unitquantreg (version 0.0.6)

ughne: The unit-Half-Normal-E distribution

Description

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

Usage

dughne(x, mu, theta, tau = 0.5, log = FALSE)
pughne(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qughne(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rughne(n, mu, theta, tau = 0.5)

Value

dughne gives the density, pughne gives the distribution function, qughne gives the quantile function and rughne 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 $$f(y\mid \alpha ,\theta )=\sqrt{\frac{2}{\pi }}\frac{\theta }{y\left[ -\log\left( y\right) \right] }\left( -{\frac{\log \left( y\right) }{\alpha }} \right)^{\theta }\mathrm{\exp }\left\{ -\frac{1}{2}\left[ -{\frac{\log \left( y\right) }{\alpha }}\right]^{2\theta }\right\}$$

Cumulative distribution function $$F(y\mid \alpha ,\theta )=2\Phi \left[ -\left( -{\frac{\log \left( y\right) }{\alpha }}\right)^{\theta }\right]$$

Quantile function $$Q(\tau \mid \alpha ,\theta )=\exp \left\{ -\alpha \left[ -\Phi^{-1}\left(\frac{\tau }{2}\right) \right]^{\frac{1}{\theta }}\right\}$$

Reparameterization $$\alpha=g^{-1}(\mu )=-\log \left( \mu \right) \left[ -\Phi^{-1}\left( \frac{\tau }{2}\right) \right]^{-\frac{1}{\theta }}$$

References

Korkmaz, M. C., (2020). The unit generalized half normal distribution: A new bounded distribution with inference and application. University Politehnica of Bucharest Scientific, 82(2), 133--140.

Examples

set.seed(123)
x <- rughne(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-Half-Normal-E')
lines(S, dughne(x = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pughne(q = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qughne(p = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)