ulogistic {unitquantreg}

The unit-Logistic distribution

Description

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

Usage

dulogistic(x, mu, theta, tau = 0.5, log = FALSE)

pulogistic(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qulogistic(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rulogistic(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 \exp \left( \alpha \right) \left(\frac{y}{1-y}\right) ^{\theta -1}}{\left[ 1+\exp \left( \alpha \right)\left( \frac{y}{1-y}\right) ^{\theta }\right] ^{2}}

Cumulative distribution function

F(y\mid \alpha ,\theta )=\frac{\exp \left( \alpha \right) \left( \frac{y}{1-y}\right) ^{\theta }}{1+\exp \left( \alpha \right) \left( \frac{y}{1-y}\right) ^{\theta }}

Quantile function

Q(\tau \mid \alpha ,\theta )=\frac{\exp \left( -\frac{\alpha }{\theta }\right) \left( \frac{\tau }{1-\tau }\right) ^{\frac{1}{\theta }}}{1+\exp\left( -\frac{\alpha }{\theta }\right) \left( \frac{\tau }{1-\tau }\right) ^{ \frac{1}{\theta }}}

Reparameterization

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

Value

dulogistic gives the density, pulogistic gives the distribution function, qulogistic gives the quantile function and rulogistic generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli [e-mail]

André F. B. Menezes [e-mail]

References

Paz, R. F., Balakrishnan, N. and Bazán, J. L., 2019. L-Logistic regression models: Prior sensitivity analysis, robustness to outliers and applications. Brazilian Journal of Probability and Statistics, 33(3), 455–479.

Examples

set.seed(123)
x <- rulogistic(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-Logistic')
lines(S, dulogistic(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pulogistic(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qulogistic(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)