Beta: The Mean-Precision Parameterized Beta Distribution

In FlexReg: Regression Models for Bounded Continuous and Discrete Responses

The Mean-Precision Parameterized Beta Distribution

Description

Density function, distribution function, quantile function, and random generation for the (augmented) beta distribution with the mean-precision parameterization.

Usage

dBeta(x, mu, phi, q0 = NULL, q1 = NULL, log = FALSE)

qBeta(prob, mu, phi, q0 = NULL, q1 = NULL, log.prob = FALSE)

pBeta(q, mu, phi, q0 = NULL, q1 = NULL, log.prob = FALSE)

rBeta(n, mu, phi, 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.
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 beta distribution has density

f_B(x;\mu,\phi)=\frac{\Gamma{(\phi)}}{\Gamma{(\mu\phi)}\Gamma{((1-\mu)\phi)}}x^{\mu\phi-1}(1-x)^{(1-\mu)\phi-1}

for 0<x<1, where 0<\mu<1 identifies the mean and \phi>0 is the precision parameter.

The augmented beta distribution has density

• q_0, if x=0

• q_1, if x=1

• (1-q_0-q_1)f_B(x;\mu,\phi), 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

The function dBeta returns a vector with the same length as x containing the density values. The function pBeta returns a vector with the same length as q containing the values of the distribution function. The function qBeta returns a vector with the same length as prob containing the quantiles. The function rBeta returns a vector of length n containing the generated random values.

References

Ferrari, S.L.P., Cribari-Neto, F. (2004). Beta Regression for Modeling Rates and Proportions. Journal of Applied Statistics, 31(7), 799–815. doi:10.1080/0266476042000214501

Examples

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

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

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

rBeta(n = 100, mu = .5, phi = 30)
rBeta(n = 100, mu = .5, phi = 30, q0 = .2, q1 = .1)

FlexReg documentation built on Sept. 9, 2025, 5:49 p.m.