# cProb: Estimator of small exceedance probabilities and large return... In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

## Description

Computes estimates of a small exceedance probability P(X>q) or large return period 1/P(X>q) using the estimates for the EVI obtained from the Hill estimator adapted for right censoring.

## Usage

 1 2 3 4 5 cProb(data, censored, gamma1, q, plot = FALSE, add = FALSE, main = "Estimates of small exceedance probability", ...) cReturn(data, censored, gamma1, q, plot = FALSE, add = FALSE, main = "Estimates of large return period", ...) 

## Arguments

 data Vector of n observations. censored A logical vector of length n indicating if an observation is censored. gamma1 Vector of n-1 estimates for the EVI obtained from cHill. q The used large quantile (we estimate P(X>q) or 1/P(X>q) for q large). 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" for cProb and "Estimates of large return period" for cReturn. ... Additional arguments for the plot function, see plot for more details.

## Details

The probability is estimated as

\hat{P}(X>q)=(1-km) \times (q/Z_{n-k,n})^{-1/H_{k,n}^c}

with Z_{i,n} the i-th order statistic of the data, H_{k,n}^c the Hill estimator adapted for right censoring and km the Kaplan-Meier estimator for the CDF evaluated in Z_{n-k,n}.

## Value

A list with following components:

 k Vector of the values of the tail parameter k. P Vector of the corresponding probability estimates, only returned for cProb. R Vector of the corresponding estimates for the return period, only returned for cReturn. q The used large quantile.

Tom Reynkens

## References

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.

cHill, cQuant, Prob, KaplanMeier
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 # 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) # Small exceedance probability q <- 10 cProb(Z, censored=censored, gamma1=chill$gamma1, q=q, plot=TRUE) # Return period cReturn(Z, censored=censored, gamma1=chill$gamma1, q=q, plot=TRUE)