gpd {eva}

The Generalized Pareto Distribution (GPD)

Description

Density, distribution function, quantile function and random number generation for the Generalized Pareto distribution with location, scale, and shape parameters.

Usage

dgpd(x, loc = 0, scale = 1, shape = 0, log.d = FALSE)

rgpd(n, loc = 0, scale = 1, shape = 0)

qgpd(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)

pgpd(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)

Arguments

x

Vector of observations.

loc, scale, shape

Location, scale, and shape parameters. Can be vectors, but the lengths must be appropriate.

log.d

Logical; if TRUE, the log density is returned.

n

Number of observations.

p

Vector of probabilities.

lower.tail

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

log.p

Logical; if TRUE, probabilities p are given as log(p).

q

Vector of quantiles.

Details

The Generalized Pareto distribution function is given (Pickands, 1975) by

H(y) = 1 - \Big[1 + \frac{\xi (y - \mu)}{\sigma}\Big]^{-1/\xi}

defined on \{y : y > 0, (1 + \xi (y - \mu) / \sigma) > 0 \}, with location \mu, scale \sigma > 0, and shape parameter \xi.

References

Pickands III, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics, 119-131.

Examples

dgpd(2:4, 1, 0.5, 0.01)
dgpd(2, -2:1, 0.5, 0.01)
pgpd(2:4, 1, 0.5, 0.01)
qgpd(seq(0.9, 0.6, -0.1), 2, 0.5, 0.01)
rgpd(6, 1, 0.5, 0.01)

# Generate sample with linear trend in location parameter
rgpd(6, 1:6, 0.5, 0.01)

# Generate sample with linear trend in location and scale parameter
rgpd(6, 1:6, seq(0.5, 3, 0.5), 0.01)

p <- (1:9)/10
pgpd(qgpd(p, 1, 2, 0.8), 1, 2, 0.8)

# Incorrect syntax (parameter vectors are of different lengths other than 1)
## Not run: 
rgpd(1, 1:8, 1:5, 0)
rgpd(10, 1:8, 1, 0.01)

## End(Not run)