unitquantreg (version 0.0.6)

leeg: The Log-extended exponential-geometric distribution

Description

Density function, distribution function, quantile function and random number generation function for the Log-extended exponential-geometric distribution reparametrized in terms of the \(\tau\)-th quantile, \(\tau \in (0, 1)\).

Usage

dleeg(x, mu, theta, tau = 0.5, log = FALSE)
pleeg(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qleeg(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rleeg(n, mu, theta, tau = 0.5)

Value

dleeg gives the density, pleeg gives the distribution function, qleeg gives the quantile function and rleeg generates random deviates.

Invalid arguments will return an error message.

Arguments

x, q

vector of positive quantiles.

mu

location parameter indicating the \(\tau\)-th quantile, \(\tau \in (0, 1)\).

theta

nonnegative shape parameter.

tau

the parameter to specify which quantile is to be used.

log, log.p

logical; If TRUE, probabilities p are given as log(p).

lower.tail

logical; If TRUE, (default), \(P(X \leq{x})\) are returned, otherwise \(P(X > x)\).

p

vector of probabilities.

n

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

Author

Josmar Mazucheli

André F. B. Menezes

Details

Probability density function $$f(y\mid \alpha ,\theta )=\frac{\theta \left( 1+\alpha \right) y^{\theta -1}}{\left( 1+\alpha y^{\theta }\right) ^{2}}$$

Cumulative distribution function $$F(y\mid \alpha ,\theta )=\frac{\left( 1+\alpha \right) y^{\theta }}{1+\alpha y^{\theta }}$$

Quantile function $$Q(\tau \mid \alpha ,\theta )=\left[ \frac{\tau }{1+\alpha \left( 1-\tau\right) }\right] ^{\frac{1}{\theta }}$$

Reparameterization $$\alpha=g^{-1}(\mu )=-\frac{1-\tau \mu ^{\theta }}{\left( 1-\tau \right) }$$

References

Jodrá, P. and Jiménez-Gamero, M. D., (2020). A quantile regression model for bounded responses based on the exponential-geometric distribution. Revstat - Statistical Journal, 18(4), 415--436.

Examples

set.seed(123)
x <- rleeg(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'Log-extended exponential-geometric')
lines(S, dleeg(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pleeg(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qleeg(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)