mean_excess {qrmtools}

Mean Excess

Description

Sample mean excess function, mean excess function of a GPD and sample mean excess plot.

Usage

mean_excess_np(x, omit = 3)
mean_excess_plot(x, omit = 3,
                 xlab = "Threshold", ylab = "Mean excess over threshold", ...)
mean_excess_GPD(x, shape, scale)

Arguments

x

mean_excess_GPD():

numeric vector of evaluation points of the mean excess function of the GPD.

otherwise:

numeric vector of data.

omit

number \ge 1 of unique last observations to be omitted from the sorted data (as mean excess plot becomes unreliable for these observations as thresholds).

xlab

x-axis label.

ylab

y-axis label.

...

additional arguments passed to the underlying plot().

shape

GPD shape parameter \xi.

scale

GPD scale parameter \beta.

Details

Mean excess plots can be used in the peaks-over-threshold method for choosing a threshold. To this end, one chooses the smallest threshold above which the mean excess plot is roughly linear.

Value

mean_excess_np() returns a two-column matrix giving the sorted data without the omit-largest unique values (first column) and the corresponding values of the sample mean excess function (second column). It is mainly used in mean_excess_plot().

mean_excess_plot() returns invisible().

mean_excess_GPD() returns the mean excess function of a generalized Pareto distribution evaluated at x.

Author(s)

Marius Hofert

Examples

## Generate losses to work with
set.seed(271)
X <- rt(1000, df = 3.5) # in MDA(H_{1/df}); see MFE (2015, Section 16.1.1)

## (Sample) mean excess plot and threshold choice
mean_excess_plot(X[X > 0]) # we only use positive values here to see 'more'
## => Any value in [0.8, 2] seems reasonable as threshold at first sight
##    but 0.8 to 1 turns out to be too small for the degrees of
##    freedom implied by the GPD estimator to be close to the true value 3.5.
## => We go with threshold 1.5 here.
u <- 1.5 # thresholds

## An alternative way
ME <- mean_excess_np(X[X > 0])
plot(ME, xlab = "Threshold", ylab = "Mean excess over threshold")

## Mean excess plot with mean excess function of the fitted GPD
fit <- fit_GPD_MLE(X[X > u] - u)
q <- seq(u, ME[nrow(ME),"x"], length.out = 129)
MEF.GPD <- mean_excess_GPD(q-u, shape = fit$par[["shape"]], scale = fit$par[["scale"]])
mean_excess_plot(X[X > 0]) # mean excess plot for positive losses...
lines(q, MEF.GPD, col = "royalblue", lwd = 1.4) # ... with mean excess function of the fitted GPD