dVIB: Probability density function of the variance-inflated beta distribution
Description
The function computes the probability density function of the variance-inflated beta distribution.
It can also compute the probability density function of the augmented variance-inflated beta distribution by assigning positive probabilities to zero and/or one values and a (continuous) variance-inflated beta density to the interval (0,1).
Usage
dVIB(x, mu, phi, p, k, q0 = NULL, q1 = NULL)
Value
A vector with the same length as x.
Arguments
x
a vector of quantiles.
mu
the mean parameter. It must lie in (0, 1).
phi
the precision parameter. It must be a real positive value.
p
the mixing weight. It must lie in (0, 1).
k
the extent of the variance inflation. It must lie in (0, 1).
q0
the probability of augmentation in zero. It must lie in (0, 1). In case of no augmentation, it is NULL (default).
q1
the probability of augmentation in one. It must lie in (0, 1). In case of no augmentation, it is NULL (default).
Details
The VIB distribution is a special mixture of two beta distributions with probability density function
$$f_{VIB}(x;\mu,\phi,p,k)=p f_B(x;\mu,\phi k)+(1-p)f_B(x;\mu,\phi),$$
for \(0<x<1\), where \(f_B(x;\cdot,\cdot)\) is the beta density with a mean-precision parameterization.
Moreover, \(0<p<1\) is the mixing weight, \(0<\mu<1\) represents the overall (as well as mixture component)
mean, \(\phi>0\) is a precision parameter, and \(0<k<1\) determines the extent of the variance inflation.
The augmented VIB distribution has density
\(q_0\), if \(x=0\)
\(q_1\), if \(x=1\)
\((1-q_0-q_1)f_{VIB}(x;\mu,\phi,p,k)\), if \(0<x<1\)
where \(0<q_0<1\) identifies the augmentation in zero, \(0<q_1<1\) identifies the augmentation in one,
and \(q_0+q_1<1\).
References
Di Brisco, A. M., Migliorati, S., Ongaro, A. (2020). Robustness against outliers: A new variance inflated regression model for proportions. Statistical Modelling, 20(3), 274--309.
doi:10.1177/1471082X18821213