unitquantreg (version 0.0.6)

johnsonsb: The Johnson SB distribution

Description

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

Usage

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)

Value

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.

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

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

References

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.

Examples

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)