tea (version 1.1)

hall: A Single Bootstrap Procedure for Choosing the Optimal Sample Fraction

Description

An Implementation of the procedure proposed in Hall (1990) for selecting the optimal sample fraction in tail index estimation

Usage

hall(data, B = 1000, epsilon = 0.955, kaux = 2 * sqrt(length(data)))

Arguments

data

vector of sample data

B

number of Bootstrap replications

epsilon

gives the amount of the first resampling size n1 by choosing n1 = n^epsilon. Default is set to epsilon=0.955

kaux

tuning parameter for the hill estimator

Value

k0

optimal number of upper order statistics, i.e. number of exceedances or data in the tail

threshold

the corresponding threshold

tail.index

the corresponding tail index

Details

The Bootstrap procedure simulates the AMSE criterion of the Hill estimator. The unknown theoretical parameter of the inverse tail index gamma is replaced by a consistent estimation using a tuning parameter kaux for the Hill estimator. Minimizing this statistic gives a consistent estimator of the sample fraction k/n with k the optimal number of upper order statistics. This number, denoted k0 here, is equivalent to the number of extreme values or, if you wish, the number of exceedances in the context of a POT-model like the generalized Pareto distribution. k0 can then be associated with the unknown threshold u of the GPD by choosing u as the n-k0th upper order statistic. For more information see references.

References

Hall, P. (1990). Using the Bootstrap to Estimate Mean Squared Error and Select Smoothing Parameter in Nonparametric Problems. Journal of Multivariate Analysis, 32, 177--203.

Examples

# NOT RUN {
data(danish)
hall(danish)
# }