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 \(k\).
Hill(data, k = TRUE, logk = FALSE, plot = FALSE, add = FALSE,
main = "Hill estimates of the EVI", ...)
A list with following components:
Vector of the values of the tail parameter \(k\).
Vector of the corresponding Hill estimates.
Vector of \(n\) observations.
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
.
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
.
Logical indicating if the estimates should be plotted as a function of \(k\), default is FALSE
.
Logical indicating if the estimates should be added to an existing plot, default is FALSE
.
Title for the plot, default is "Hill estimates of the EVI"
.
Additional arguments for the plot
function, see plot
for more details.
Tom Reynkens based on S-Plus
code from Yuri Goegebeur.
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 $$H_{k,n}=1/k\sum_{j=1}^k \log X_{n-j+1,n}- \log X_{n-k,n}.$$
See Section 4.2.1 of Albrecher et al. (2017) for more details.
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.
ParetoQQ
, Hill.2oQV
, genHill
data(norwegianfire)
# Plot Hill estimates as a function of k
Hill(norwegianfire$size[norwegianfire$year==76],plot=TRUE)
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