Burr: The Burr distribution In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

Description

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

Usage

 1 2 3 4 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 α parameter of the Burr distribution, a strictly positive number. rho The ρ parameter of the Burr distribution, a strictly negative number. eta The η 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≤ 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-((η+x^{-ρ\timesα})/η)^{1/ρ} for all x ≥ 0 and F(x)=0 otherwise. We need that α>0, ρ<0 and η>0.

Beirlant et al. (2004) uses parameters η, τ, λ which correspond to η, τ=-ρ\timesα and λ=-1/ρ.

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.

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.