The Generalized Pareto Distribution

Density function, distribution function, quantile function and random generation for the generalized Pareto distribution (GenPareto) with location, scale and shape parameters.

dGenPareto(x, loc=0, scale=1, shape=0, log = FALSE)
pGenPareto(q, loc=0, scale=1, shape=0, lower.tail = TRUE)
qGenPareto(p, loc=0, scale=1, shape=0, lower.tail = TRUE)
rGenPareto(n, loc=0, scale=1, shape=0)

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

loc, scale, shape

Location, scale and shape parameters; the shape argument cannot be a vector (must have length one).

log

Logical; if TRUE, the log density is returned.

lower.tail

Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

Details

The generalized Pareto distribution function (Pickands, 1975) with parameters \(loc = a\), \(scale = b\) and \(shape = s\) is $$G(z) = 1 - \{1+s(z-a)/b\}^{-1/s}$$ for \(1+s(z-a)/b > 0\) and \(z > a\), where \(b > 0\). If \(s = 0\) the distribution is defined by continuity.

Value

dGenPareto gives the density function, pGenPareto gives the distribution function, qGenPareto gives the quantile function, and rGenPareto generates random deviates.

References

Pickands, J. (1975) Statistical inference using Extreme Order statistics. Annals of Statistics, 3, 119--131.

See also

rGenExtrVal

Author

Alec Stephenson <alec_stephenson@hotmail.com>

Examples

dGenPareto(2:4, 1, 0.5, 0.8)
#> [1] 0.23299144 0.07919889 0.03831043
pGenPareto(2:4, 1, 0.5, 0.8)
#> [1] 0.6971111 0.8336823 0.8888998
qGenPareto(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8)
#> [1] 5.318483 3.639936 3.012506 2.675864
rGenPareto(6, 1, 0.5, 0.8)
#> [1] 4.320519 1.177982 1.073342 1.434377 1.367703 1.503896
p <- (1:9)/10
pGenPareto(qGenPareto(p, 1, 2, 0.8), 1, 2, 0.8)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9