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)
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 |
log |
Logical indicating if the densities are given as \log(f), default is |
lower.tail |
Logical indicating if the probabilities are of the form P(X≤ x) ( |
log.p |
Logical indicating if the probabilities are given as \log(p), default is |
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.
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.
Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.
# 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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