ugumbel {unitquantreg}

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)

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.

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.

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.

Author(s)

Josmar Mazucheli

Andre F. B. Menezes

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)