Hill {ReIns}

Hill estimator

Description

Computes the Hill estimator for positive extreme value indices (Hill, 1975) as a function of the tail parameter k. Optionally, these estimates are plotted as a function of k.

Usage

Hill(data, k = TRUE, logk = FALSE, plot = FALSE, add = FALSE, 
     main = "Hill estimates of the EVI", ...)

Arguments

data

Vector of n observations.

k

Logical indicating if the Hill estimates are plotted as a function of the tail parameter k (k=TRUE) or as a function of \log(X_{n-k}). Default is TRUE.

logk

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

plot

Logical indicating if the estimates should be plotted as a function of k, default is FALSE.

add

Logical indicating if the estimates should be added to an existing plot, default is FALSE.

main

Title for the plot, default is "Hill estimates of the EVI".

...

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

Details

The Hill estimator can be seen as the estimator of slope in the upper right corner (k last points) of the Pareto QQ-plot when using constrained least squares (the regression line has to pass through the point (-\log((k+1)/(n+1)),\log X_{n-k})). It is given by

H_{k,n}=1/k\sum_{j=1}^k \log X_{n-j+1,n}- \log X_{n-k,n}.

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

Value

A list with following components:

k

Vector of the values of the tail parameter k.

gamma

Vector of the corresponding Hill estimates.

Author(s)

Tom Reynkens based on S-Plus code from Yuri Goegebeur.

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.

Hill, B. M. (1975). "A simple general approach to inference about the tail of a distribution." Annals of Statistics, 3, 1163–1173.

See Also

ParetoQQ, Hill.2oQV, genHill

Examples

data(norwegianfire)

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