ReIns (version 1.0.10)

trProb: Estimator of small exceedance probabilities using truncated Hill

Description

Computes estimates of a small exceedance probability \(P(X>q)\) using the estimates for the EVI obtained from the Hill estimator adapted for upper truncation.

Usage

trProb(data, r = 1, gamma, q, warnings = TRUE, plot = FALSE, add = FALSE, 
       main = "Estimates of small exceedance probability", ...)

Value

A list with following components:

k

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

P

Vector of the corresponding probability estimates.

q

The used large quantile.

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.

q

The used large quantile (we estimate \(P(X>q)\) for \(q\) large).

warnings

Logical indicating if warnings are shown, default is TRUE.

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 "Estimates of small exceedance probability".

...

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

Author

Tom Reynkens based on R code of Dries Cornilly.

Details

The probability is estimated as $$\hat{P}(X>q)=(k+1)/(n+1) ( (q/X_{n-k,n})^{-1/\gamma_k} - R_{r,k,n}^{1/\hat{\gamma}_k} ) / (1- R_{r,k,n}^{1/\hat{\gamma}_k})$$ with \(R_{r,k,n} = X_{n-k,n} / X_{n-r+1,n}\) and \(\hat{\gamma}_k\) 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, trQuant, Prob, trProbMLE

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))

# Small probability
trProb(X, gamma=trh$gamma, q=8, plot=TRUE)