ubs {unitquantreg} | R Documentation |
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)
Arguments
x , q |
vector of positive quantiles. |
mu |
location parameter indicating the |
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 |
vector of probabilities. |
n |
number of observations. If |
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\} .
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.
Author(s)
Josmar Mazucheli jmazucheli@gmail.com
André F. B. Menezes andrefelipemaringa@gmail.com
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)