Burr function

The Burr distribution

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

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 $\lambda=-1/\rho$.

Returns

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 $n$.

References

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

Author(s)

Tom Reynkens.

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")