qrmtools (version 0.0-17)

GPDtail: GPD-Based Tail Distribution (POT method)

Description

Density, distribution function, quantile function and random variate generation for the GPD-based tail distribution in the POT method.

Usage

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)

Value

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.

Arguments

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 mean(x > threshold) for x being the data.

shape

GPD shape parameter \(\xi\) (a real number).

scale

GPD scale parameter \(\beta\) (a positive number).

lower.tail

logical; if TRUE (default) probabilities are \(P(X \le x)\) otherwise, \(P(X > x)\).

log, log.p

logical; if TRUE, probabilities p are given as log(p).

Author

Marius Hofert

Details

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 \(P(X \le u)\) (1-p.exceed) at \(u\).

References

McNeil, A. J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.

Examples

## 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]