# GPD-Based Tail Distribution (POT method) Density, distribution function, quantile function and random variate generation for the GPD-based tail distribution in the POT method. ```r 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) ``` ## 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 $P(X <= x)$ otherwise, $P(X > x)$. - `log, log.p`: logical; if `TRUE`, probabilities `p` are given as `log(p)`. ## Returns `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. ## 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 $P(X <=u)$ (`1-p.exceed`) at $u$. ## Author(s) Marius Hofert ## References McNeil, A. J., Frey, R., and Embrechts, P. (2015). **Quantitative Risk Management: Concepts, Techniques, Tools**. Princeton University Press. ## Examples ```r ## 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] ```

GPDtail function