VGAM (version 0.9-1)

gpdUC: The Generalized Pareto Distribution

Description

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.

Usage

dgpd(x, location = 0, scale = 1, shape = 0, log = FALSE,
     tolshape0 = sqrt(.Machine$double.eps),
     oobounds.log = -Inf, giveWarning = FALSE)
pgpd(q, location = 0, scale = 1, shape = 0)
qgpd(p, location = 0, scale = 1, shape = 0)
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.

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.

oobounds.log, giveWarning

Numeric and logical. The GPD distribution has support in the region satisfying (x-location)/scale > 0 and 1+shape*(x-location)/scale > 0. Outside that region, the logarithm of the density is assigned oobounds.lo

Value

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

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.

References

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

See Also

gpd.

Examples

loc <- 2; sigma <- 1; xi <- -0.4
x <- seq(loc - 0.2, loc + 3, by = 0.01)
plot(x, dgpd(x, loc, sigma, xi), type = "l", col = "blue", ylim = c(0, 1),
     main = "Blue is density, red is cumulative distribution function",
     sub = "Purple are 5,10,...,95 percentiles", ylab = "", las = 1)
abline(h = 0, col = "blue", lty = 2)
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")
lines(x, pgpd(x, loc, sigma, xi), type = "l", col = "red")
abline(h = 0, lty = 2)

pgpd(qgpd(seq(0.05, 0.95, by = 0.05), loc, sigma, xi), loc, sigma, xi)