dVIB function

Probability density function of the variance-inflated beta distribution

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).

dVIB(x, mu, phi, p, k, q0 = NULL, q1 = NULL)

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).

Returns

A vector with the same length as x.

Details

The VIB distribution is a special mixture of two beta distributions with probability density function

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 $q_0+q_1<1$.

Examples

dVIB(x = c(.5,.7,.8), mu = .3, phi = 20, p = .5, k= .5)
dVIB(x = c(.5,.7,.8), mu = .3, phi = 20, p = .5, k= .5, q1 = .1)
dVIB(x = c(.5,.7,.8), mu = .3, phi = 20, p = .5, k= .5, q0 = .2, q1 = .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