phi)+(1-p)f_B(x;\lambda_2,\phi),$$ for \(0<x<1\), where \(f_B(x;\cdot,\cdot)\) is the beta density with a mean-precision parameterization. Moreover, \(0<\mu=p\lambda_1+(1-p)\lambda_2<1\) is the overall mean, \(\phi>0\) is a precision parameter, \(0<p<1\) is the mixing weight, \(0<w<1\) is the normalized distance between component means, and \(\lambda_1=\mu+(1-p)w\) and \(\lambda_2=\mu-pw\) are the means of the first and second component of the mixture, respectively.
The augmented FB distribution has density
\(q_0\), if \(x=0\)
\(q_1\), if \(x=1\)
\((1-q_0-q_1)f_{FB}(x;\mu,\phi,p,w)\), if \(0<x<1 \)
where \(0<q_0<1\) identifies the augmentation in zero, \(0<q_1<1\) identifies the augmentation in one, and \(q_0+q_1<1\).
Di Brisco, A. M., Migliorati, S. (2020). A new mixed-effects mixture model for constrained longitudinal data. Statistics in Medicine, 39(2), 129--145. doi:10.1002/sim.8406
Migliorati, S., Di Brisco, A. M., Ongaro, A. (2018). A New Regression Model for Bounded Responses. Bayesian Analysis, 13(3), 845--872. doi:10.1214/17-BA1079
dFB(x = c(.5,.7,.8), mu = .3, phi = 20, p = .5, w = .5)
dFB(x = c(.5,.7,.8), mu = .3, phi = 20, p = .5, w = .5, q0 = .2)
dFB(x = c(.5,.7,.8), mu = .3, phi = 20, p = .5, w = .5, q0 = .2, q1 = .1)
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