FlexReg (version 1.4.2)

dVIB: The Variance-Inflated Beta Distribution

Description

Density function, distribution function, quantile function, and random generation for the (augmented) variance-inflated beta distribution.

Usage

dVIB(x, mu, phi, p, k, q0 = NULL, q1 = NULL, log = FALSE)
qVIB(prob, mu, phi, p, k, q0 = NULL, q1 = NULL, log.prob = FALSE)
pVIB(q, mu, phi, p, k, q0 = NULL, q1 = NULL, log.prob = FALSE)
rVIB(n, mu, phi, p, k, q0 = NULL, q1 = NULL)

Value

The function dVIB returns a vector with the same length as x containing the density values. The function pVIB returns a vector with the same length as q containing the values of the distribution function. The function qVIB returns a vector with the same length as prob containing the quantiles. The function rVIB returns a vector of length n containing the generated random values.

Arguments

x, q

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).

log

logical; if TRUE, densities are returned on log-scale.

prob

a vector of probabilities.

log.prob

logical; if TRUE, probabilities prob are given as log(prob).

n

the number of values to generate. If length(n) > 1, the length is taken to be the number required.

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

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)

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

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

rVIB(n = 100, mu = .5, phi = 30, p = .3, k = .6)
rVIB(n = 100, mu = .5, phi = 30, p = .3, k = .6, q0 = .2, q1 = .1)