unitquantreg (version 0.0.6)

kum: The Kumaraswamy distribution

Description

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

Usage

dkum(x, mu, theta, tau = 0.5, log = FALSE)
pkum(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qkum(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rkum(n, mu, theta, tau = 0.5)

Value

dkum gives the density, pkum gives the distribution function, qkum gives the quantile function and rkum generates random deviates.

Invalid arguments will return an error message.

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.

Author

Josmar Mazucheli

André F. B. Menezes

Details

Probability density function $$f(y\mid \alpha ,\theta )=\alpha \theta y^{\theta -1}(1-y^{\theta })^{\alpha-1}$$

Cumulative distribution function $$F(y\mid \alpha ,\theta )=1-\left( 1-y^{\theta }\right) ^{\alpha }$$

Quantile function $$Q(\tau \mid \alpha ,\theta )=\left[ 1-\left( 1-\tau \right) ^{\frac{1}{\alpha }}\right] ^{\frac{1}{\theta }}$$

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

References

Kumaraswamy, P., (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1), 79--88.

Jones, M. C., (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.

Examples

set.seed(123)
x <- rkum(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 = 'Kumaraswamy')
lines(S, dkum(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pkum(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qkum(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)