Mean excess plot (also known as a mean residual life plot), a diagnostic plot for the generalized Pareto distribution (GPD).
meplot(object, ...)
meplot.default(y, main = "Mean Excess Plot",
xlab = "Threshold", ylab = "Mean Excess", lty = c(2, 1:2),
conf = 0.95, col = c("blue", "black", "blue"), type = "l", ...)
meplot.vlm(object, ...)A list is returned invisibly with the following components.
The x axis values.
The y axis values. Each value is a sample mean minus a value \(u\).
The amount which is added or subtracted
from the mean excess to give the confidence interval.
The last value is a NA because it is based
on one observation.
A numerical vector. NAs etc. are not allowed.
Character. Overall title for the plot, and titles for the x- and y-axes.
Line type. The second value is for the mean excess value, the first and third values are for the envelope surrounding the confidence interval.
Confidence level. The default results in approximate 95 percent confidence intervals for each mean excess value.
Colour of the three lines.
Type of plot. The default means lines are joined between the mean excesses and also the upper and lower limits of the confidence intervals.
An object that inherits class "vlm",
usually of class vglm-class or
vgam-class.
Graphical argument passed into
plot.
See par
for an exhaustive list.
The arguments xlim and ylim are particularly useful.
T. W. Yee
If \(Y\) has a GPD with scale parameter
\(\sigma\) and shape parameter \(\xi<1\),
and if \(y>0\), then
$$E(Y-u|Y>u) = \frac{\sigma+\xi u}{1-\xi}.$$
It is a linear function in \(u\), the threshold.
Note that \(Y-u\) is called the excess and
values of \(Y\) greater than \(u\) are
called exceedances.
The empirical versions used by these functions is to use
sample means to estimate the left hand side of the equation.
Values of \(u\) in the plot are the values of \(y\) itself.
If the plot is roughly a straight line then the GPD is a good
fit; this plot can be used to select an appropriate threshold
value. See gpd for more details.
If the plot is flat then the data may be exponential,
and if it is curved then it may be Weibull or gamma.
There is often a lot of variance/fluctuation at the RHS of the
plot due to fewer observations.
The function meplot is generic, and
meplot.default and meplot.vlm are some
methods functions for mean excess plots.
Davison, A. C. and Smith, R. L. (1990). Models for exceedances over high thresholds (with discussion). Journal of the Royal Statistical Society, Series B, Methodological, 52, 393--442.
Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.
gpd.
if (FALSE) meplot(with(venice90, sealevel), las = 1) -> ii
names(ii)
abline(h = ii$meanExcess[1], col = "orange", lty = "dashed")
par(mfrow = c(2, 2))
for (ii in 1:4)
meplot(rgpd(1000), col = c("orange", "blue", "orange"))
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