ReIns (version 1.0.7)

genHill: Generalised Hill estimator

Description

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

Usage

genHill(data, gamma, logk = FALSE, plot = FALSE, add = FALSE, 
        main = "Generalised Hill estimates of the EVI", ...)

Arguments

data

Vector of \(n\) observations.

gamma

Vector of \(n-1\) estimates for the EVI, typically Hill estimates are used.

logk

Logical indicating if the estimates are plotted as a function of \(\log(k)\) (logk=TRUE) or as a function of \(k\). 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 "Generalised Hill estimates of the EVI".

…

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 \(k\).

gamma

Vector of the corresponding generalised Hill estimates.

Details

The generalised Hill estimator is an estimator for the slope of the \(k\) last points of the generalised QQ-plot: $$\hat{\gamma}^{GH}_{k,n}=1/k\sum_{j=1}^k \log UH_{j,n}- \log UH_{k+1,n}$$ with \(UH_{j,n}=X_{n-j,n}H_{j,n}\) the UH scores and \(H_{j,n}\) the Hill estimates. This is analogous to the (ordinary) Hill estimator which is the estimator of the slope of the \(k\) last points of the Pareto QQ-plot when using constrained least squares.

See Section 4.2.2 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.L. (1996). "Excess Function and Estimation of the Extreme-value Index". Bernoulli, 2, 293--318.

See Also

Hill, genQQ, Moment

Examples

# NOT RUN {
data(soa)

# Hill estimator
H <- Hill(soa$size, plot=FALSE)
# Moment estimator
M <- Moment(soa$size)
# Generalised Hill estimator
gH <- genHill(soa$size, gamma=H$gamma)

# Plot estimates
plot(H$k[1:5000], M$gamma[1:5000], xlab="k", ylab=expression(gamma), type="l", ylim=c(0.2,0.5))
lines(H$k[1:5000], gH$gamma[1:5000], lty=2)
legend("topright", c("Moment", "Generalised Hill"), lty=1:2)
# }