ReIns: Hill – R documentation

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Hill

Hill estimator

Description

Computes the Hill estimator for positive extreme value indices (Hill, 1975) as a function of the tail parameter k. Optionally, these estimates are plotted as a function of k.

Usage

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

Arguments

`data`	Vector of n observations.
`k`	Logical indicating if the Hill estimates are plotted as a function of the tail parameter k (`k=TRUE`) or as a function of \log(X_{n-k}). Default is `TRUE`.
`logk`	Logical indicating if the Hill estimates are plotted as a function of \log(k) (`logk=TRUE`) or as a function of k (`logk=FALSE`) when `k=TRUE`. 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 `"Hill estimates of the EVI"`.
`...`	Additional arguments for the `plot` function, see `plot` for more details.

Details

The Hill estimator can be seen as the estimator of slope in the upper right corner (k last points) of the Pareto QQ-plot when using constrained least squares (the regression line has to pass through the point (-\log((k+1)/(n+1)),\log X_{n-k})). It is given by

H_{k,n}=1/k∑_{j=1}^k \log X_{n-j+1,n}- \log X_{n-k,n}.

See Section 4.2.1 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 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.

Hill, B. M. (1975). "A simple general approach to inference about the tail of a distribution." Annals of Statistics, 3, 1163–1173.

Examples

data(norwegianfire)

# Plot Hill estimates as a function of k
Hill(norwegianfire$size[norwegianfire$year==76],plot=TRUE)

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