ReIns: genHill – R documentation

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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.

Details

The generalised Hill estimator is an estimator for the slope of the k last points of the generalised QQ-plot:

\hat{γ}^{GH}_{k,n}=1/k∑_{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.

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)

ReIns

Functions from "Reinsurance: Actuarial and Statistical Aspects"

v1.0.10

GPL (>= 2)

Authors

Tom Reynkens [aut, cre] (<https://orcid.org/0000-0002-5516-5107>), Roel Verbelen [aut] (R code for Mixed Erlang distribution, <https://orcid.org/0000-0002-2347-9240>), Anastasios Bardoutsos [ctb] (Original R code for cEPD estimator), Dries Cornilly [ctb] (Original R code for EVT estimators for truncated data), Yuri Goegebeur [ctb] (Original S-Plus code for basic EVT estimators), Klaus Herrmann [ctb] (Original R code for GPD estimator)

Initial release

2020-05-16