Burr: 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)

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 \(n\).

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.

Author

Tom Reynkens.

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 \(\lambda=-1/\rho\).

References

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

Examples

Run this code

# 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")

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