ughnx {unitquantreg}

The unit-Half-Normal-X distribution

Description

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

Usage

dughnx(x, mu, theta, tau = 0.5, log = FALSE)

pughnx(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qughnx(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rughnx(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.

Details

Probability density function

f(y\mid \alpha ,\theta )=\sqrt{\frac{2}{\pi }}\frac{\theta }{y\left(1-y\right) }\left( {\frac{y}{\alpha \left( 1-y\right) }}\right) ^{\theta }\mathrm{\exp }\left\{ -\frac{1}{2}\left[ {\frac{y}{\alpha \left( 1-y\right) }}\right] ^{2\theta }\right\}

Cumulative density function

F(y\mid \alpha ,\theta )=2\Phi \left[ \left( \frac{y}{\alpha \left(1-y\right) }\right) ^{\theta }\right] -1

Quantile Function

Q(\tau \mid \alpha )=\frac{\alpha \left[ \Phi ^{-1}\left( \frac{\tau +1}{2}\right) \right] ^{\frac{1}{\theta }}}{1+\alpha \left[ \Phi ^{-1}\left( \frac{ \tau +1}{2}\right) \right] ^{\frac{1}{\theta }}}

Reparametrization

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

Value

dughnx gives the density, pughnx gives the distribution function, qughnx gives the quantile function and rughnx generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli [e-mail]

André F. B. Menezes [e-mail]

References

Bakouch, H. S., Nik, A. S., Asgharzadeh, A. and Salinas, H. S., (2021). A flexible probability model for proportion data: Unit-Half-Normal distribution. Communications in Statistics: CaseStudies, Data Analysis and Applications, 0(0), 1–18.

Examples

set.seed(123)
x <- rughnx(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-X')
lines(S, dughnx(x = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pughnx(q = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qughnx(p = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)