MEplot {AoE} | R Documentation |
Draws the mean-excess plot for a given sample.
MEplot(data, omit = 0, ...)
data |
The sample. |
omit |
Number of largest observations for which the mean-excess function will not be plotted. |
... |
Further arguments passed on to plot . |
The mean-excess function of the distribution of the random variable X is defined as
m(x) = E[X - x | X > x]
Its empirical counterpart, the empirical mean-excess function hat{m}(x), is defined by taking expectations with respect to the empirical distribution: for x < max_i (X_i),
hat{m}(x) = Σ_i max(X_i - x, 0) / sum_i I(X_i > x)
The mean-excess plot is the plot of the pairs
(X_i, hat{m}(X_i))
for i = 1, ..., n-1. Often, the points corresponding to the largest order statistics are omitted from the plot; this is the purpose of the argument omit
.
For a distribution with extreme-value index gamma < 1,
m(x)/x -> max(gamma/(1-gamma), 0), x -> infinity.
As a consequence, if the empirical mean-excess function is increasing for large x, then this is an indication that the underlying distribution has a heavy tail.
The function is mainly used for its side-effect, which is to plot the mean-excess function. The function invisibly returns a list with two components:
x |
The x-coordinates of the points in the mean-excess plot. |
me |
The y-coordinates of the points in the mean-excess plot. |
# for exponential data, the mean-excess function is approx. constant: x <- rexp(n = 100, rate = 1) MEplot(x) # for heavy-tailed data, the mean-excess function is increasing: x <- rburr(n = 100, gamma = 0.5, rho = -1) MEplot(x, omit = 5) # the Loss data look heavy-tailed: data(Loss) MEplot(Loss)