Moment function

Moment estimator

Compute the moment estimates for real extreme value indices as a function of the tail parameter $k$. Optionally, these estimates are plotted as a function of $k$.

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

Arguments

• data: Vector of $n$ observations.
• 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 "Moment estimates of the EVI".
• ...: Additional arguments for the plot function, see plot for more details.

Details

The moment estimator for the EVI is introduced by Dekkers et al. (1989) and is a generalisation of the Hill estimator.

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 moment 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.

Dekkers, A.L.M, Einmahl, J.H.J. and de Haan, L. (1989). "A Moment Estimator for the Index of an Extreme-value Distribution." Annals of Statistics, 17, 1833--1855.

Author(s)

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

See Also

Hill, genHill

Examples

data(soa)

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

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)