GPDtail {qrmtools} | R Documentation |
Density, distribution function, quantile function and random variate generation for the GPD-based tail distribution in the POT method.
dGPDtail(x, threshold, p.exceed, shape, scale, log = FALSE) pGPDtail(q, threshold, p.exceed, shape, scale, lower.tail = TRUE, log.p = FALSE) qGPDtail(p, threshold, p.exceed, shape, scale, lower.tail = TRUE, log.p = FALSE) rGPDtail(n, threshold, p.exceed, shape, scale)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. |
threshold |
threshold u in the POT method. |
p.exceed |
probability of exceeding the threshold u; for the
Smith estimator, this is |
shape |
GPD shape parameter xi (a real number). |
scale |
GPD scale parameter beta (a positive number). |
lower.tail |
|
log, log.p |
logical; if |
Let u denote the threshold (threshold
), p_u the exceedance
probability (p.exceed
) and F_{GPD} the GPD
distribution function. Then the distribution function of the GPD-based tail
distribution is given by
F(q) = 1-p_u(1-F_{GPD}(q - u))
. The quantile function is
F^{-1}(p) = u + F_GPD^{-1}(1-(1-p)/p_u)
and the density is
f(x) = p_u f_{GPD}(x - u)
, where f_{GPD} denotes the GPD density.
Note that the distribution function has a jumpt of height P(X <=u) (1-p.exceed
) at u.
dGPDtail()
computes the density, pGPDtail()
the distribution
function, qGPDtail()
the quantile function and rGPDtail()
random
variates of the GPD-based tail distribution in the POT method.
Marius Hofert
McNeil, A. J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.
## Generate data to work with set.seed(271) X <- rt(1000, df = 3.5) # in MDA(H_{1/df}); see MFE (2015, Section 16.1.1) ## Determine thresholds for POT method mean_excess_plot(X[X > 0]) abline(v = 1.5) u <- 1.5 # threshold ## Fit GPD to the excesses (per margin) fit <- fit_GPD_MLE(X[X > u] - u) fit$par 1/fit$par["shape"] # => close to df ## Estimate threshold exceedance probabilities p.exceed <- mean(X > u) ## Define corresponding densities, distribution function and RNG dF <- function(x) dGPDtail(x, threshold = u, p.exceed = p.exceed, shape = fit$par["shape"], scale = fit$par["scale"]) pF <- function(q) pGPDtail(q, threshold = u, p.exceed = p.exceed, shape = fit$par["shape"], scale = fit$par["scale"]) rF <- function(n) rGPDtail(n, threshold = u, p.exceed = p.exceed, shape = fit$par["shape"], scale = fit$par["scale"]) ## Basic check of dF() curve(dF, from = u - 1, to = u + 5) ## Basic check of pF() curve(pF, from = u, to = u + 5, ylim = 0:1) # quite flat here abline(v = u, h = 1-p.exceed, lty = 2) # mass at u is 1-p.exceed (see 'Details') ## Basic check of rF() set.seed(271) X. <- rF(1000) plot(X., ylab = "Losses generated from the fitted GPD-based tail distribution") stopifnot(all.equal(mean(X. == u), 1-p.exceed, tol = 7e-3)) # confirms the above ## Pick out 'continuous part' X.. <- X.[X. > u] plot(pF(X..), ylab = "Probability-transformed tail losses") # should be U[1-p.exceed, 1]