DescTools (version 0.99.53)

GenPareto: The Generalized Pareto Distribution

Description

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

Usage

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)

Value

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

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]

Author

Alec Stephenson <[e-mail]>

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.

References

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

See Also

rGenExtrVal

Examples

dGenPareto(2:4, 1, 0.5, 0.8)
pGenPareto(2:4, 1, 0.5, 0.8)
qGenPareto(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8)
rGenPareto(6, 1, 0.5, 0.8)
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