johnsonsb {unitquantreg}

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)

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.

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)

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.

Author(s)

Josmar Mazucheli

André F. B. Menezes

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)