genHill: Generalised Hill estimator

In TReynkens/ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

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.

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.

Value

A list with following components:

k	Vector of the values of the tail parameter k.
gamma	Vector of the corresponding generalised 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.

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

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)

TReynkens/ReIns documentation built on Nov. 9, 2023, 1:29 p.m.