EGB {robustbetareg}

The Exponential Generalized Beta of the Second Type Distribution

Description

Density, distribution function, quantile function and random generation for exponential generalized beta of the second type distribution.

Usage

dEGB(y_star, mu, phi, log = FALSE)

pEGB(q, mu, phi)

qEGB(p, mu, phi)

rEGB(n, mu, phi)

Arguments

y_star, q

vector of quantiles.

mu

mu parameter.

phi

phi parameter.

log

logical; if TRUE, probabilities p are given as log(p). Default is FALSE.

p

vector of probabilities.

n

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

Details

The EGB distribution with parameters mu = \mu and phi = \phi has density

f(y^\star;\mu,\phi)= B^{-1}(\mu\phi,(1-\mu)\phi) \exp\{-y^\star(1-\mu)\phi\}/ (1+\exp\{-y^\star\})^{\phi},

with \mu\in(0,1),\phi>0 and y^\star \in (-\infty, \infty). For this distribution, E(y^\star)=\psi(\mu\phi)-\psi((1-\mu)\phi) and Var(y^\star)=\psi'(\mu\phi)+\psi'((1-\mu)\phi), where \psi is the digamma function. See Kerman and McDonald (2015) for additional details. If y \sim beta(\mu, \phi), with \mu and \phi representing the mean and precision of y, then y^\star = \log(y/(1-y)) \sim EGB(\mu, \phi) with the density given above.

Value

dEGB gives the density, pEGB gives the distribution function, qEGB gives the quantile function, and rEGB generates random variables.

Author(s)

Yuri S. Maluf ([e-mail]), Francisco F. Queiroz ([e-mail]) and Silvia L. P. Ferrari.

References

Maluf, Y.S., Ferrari, S.L.P., and Queiroz, F.F. (2022). Robust beta regression through the logit transformation. arXiv:2209.11315.

Kerman, S. and McDonald, J.B. (2015). Skewness-kurtosis bounds for EGB1, EGB2, and special cases. Communications in Statistics - Theory and Methods, 44:3857-3864.

Examples

dEGB(0.2, mu = 0.3, phi = 1)
mu = 0.2; phi = 2;
set.seed(1)
EGBsample = rEGB(1000, mu, phi)
hist(EGBsample, prob = TRUE, breaks = 15, main = "", las = 1, ylim = c(0, 0.2),
     xlim = c(-20, 10))
curve(dEGB(x, mu, phi), from = -20, to = 8, add = TRUE, col = "red")


# Showing the P(Y* < -5) = 0.17, where Y* ~ EGB(0.2, 2).
x = seq(-20, 10,0.01)
y = dEGB(x, mu, phi)
plot(x, y, type = "l", lwd = 2, las = 1)
x1 = seq(-20, -5, 0.01)
y1 = dEGB(x1, mu, phi)
polygon(c(x1, -5, -5), c(y1, 0, 0), col = "lightblue")


plot(x, pEGB(x, mu, phi), type = "l", las = 1, lwd = 2,
     ylab = expression(P("Y*"<y)), xlab = "y")
p = pEGB(0, mu, phi)
q = qEGB(p, mu, phi)
points(q, p, pch = 16, col = 2, cex = 1.5)
text(2, 0.83, paste("(", 0, ",", round(p, 2), ")"), font = 2,
     cex = 0.8, col = "red")