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)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.
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.
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)
