unitquantreg (version 0.0.6)

uweibull: The unit-Weibull distribution

Description

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

Usage

duweibull(x, mu, theta, tau = 0.5, log = FALSE)
puweibull(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
quweibull(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
ruweibull(n, mu, theta, tau = 0.5)

Value

duweibull gives the density, puweibull gives the distribution function, quweibull gives the quantile function and ruweibull 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

André F. B. Menezes

Details

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

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

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

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

References

Mazucheli, J., Menezes, A. F. B and Ghitany, M. E., (2018). The unit-Weibull distribution and associated inference. Journal of Applied Probability and Statistics, 13(2), 1--22.

Mazucheli, J., Menezes, A. F. B., Fernandes, L. B., Oliveira, R. P. and Ghitany, M. E., (2020). The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. Journal of Applied Statistics, 47(6), 954--974.

Mazucheli, J., Menezes, A. F. B., Alqallaf, F. and Ghitany, M. E., (2021). Bias-Corrected Maximum Likelihood Estimators of the Parameters of the Unit-Weibull Distribution. Austrian Journal of Statistics, 50(3), 41--53.

Examples

set.seed(6969)
x <- ruweibull(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-Weibull')
lines(S, duweibull(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, puweibull(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(quweibull(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)