pBurrXII: The Extended Burr type XII distribution

Description

Density, distribution function and quantile function for the Extended (three-parameter) Burr type XII with scale parameter lambda and shape parameters k and c.

Usage

pBurrXII(x, lambda, k, c)
	dBurrXII(x, lambda, k, c)
	qBurrXII(p, lambda, k, c)

Arguments

vector of quantiles.

vector of probabilities.

lambda, k, c

respectively the scale parameter and the two shape parameters.

Value

dBurrXII gives the density, pBurrXII gives the distribution function and qBurrXII gives the quantile function. The length of the result is determined by the length of x or p vectors.

Details

The cumulative function of the extended Burr XII distribution reads: $$P(x) = 1-\left(1-k \left(\frac{x}{\lambda}\right)^c\right)^{1/k} \quad k \neq 0$$ $$P(x) = 1 - \exp \left[ -\left(\frac{x}{\lambda}\right)^c\right] \quad k = 0$$ where $x$ is the quantile, $P$ is the non-exceedance probability, $\lambda$ is the scale parameter and $k$ and $c$ are the shape parameters.

The range of the distribution is $0 \le x \le \infty$ for $k \le 0$ and $0 \le x \le \lambda k^{-1/c}$ otherwise.

References