extraDistr (version 1.8.1)

Kumaraswamy: Kumaraswamy distribution

Description

Density, distribution function, quantile function and random generation for the Kumaraswamy distribution.

Usage

dkumar(x, a = 1, b = 1, log = FALSE)
pkumar(q, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)
qkumar(p, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)
rkumar(n, a = 1, b = 1)

Arguments

x, q

vector of quantiles.

a, b

positive valued parameters.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are $P[X \le x]$ 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 $$ f(x) = abx^{a-1} (1-x^a)^{b-1} $$

Cumulative distribution function $$ F(x) = 1-(1-x^a)^b $$

Quantile function $$ F^{-1}(p) = 1-(1-p^{1/b})^{1/a} $$

References

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

Cordeiro, G.M. and de Castro, M. (2009). A new family of generalized distributions. Journal of Statistical Computation & Simulation, 1-17.

Examples

x <- rkumar(1e5, 5, 16)
xx <- seq(0, 1, by = 0.001)
hist(x, 100, freq = FALSE)
lines(xx, dkumar(xx, 5, 16), col = "red")
hist(pkumar(x, 5, 16))
plot(ecdf(x))
lines(xx, pkumar(xx, 5, 16), col = "red", lwd = 2)