ReIns (version 1.0.7)

Hill.kopt: Select optimal threshold for Hill estimator

Description

Select optimal threshold for the Hill estimator by minimising the Asymptotic Mean Squared Error (AMSE).

Usage

Hill.kopt(data, start = c(1, 1, 1), warnings = FALSE, logk = FALSE, 
          plot = FALSE, add = FALSE, main = "AMSE plot", ...)

Arguments

data

Vector of \(n\) observations.

start

A vector of length 3 containing starting values for the first numerical optimisation (see Hill.2oQV for more details). Default is c(1,1,1).

warnings

Logical indicating if possible warnings from the optimisation function are shown, default is FALSE.

logk

Logical indicating if the AMSE values are plotted as a function of \(\log(k)\) (logk=TRUE) or as a function of \(k\). Default is FALSE.

plot

Logical indicating if the AMSE values should be plotted as a function of \(k\), default is FALSE.

add

Logical indicating if the optimal value for \(k\) should be added to an existing plot, default is FALSE.

main

Title for the plot, default is "AMSE plot".

…

Additional arguments for the plot function, see plot for more details.

Value

A list with following components:

k

Vector of the values of the tail parameter \(k\).

AMSE

Vector of the AMSE values for each value of \(k\).

kopt

Optimal value of \(k\) corresponding to minimal \(AMSE\) value.

gammaopt

Optimal value of the Hill estimator corresponding to minimal \(AMSE\) value.

Details

See Section 4.2.1 of Albrecher et al. (2017) for more details.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

Beirlant J., Vynckier, P. and Teugels, J. (1996). "Tail Index Estimation, Pareto Quantile Plots, and Regression Diagnostics." Journal of the American Statistical Association, 91, 1659--1667.

See Also

Hill, Hill.2oQV

Examples

# NOT RUN {
data(norwegianfire)

# Plot Hill estimator as a function of k
Hill(norwegianfire$size[norwegianfire$year==76],plot=TRUE)

# Add optimal value of k
Hill.kopt(norwegianfire$size[norwegianfire$year==76],add=TRUE)
# }