Hill function

Hill estimator

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", ...)

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

See Section 4.2.1 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 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.

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

Author(s)

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

See Also

ParetoQQ, Hill.2oQV, genHill

Examples

data(norwegianfire)

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