unitquantreg (version 0.0.6)

ugumbel: The unit-Gumbel distribution

Description

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

Usage

dugumbel(x, mu, theta, tau = 0.5, log = FALSE)
pugumbel(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qugumbel(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rugumbel(n, mu, theta, tau = 0.5)

Value

dugumbel gives the density, pugumbel gives the distribution function, qugumbel gives the quantile function and rugumbel 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 use in the parametrization.

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

Andre F. B. Menezes

Details

Probability density function $$f(y\mid \alpha ,\theta )=\frac{\theta }{y(1-y)}\exp \left\{ -\alpha -\theta \log \left( \frac{y}{1-y}\right) -\exp \left[ -\alpha -\theta \log \left( \frac{y}{1-y}\right) \right] \right\}$$

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

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

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

where \(0<y<1\) and \(\theta >0\) is the shape parameter.

References

Mazucheli, J. and Alves, B., (2021). The unit-Gumbel Quantile Regression Model for Proportion Data. Under Review.

Gumbel, E. J., (1941). The return period of flood flows. The Annals of Mathematical Statistics, 12(2), 163--190.

Examples

set.seed(6969)
x <- rugumbel(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 = 'unit-Gumbel')
lines(S, dugumbel(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pugumbel(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qugumbel(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)