gpd: Generalised Pareto Distribution (GPD)

Description

Density, cumulative distribution function, quantile function and random number generation for the GPD conditional on being above a threshold u with parameters sigmau and xi. Unconditional quantities are provided when the probability phiu of being above the threshold u is given.

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

quantile

cumulative probability

sample size (non-negative integer)

threshold

sigmau

scale parameter (non-negative)

shape parameter

phiu

probability of being above threshold [0,1]

log

logical, if TRUE then log density

lower.tail

logical, if FALSE then upper tail probabilities

Value

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

Details

The GPD with parameters scale $\sigma_u$ and shape $\xi$ has conditional density given by

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

for non-zero $\xi$, $x > u$ and $\sigma_u > 0$. Further, $[1+\xi (x - u) / \sigma_u] > 0$ which for $\xi < 0$ implies $u < x \le u - \sigma_u/\xi$. In the special case of $\xi = 0$, which is treated as $|\xi|<1e-6$, 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) $\phi_u = P(X > u)$ giving $f(x) = \phi_u f(x | X > u)$. The syntax of these functions are similar to those of the evd package, so most code using these functions can simply 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 Based on GPD functions in the evd package.

Examples

Run this code

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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