kum {unitquantreg}

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)

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 )=\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 })}

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.

Author(s)

Josmar Mazucheli

André F. B. Menezes

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)