qrmtools (version 0.0-15)

mean_excess: 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)

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.

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 \(\beta\).

Author

Marius Hofert

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.

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