johnsonsb {unitquantreg} | R Documentation |
Density function, distribution function, quantile function and random number generation function
for the Johnson SB distribution reparametrized in terms of the \tau
-th quantile, \tau \in (0, 1)
.
djohnsonsb(x, mu, theta, tau = 0.5, log = FALSE)
pjohnsonsb(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qjohnsonsb(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rjohnsonsb(n, mu, theta, tau = 0.5)
x , q |
vector of positive quantiles. |
mu |
location parameter indicating the |
theta |
nonnegative shape parameter. |
tau |
the parameter to specify which quantile is to 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 |
Probability density function
f(y\mid \alpha ,\theta )=\frac{\theta }{\sqrt{2\pi }}\frac{1}{y(1-y)}\exp\left\{ -\frac{1}{2}\left[\alpha +\theta \log\left(\frac{y}{1-y}\right)\right] ^{2}\right\}
Cumulative distribution function
F(y\mid \alpha ,\theta )=\Phi \left[ \alpha +\theta \log \left( \frac{y}{1-y}\right) \right]
Quantile function
Q(\tau \mid \alpha ,\theta )=\frac{\exp \left[ \frac{\Phi ^{-1}(\tau)-\alpha }{\theta }\right] }{1+\exp \left[ \frac{\Phi ^{-1}(\tau )-\alpha }{\theta }\right] }
Reparameterization
\alpha =g^{-1}(\mu )=\Phi ^{-1}(\tau )-\theta \log \left( \frac{\mu }{1-\mu }\right)
djohnsonsb
gives the density, pjohnsonsb
gives the distribution function,
qjohnsonsb
gives the quantile function and rjohnsonsb
generates random deviates.
Invalid arguments will return an error message.
Josmar Mazucheli
André F. B. Menezes
Lemonte, A. J. and Bazán, J. L., (2015). New class of Johnson SB distributions and its associated regression model for rates and proportions. Biometrical Journal, 58(4), 727–746.
Johnson, N. L., (1949). Systems of frequency curves generated by methods of translation. Biometrika, 36(1), 149–176.
set.seed(123)
x <- rjohnsonsb(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 = 'Johnson SB')
lines(S, djohnsonsb(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pjohnsonsb(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qjohnsonsb(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)