genHill function

Generalised Hill estimator

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 $k$.

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:

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.

Returns

A list with following components: - k: Vector of the values of the tail parameter $k$.

• gamma: Vector of the corresponding generalised Hill estimates.

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.

Author(s)

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

See Also

Hill, genQQ, Moment

Examples

data(soa)

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

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)