ReIns (version 1.0.15)

trEndpoint: Estimator of endpoint

Description

Estimator of endpoint using truncated Hill estimates.

Usage

trEndpoint(data, r = 1, gamma, plot = FALSE, add = FALSE,
           main = "Estimates of endpoint", ...)

Value

A list with following components:

k

Vector of the values of the tail parameter \(k\).

Tk

Vector of the corresponding estimates for the endpoint \(T\).

Arguments

data

Vector of \(n\) observations.

r

Trimming parameter, default is 1 (no trimming).

gamma

Vector of \(n-1\) estimates for the EVI obtained from trHill.

plot

Logical indicating if the estimates of \(T\) should be plotted as a function of \(k\), default is FALSE.

add

Logical indicating if the estimates of \(T\) should be added to an existing plot, default is FALSE.

main

Title for the plot, default is "Estimates of endpoint".

...

Additional arguments for the plot function, see plot for more details.

Author

Tom Reynkens based on R code of Dries Cornilly.

Details

The endpoint is estimated as $$\hat{T}_{k,n} = \max\{X_{n-k,n} ( ((X_{n-k,n}/X_{n,n})^{1/\hat{\gamma}_k} - 1/(k+1)) / (1-1/(k+1)))^{-\hat{\gamma}_k}, X_{n,n}\}$$ with \(\hat{\gamma}_k\) the Hill estimates adapted for truncation.

See Beirlant et al. (2016) or Section 4.2.3 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., 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

trHill, trDT, trEndpointMLE

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))

# Endpoint
trEndpoint(X, gamma=trh$gamma, plot=TRUE, ylim=c(8,12))
abline(h=qpareto(0.99, shape=shape), lty=2)