FlexReg (version 1.3.1)

dBeta: Probability density function of the beta distribution

Description

The function computes the probability density function of the beta distribution with a mean-precision parameterization. It can also compute the probability density function of the augmented beta distribution by assigning positive probabilities to zero and/or one values and a (continuous) beta density to the interval (0,1).

Usage

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

Value

A vector with the same length as x.

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.

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 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 \(q_0+q_1<1\).

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)