unitquantreg (version 0.0.6)

ubs: The unit-Birnbaum-Saunders distribution

Description

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

Usage

dubs(x, mu, theta, tau = 0.5, log = FALSE)
pubs(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qubs(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rubs(n, mu, theta, tau = 0.5)

Value

dubs gives the density, pubs gives the distribution function, qubs gives the quantile function and rubs 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 $$f(y\mid \alpha ,\theta )=\frac{1}{2y\alpha \theta \sqrt{2\pi }}\left[\left( -\frac{\alpha }{\log (y)}\right) ^{\frac{1}{2}}+\left( -\frac{\alpha}{\log (y)}\right) ^{\frac{3}{2}}\right] \exp \left[ \frac{1}{2\theta ^{2}}\left( 2+\frac{\log (y)}{\alpha }+\frac{\alpha }{\log (y)}\right) \right]$$

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

Quantile function $$Q\left( \tau \mid \alpha ,\theta \right) ={\exp }\left\{ -{\frac{2\alpha}{2+\left[ {\theta }\Phi ^{-1}\left( 1-\tau \right) \right] ^{2}-{\theta } \Phi ^{-1}\left( 1-\tau \right) \sqrt{4+\left[ {\theta }\Phi ^{-1}\left(1-\tau \right) \right] ^{2}}}}\right\}$$

Reparameterization $$\alpha=g^{-1}(\mu )=\log \left( \mu \right) g\left( \theta ,\tau \right)$$ where \(g\left( \theta ,\tau \right) =-\frac{1}{2}\left\{ 2+\left[ {\theta }\Phi^{-1}\left( 1-\tau \right) \right] ^{2}-{\theta }\Phi ^{-1}\left( 1-\tau\right) \sqrt{4+{\theta }\Phi ^{-1}\left( 1-\tau \right) }\right\} .\)

References

Birnbaum, Z. W. and Saunders, S. C., (1969). A new family of life distributions. Journal of Applied Probability, 6(2), 637--652. Mazucheli, J., Menezes, A. F. B. and Dey, S., (2018). The unit-Birnbaum-Saunders distribution with applications. Chilean Journal of Statistics, 9(1), 47--57.

Mazucheli, J., Alves, B. and Menezes, A. F. B., (2021). A new quantile regression for modeling bounded data under a unit Birnbaum-Saunders distribution with applications. Simmetry, (), 1--28.

Examples

set.seed(123)
x <- rubs(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-Birnbaum-Saunders')
lines(S, dubs(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pubs(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qubs(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)