Hill.diagnostic {AoE}R Documentation

Diagnostic Plot for the Hill Estimator

Description

Computes the Hill estimator at a certain threshold and shows a quantile-quantile plot of the log-excesses over the threshold versus the exponential distribution.

Usage

Hill.diagnostic(data, k)

Arguments

data A numeric vector containing the observations.
k An integer between 1 and n-1, where n is the sample size, that is, the length of the data vector.

Details

For a sample X_1, ..., X_n with order statistics X_{1:n} <= ... <= X_{n:n} and for the given value of k = 1, ..., n-1, a scatterplot is shown of the points

(log(k+1) - log(i), log X_{n-i+1:n} - log X_{n-k:n})

for i = 1, ..., k. This is in fact a quantile-quantile plot of the excesses of the log-data over the threshold log X_{n-k:n} versus the standard exponential distribution.

The Hill estimator is defined as the mean of these log-excesses and is an estimator of the positive extreme-value index; see Hill. If the assumptions that justify the Hill estimator are justified, the points in the above scatterplot should be scattered around the line with intercept zero and slope equal to the Hill estimator. This heuristic can be transformed into an (experimental) method for automated threshold selection: see ChooseK.

Value

The function is primarily called for its side-effect, which is to show the above diagnostic plot. It silently returns a list with the following components:

x The x-coordinates of the points in the scatterplot.
y The y-coordinates of the points in the scatterplot.
H The Hill estimate.

References

Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3, 1163-1174.

See Also

ChooseK, GPD_par, Hill

Examples

x <- rburr(n = 1000, gamma = 0.5, rho = -0.5)
Hill.diagnostic(x, k = 20)  # high threshold, good fit
Hill.diagnostic(x, k = 200) # low threshold, bad fit

[Package AoE version 1.0.1 Index]