# cProbGH: 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 generalised Hill estimator adapted for right censoring.

## Usage

 1 2 3 4 5 cProbGH(data, censored, gamma1, q, plot = FALSE, add = FALSE, main = "Estimates of small exceedance probability", ...) cReturnGH(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 cgenHill. 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 cProbGH and "Estimates of large return period" for cReturnGH. ... Additional arguments for the plot function, see plot for more details.

## Details

The probability is estimated as

\hat{P}(X>q)=(1-km) \times (1+ \hat{γ}_1/a_{k,n} \times (q-Z_{n-k,n}))^{-1/\hat{γ}_1}

with Z_{i,n} the i-th order statistic of the data, \hat{γ}_1 the generalised Hill estimator adapted for right censoring and km the Kaplan-Meier estimator for the CDF evaluated in Z_{n-k,n}. The value a is defined as

a_{k,n} = Z_{n-k,n} H_{k,n} (1-\min(\hat{γ}_1,0)) / \hat{p}_k

with H_{k,n} the ordinary Hill estimator and \hat{p}_k the proportion of the k largest observations that is non-censored.

## 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 cProbGH. R Vector of the corresponding estimates for the return period, only returned for cReturnGH. q The used large quantile.

Tom Reynkens

## References

Einmahl, J.H.J., Fils-Villetard, A. and Guillou, A. (2008). "Statistics of Extremes Under Random Censoring." Bernoulli, 14, 207–227.

cQuantGH, cgenHill, ProbGH, cProbMOM, 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) # Generalised Hill estimator adapted for right censoring cghill <- cgenHill(Z, censored=censored, plot=TRUE) # Small exceedance probability q <- 10 cProbGH(Z, censored=censored, gamma1=cghill$gamma1, q=q, plot=TRUE) # Return period cReturnGH(Z, censored=censored, gamma1=cghill$gamma1, q=q, plot=TRUE)