gpd: Generalised Pareto Distribution (GPD)

Description

Density, cumulative distribution function, quantile function and random number generation for the generalised Pareto distribution, either as a conditional on being above the threshold u or unconditional.

Usage

dgpd(x, u = 0, sigmau = 1, xi = 0, phiu = 1, log = FALSE)
pgpd(q, u = 0, sigmau = 1, xi = 0, phiu = 1, lower.tail = TRUE)
qgpd(p, u = 0, sigmau = 1, xi = 0, phiu = 1, lower.tail = TRUE)
rgpd(n = 1, u = 0, sigmau = 1, xi = 0, phiu = 1)

Arguments

quantiles

threshold

sigmau

scale parameter (positive)

shape parameter

phiu

probability of being above threshold $[0, 1]$

log

logical, if TRUE then log density

quantiles

lower.tail

logical, if FALSE then upper tail probabilities

cumulative probabilities

sample size (positive integer)

Value

dgpd gives the density, pgpd gives the cumulative distribution function, qgpd gives the quantile function and rgpd gives a random sample.

Acknowledgments

Based on the gpd functions in the evd package for which their author's contributions are gratefully acknowledged. They are designed to have similar syntax and functionality to simplify the transition for users of these packages.

Details

The GPD with parameters scale $σ_{u}$ and shape $ξ$ has conditional density of being above the threshold u given by

$f (x | X > u) = 1 / σ_{u} [1 + ξ (x - u) / σ_{u}]^{- 1 / ξ - 1}$

for non-zero $ξ$ , $x > u$ and $σ_{u} > 0$ . Further, $[1 + ξ (x - u) / σ_{u}] > 0$ which for $ξ < 0$ implies $u < x \leq u - σ_{u} / ξ$ . In the special case of $ξ = 0$ considered in the limit $ξ \to 0$ , which is treated here as $| ξ | < 1 e - 6$ , it reduces to the exponential:

$f (x | X > u) = 1 / σ_{u} e x p (- (x - u) / σ_{u}) .$

The unconditional density is obtained by mutltiplying this by the survival probability (or tail fraction) $ϕ_{u} = P (X > u)$ giving $f (x) = ϕ_{u} f (x | X > u)$ .

The syntax of these functions are similar to those of the evd package, so most code using these functions can be reused. The key difference is the introduction of phiu to permit output of unconditional quantities.

References

http://en.wikipedia.org/wiki/Generalized_Pareto_distribution

Hu Y. and Scarrott, C.J. (2018). evmix: An R Package for Extreme Value Mixture Modeling, Threshold Estimation and Boundary Corrected Kernel Density Estimation. Journal of Statistical Software 84(5), 1-27. doi: 10.18637/jss.v084.i05.

Coles, S.G. (2001). An Introduction to Statistical Modelling of Extreme Values. Springer Series in Statistics. Springer-Verlag: London.

Examples

Run this code

# NOT RUN {
set.seed(1)
par(mfrow = c(2, 2))

x = rgpd(1000) # simulate sample from GPD
xx = seq(-1, 10, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dgpd(xx))

# three tail behaviours
plot(xx, pgpd(xx), type = "l")
lines(xx, pgpd(xx, xi = 0.3), col = "red")
lines(xx, pgpd(xx, xi = -0.3), col = "blue")
legend("bottomright", paste("xi =",c(0, 0.3, -0.3)),
  col=c("black", "red", "blue"), lty = 1)

# GPD when xi=0 is exponential, and demonstrating phiu
x = rexp(1000)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dgpd(xx, u = 0, sigmau = 1, xi = 0), lwd = 2)
lines(xx, dgpd(xx, u = 0.5, phiu = 1 - pexp(0.5)), col = "red", lwd = 2)
lines(xx, dgpd(xx, u = 1.5, phiu = 1 - pexp(1.5)), col = "blue", lwd = 2)
legend("topright", paste("u =",c(0, 0.5, 1.5)),
  col=c("black", "red", "blue"), lty = 1, lwd = 2)

# Quantile function and phiu
p = pgpd(xx)
plot(qgpd(p), p, type = "l")
lines(xx, pgpd(xx, u = 2), col = "red")
lines(xx, pgpd(xx, u = 5, phiu = 0.2), col = "blue")
legend("bottomright", c("u = 0 phiu = 1","u = 2 phiu = 1","u = 5 phiu = 0.2"),
  col=c("black", "red", "blue"), lty = 1)
  
# }

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