trHill: Hill estimator for upper truncated data

In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

Hill estimator for upper truncated data

Description

Computes the Hill estimator for positive extreme value indices, adapted for upper truncation, as a function of the tail parameter k (Aban et al. 2006; Beirlant et al., 2016). Optionally, these estimates are plotted as a function of k.

Usage

trHill(data, r = 1, tol = 1e-08, maxiter = 100, logk = FALSE,
       plot = FALSE, add = FALSE, main = "Estimates of the EVI", ...)

Arguments

data	Vector of n observations.
r	Trimming parameter, default is 1 (no trimming).
tol	Numerical tolerance for stopping criterion used in Newton-Raphson iterations, default is 1e-08.
maxiter	Maximum number of Newton-Raphson iterations, default is 100.
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 should be plotted as a function of k, default is FALSE.
add	Logical indicating if the estimates of \gamma should be added to an existing plot, default is FALSE.
main	Title for the plot, default is "Estimates of the EVI".
...	Additional arguments for the plot function, see plot for more details.

Details

The truncated Hill estimator is the MLE for \gamma under the truncated Pareto distribution.

To estimate the EVI using the truncated Hill estimator an equation needs to be solved. Beirlant et al. (2016) propose to use Newton-Raphson iterations to solve this equation. We take the trimmed Hill estimates as starting values for this algorithm. The trimmed Hill estimator is defined as

H_{r,k,n} = 1/(k-r+1) \sum_{j=r}^k \log(X_{n-j+1,n})-\log(X_{n-k,n})

for 1 \le r < k < n and is a basic extension of the Hill estimator for upper truncated data (the ordinary Hill estimator is obtained for r=1).

The equation that needs to be solved is

H_{r,k,n} = \gamma + R_{r,k,n}^{1/\gamma} \log(R_{r,k,n}) / (1-R_{r,k,n}^{1/\gamma})

with R_{r,k,n} = X_{n-k,n} / X_{n-r+1,n}.

See Beirlant et al. (2016) or Section 4.2.3 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 estimates for \gamma.
H	Vector of corresponding trimmed Hill estimates.

Author(s)

Tom Reynkens based on R code of Dries Cornilly.

References

Aban, I.B., Meerschaert, M.M. and Panorska, A.K. (2006). "Parameter Estimation for the Truncated Pareto Distribution." Journal of the American Statistical Association, 101, 270–277.

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

Beirlant, J., Fraga Alves, M.I. and Gomes, M.I. (2016). "Tail fitting for Truncated and Non-truncated Pareto-type Distributions." Extremes, 19, 429–462.

See Also

Hill, trDT, trEndpoint, trProb, trQuant, trMLE

Examples

# Sample from truncated Pareto distribution.
# truncated at 99% quantile
shape <- 2
X <- rtpareto(n=1000, shape=shape, endpoint=qpareto(0.99, shape=shape))

# Truncated Hill estimator
trh <- trHill(X, plot=TRUE, ylim=c(0,2))

ReIns documentation built on Nov. 3, 2023, 5:08 p.m.