ulogistic | R Documentation |
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)
.
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)
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 \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)
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.
Josmar Mazucheli jmazucheli@gmail.com
André F. B. Menezes andrefelipemaringa@gmail.com
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.
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)
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