The Generalized Pareto Distribution

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

dgpd(x, location = 0, scale = 1, shape = 0, log = FALSE,
     tolshape0 = sqrt(.Machine$double.eps))
pgpd(q, location = 0, scale = 1, shape = 0,
     lower.tail = TRUE, log.p = FALSE)
qgpd(p, location = 0, scale = 1, shape = 0,
     lower.tail = TRUE, log.p = FALSE)
rgpd(n, location = 0, scale = 1, shape = 0)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

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

location

the location parameter \(\mu\).

scale

the (positive) scale parameter \(\sigma\).

shape

the shape parameter \(\xi\).

log

Logical. If log = TRUE then the logarithm of the density is returned.

lower.tail, log.p

Same meaning as in punif or qunif.

tolshape0

Positive numeric. Threshold/tolerance value for resting whether \(\xi\) is zero. If the absolute value of the estimate of \(\xi\) is less than this value then it will be assumed zero and an exponential distribution will be used.

Value

dgpd gives the density, pgpd gives the distribution function, qgpd gives the quantile function, and rgpd generates random deviates.

References

Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.

Author

T. W. Yee and Kai Huang

Details

See gpd, the VGAM family function for estimating the two parameters by maximum likelihood estimation, for formulae and other details. Apart from n, all the above arguments may be vectors and are recyled to the appropriate length if necessary.

Note

The default values of all three parameters, especially \(\xi = 0\), means the default distribution is the exponential.

Currently, these functions have different argument names compared with those in the evd package.

See also

gpd, Exponential.

Examples

if (FALSE)  loc <- 2; sigma <- 1; xi <- -0.4
x <- seq(loc - 0.2, loc + 3, by = 0.01)
#> Error: object 'loc' not found
plot(x, dgpd(x, loc, sigma, xi), type = "l", col = "blue",
     main = "Blue is density, red is the CDF", ylim = c(0, 1),
     sub = "Purple are 5,10,...,95 percentiles", ylab = "", las = 1)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'x' in selecting a method for function 'plot': object 'x' not found
abline(h = 0, col = "blue", lty = 2)
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet
lines(qgpd(seq(0.05, 0.95, by = 0.05), loc, sigma, xi),
  dgpd(qgpd(seq(0.05, 0.95, by = 0.05), loc, sigma, xi), loc, sigma, xi),
      col = "purple", lty = 3, type = "h")
#> Error: object 'loc' not found
lines(x, pgpd(x, loc, sigma, xi), type = "l", col = "red")
#> Error: object 'x' not found
abline(h = 0, lty = 2)
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet

pgpd(qgpd(seq(0.05, 0.95, by = 0.05), loc, sigma, xi), loc, sigma, xi)
#> Error: object 'loc' not found
 # \dontrun{}