The Generalised Pareto Distribution

Density function, distribution function, quantile function and random generation for the generalised Pareto (GP) distribution.

dgp(x, loc = 0, scale = 1, shape = 0, log = FALSE)

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

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

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

Arguments

x, q

Numeric vectors of quantiles. All elements of x and q must be non-negative.

loc, scale, shape

Numeric vectors. Location, scale and shape parameters. All elements of scale must be positive.

log, log.p

A logical scalar; if TRUE, probabilities p are given as log(p).

lower.tail

A logical scalar. If TRUE (default), probabilities are \(P[X \leq x]\), otherwise, \(P[X > x]\).

p

A numeric vector of probabilities in [0,1].

n

Numeric scalar. The number of observations to be simulated. If length(n) > 1 then length(n) is taken to be the number required.

Value

dgp gives the density function, pgp gives the distribution function, qgp gives the quantile function, and rgp generates random deviates.

Details

The distribution function of a GP distribution with parameters location = \(\mu\), scale = \(\sigma (> 0)\) and shape = \(\xi\) is $$F(x) = 1 - [1 + \xi (x - \mu) / \sigma] ^ {-1/\xi}$$ for \(1 + \xi (x - \mu) / \sigma > 0\). If \(\xi = 0\) the distribution function is defined as the limit as \(\xi\) tends to zero. The support of the distribution depends on \(\xi\): it is \(x \geq \mu\) for \(\xi \geq 0\); and \(\mu \leq x \leq \mu - \sigma / \xi\) for \(\xi < 0\). Note that if \(\xi < -1\) the GP density function becomes infinite as \(x\) approaches \(\mu - \sigma/\xi\).

If lower.tail = TRUE then if p = 0 (p = 1) then the lower (upper) limit of the distribution is returned. The upper limit is Inf if shape is non-negative. Similarly, but reversed, if lower.tail = FALSE.

See https://en.wikipedia.org/wiki/Generalized_Pareto_distribution for further information.

References

Pickands, J. (1975) Statistical inference using extreme order statistics. Annals of Statistics, 3, 119-131. doi:10.1214/aos/1176343003

Coles, S. G. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London. Chapter 4: doi:10.1007/978-1-4471-3675-0_4

Examples

dgp(0:4, scale = 0.5, shape = 0.8)
#> [1] 2.00000000 0.23299144 0.07919889 0.03831043 0.02214413
dgp(1:6, scale = 0.5, shape = -0.2, log = TRUE)
#> [1] -1.350155 -5.744604      -Inf      -Inf      -Inf      -Inf
dgp(1, scale = 1, shape = c(-0.2, 0.4))
#> [1] 0.4096000 0.3080008

pgp(0:4, scale = 0.5, shape = 0.8)
#> [1] 0.0000000 0.6971111 0.8336823 0.8888998 0.9180667
pgp(1:6, scale = 0.5, shape = -0.2)
#> [1] 0.92224 0.99968 1.00000 1.00000 1.00000 1.00000
pgp(1, scale = c(1, 2), shape = c(-0.2, 0.4))
#> [1] 0.6723200 0.3660619
pgp(7, scale = 1, shape = c(-0.2, 0.4))
#> [1] 1.0000000 0.9644744

qgp((0:9)/10, scale = 0.5, shape = 0.8)
#>  [1] 0.00000000 0.05496414 0.12215039 0.20638385 0.31550047 0.46318820
#>  [7] 0.67586439 1.01250643 1.63993645 3.31848340
qgp(0.5, scale = c(0.5, 1), shape = c(-0.5, 0.5))
#> [1] 0.2928932 0.8284271

p <- (1:9)/10
pgp(qgp(p, scale = 2, shape = 0.8), scale = 2, shape = 0.8)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

rgp(6, scale = 0.5, shape = 0.8)
#> [1] 1.2688250 0.4340321 2.5833754 0.3176955 0.1650756 0.4685481