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, \(\tau \in (0, 1)\).
Usage
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)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.
Value
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.
Details
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 $$\alpha=g^{-1}(\mu)=\frac{\log\left ( \tau^{-1} \right )}{\log\left [ 1+\log\left ( \frac{1}{\mu} \right )^\theta \right ]}$$
References
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.
Examples
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)
