uburrxii | R Documentation |
Density function, distribution function, quantile function and random number generation function
for the unit-Burr-XII distribution reparametrized in terms of the \tau
-th quantile, \tau \in (0, 1)
.
duburrxii(x, mu, theta, tau = 0.5, log = FALSE)
puburrxii(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
quburrxii(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
ruburrxii(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{\alpha \theta }{y}\left[ -\log (y)\right]^{\theta -1}\left\{ 1+\left[ -\log (y)\right] ^{\theta }\right\} ^{-\alpha -1}
Cumulative distribution function
F(y\mid \alpha, \theta )=\left\{ 1+\left[ -\log (y)\right] ^{\theta}\right\} ^{-\alpha }
Quantile function
Q(\tau \mid \alpha, \theta )=\exp \left[ -\left( \tau ^{-\frac{1}{\alpha }}-1\right)^{\frac{1}{\theta }} \right]
Reparameterization
\alpha=g^{-1}(\mu)=\frac{\log\left ( \tau^{-1} \right )}{\log\left [ 1+\log\left ( \frac{1}{\mu} \right )^\theta \right ]}
duburrxii
gives the density, puburrxii
gives the distribution function,
quburrxii
gives the quantile function and ruburrxii
generates random deviates.
Invalid arguments will return an error message.
Josmar Mazucheli jmazucheli@gmail.com
André F. B. Menezes andrefelipemaringa@gmail.com
Korkmaz M. C. and Chesneau, C., (2021). On the unit Burr-XII distribution with the quantile regression modeling and applications. Computational and Applied Mathematics, 40(29), 1–26.
set.seed(123)
x <- ruburrxii(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-Burr-XII')
lines(S, duburrxii(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, puburrxii(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(quburrxii(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
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