cHill: Hill estimator for right censored data

Description

Computes the Hill estimator for positive extreme value indices, adapted for right censoring, as a function of the tail parameter \(k\) (Beirlant et al., 2007). Optionally, these estimates are plotted as a function of \(k\).

Usage

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

Value

A list with following components:

k: Vector of the values of the tail parameter \(k\).
gamma1: Vector of the corresponding Hill estimates.

Arguments

data: Vector of \(n\) observations.
censored: A logical vector of length \(n\) indicating if an observation is censored.
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 of \(\gamma_1\) should be plotted as a function of \(k\), default is FALSE.
add: Logical indicating if the estimates of \(\gamma_1\) 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.

Author

Tom Reynkens

Details

The Hill estimator adapted for right censored data is equal to the ordinary Hill estimator \(H_{k,n}\) divided by the proportion of the \(k\) largest observations that is non-censored.

This estimator is only suitable for right censored data, use icHill for interval censored data.

See Section 4.3.2 of Albrecher et al. (2017) for more details.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Beirlant, J., Guillou, A., Dierckx, G. and Fils-Villetard, A. (2007). "Estimation of the Extreme Value Index and Extreme Quantiles Under Random Censoring." Extremes, 10, 151--174.

Examples

Run this code

# Set seed
set.seed(29072016)

# Pareto random sample
X <- rpareto(500, shape=2)

# Censoring variable
Y <- rpareto(500, shape=1)

# Observed sample
Z <- pmin(X, Y)

# Censoring indicator
censored <- (X>Y)

# Hill estimator adapted for right censoring
chill <- cHill(Z, censored=censored, plot=TRUE)

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