# trProb: Estimator of small exceedance probabilities using truncated... In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

## 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

 1 2 trProb(data, r = 1, gamma, q, warnings = TRUE, plot = FALSE, add = FALSE, main = "Estimates of small exceedance probability", ...) 

## 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.

## Details

The probability is estimated as

\hat{P}(X>q)=(k+1)/(n+1) ( (q/X_{n-k,n})^{-1/γ_k} - R_{r,k,n}^{1/\hat{γ}_k} ) / (1- R_{r,k,n}^{1/\hat{γ}_k})

with R_{r,k,n} = X_{n-k,n} / X_{n-r+1,n} and \hat{γ}_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.

## 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.

## Author(s)

Tom Reynkens based on R code of Dries Cornilly.

## 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.

trHill, trQuant, Prob, trProbMLE
  1 2 3 4 5 6 7 8 9 10 # 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)