Burr: The Burr distribution

In TReynkens/ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

The Burr distribution

Description

Density, distribution function, quantile function and random generation for the Burr distribution (type XII).

Usage

dburr(x, alpha, rho, eta = 1, log = FALSE)
pburr(x, alpha, rho, eta = 1, lower.tail = TRUE, log.p = FALSE)
qburr(p, alpha, rho, eta = 1, lower.tail = TRUE, log.p = FALSE)
rburr(n, alpha, rho, eta = 1)

Arguments

x	Vector of quantiles.
p	Vector of probabilities.
n	Number of observations.
alpha	The \alpha parameter of the Burr distribution, a strictly positive number.
rho	The \rho parameter of the Burr distribution, a strictly negative number.
eta	The \eta parameter of the Burr distribution, a strictly positive number. The default value is 1.
log	Logical indicating if the densities are given as \log(f), default is FALSE.
lower.tail	Logical indicating if the probabilities are of the form P(X\le x) (TRUE) or P(X>x) (FALSE). Default is TRUE.
log.p	Logical indicating if the probabilities are given as \log(p), default is FALSE.

Details

The Cumulative Distribution Function (CDF) of the Burr distribution is equal to F(x) = 1-((\eta+x^{-\rho\times\alpha})/\eta)^{1/\rho} for all x \ge 0 and F(x)=0 otherwise. We need that \alpha>0, \rho<0 and \eta>0.

Beirlant et al. (2004) uses parameters \eta, \tau, \lambda which correspond to \eta, \tau=-\rho\times\alpha and \lambda=-1/\rho.

Value

dburr gives the density function evaluated in x, pburr the CDF evaluated in x and qburr the quantile function evaluated in p. The length of the result is equal to the length of x or p.

rburr returns a random sample of length n.

Author(s)

Tom Reynkens.

References

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

See Also

tBurr, Distributions

Examples

# Plot of the PDF
x <- seq(0, 10, 0.01)
plot(x, dburr(x, alpha=2, rho=-1), xlab="x", ylab="PDF", type="l")

# Plot of the CDF
x <- seq(0, 10, 0.01)
plot(x, pburr(x, alpha=2, rho=-1), xlab="x", ylab="CDF", type="l")

TReynkens/ReIns documentation built on Nov. 30, 2024, 3:47 p.m.