# The Variance-Inflated Beta Distribution Density function, distribution function, quantile function, and random generation for the (augmented) variance-inflated beta distribution. ```r 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) ``` ## 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. ## Returns 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. ## 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 $00$ is a precision parameter, and $0

VIB function