unitquantreg (version 0.0.6)

ugompertz: The unit-Gompertz distribution

Description

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

Usage

dugompertz(x, mu, theta, tau = 0.5, log = FALSE)
pugompertz(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qugompertz(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rugompertz(n, mu, theta, tau = 0.5)

Value

dugompertz gives the density, pugompertz gives the distribution function, qugompertz gives the quantile function and rugompertz 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 $$f(y\mid \alpha ,\theta )=\frac{\alpha \theta }{x}\exp \left\{ \alpha -\theta \log \left( y\right) -\alpha \exp \left[ -\theta \log \left( y\right) \right] \right\} $$

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

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

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

References

Mazucheli, J., Menezes, A. F. and Dey, S., (2019). Unit-Gompertz Distribution with Applications. Statistica, 79(1), 25-43.

Examples

set.seed(123)
x <- rugompertz(n = 1000, mu = 0.5, theta = 2, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'unit-Gompertz')
lines(S, dugompertz(x = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pugompertz(q = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qugompertz(p = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)