cHill function

Hill estimator for right censored data

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 $k$.

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

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.

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.

Returns

A list with following components: - k: Vector of the values of the tail parameter $k$.

• gamma1: Vector of the corresponding Hill estimates.

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.

Author(s)

Tom Reynkens

See Also

Hill, icHill, cParetoQQ, cProb, cQuant

Examples

# 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)