Mean Excess Plot

Mean excess plot (also known as a mean residual life plot), a diagnostic plot for the generalized Pareto distribution (GPD).

Usage

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, ...)

Arguments

y

A numerical vector. NAs etc. are not allowed.

main, xlab, ylab

Character. Overall title for the plot, and titles for the x- and y-axes.

lty

Line type. The second value is for the mean excess value, the first and third values are for the envelope surrounding the confidence interval.

conf

Confidence level. The default results in approximate 95 percent confidence intervals for each mean excess value.

col

Colour of the three lines.

type

Type of plot. The default means lines are joined between the mean excesses and also the upper and lower limits of the confidence intervals.

object

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.

Details

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.

Value

A list is returned invisibly with the following components.

threshold

The x axis values.

meanExcess

The y axis values. Each value is a sample mean minus a value $u$.

plusminus

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.

References

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.

Author

T. W. Yee

Note

The function is designed for speed and not accuracy, therefore huge data sets with extremely large values may cause failure (the function cumsum is used.) Ties may not be well handled.

Examples

if (FALSE) meplot(with(venice90, sealevel), las = 1) -> ii
names(ii)
#> Error: object 'ii' not found
abline(h = ii$meanExcess[1], col = "orange", lty = "dashed")
#> Error: object 'ii' not found

par(mfrow = c(2, 2))
for (ii in 1:4)
  meplot(rgpd(1000), col = c("orange", "blue", "orange"))

 # \dontrun{}