# dVIB: Probability density function of the variance-inflated beta... In FlexReg: Regression Models for Bounded Continuous and Discrete Responses

 dVIB R Documentation

## 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)


### 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.

### Value

A vector with the same length as x.

### 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

### 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)



FlexReg documentation built on Sept. 29, 2023, 9:06 a.m.