# cQuantGPD: Estimator of large quantiles using censored GPD-MLE In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

 cQuantGPD R Documentation

## Estimator of large quantiles using censored GPD-MLE

### Description

Computes estimates of large quantiles Q(1-p) using the estimates for the EVI obtained from the GPD-ML estimator adapted for right censoring.

### Usage

cQuantGPD(data, censored, gamma1, sigma1, p, plot = FALSE, add = FALSE,
main = "Estimates of extreme quantile", ...)

### 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 cGPDmle. sigma1 Vector of n-1 estimates for \sigma_1 obtained from cGPDmle. p The exceedance probability of the quantile (we estimate Q(1-p) for p small). 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 extreme quantile". ... Additional arguments for the plot function, see plot for more details.

### Details

The quantile is estimated as

\hat{Q}(1-p)= Z_{n-k,n} + a_{k,n} ( ( (1-km)/p)^{\hat{\gamma}_1} -1 ) / \hat{\gamma}_1)

ith Z_{i,n} the i-th order statistic of the data, \hat{\gamma}_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} = \hat{\sigma}_1 / \hat{p}_k

with \hat{\sigma}_1 the ML estimate for \sigma_1 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. Q Vector of the corresponding quantile estimates. p The used exceedance probability.

Tom Reynkens

### References

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

cProbGPD, cGPDmle, QuantGPD, Quant, KaplanMeier

### Examples

# 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)

# GPD-MLE estimator adapted for right censoring
cpot <- cGPDmle(Z, censored=censored, plot=TRUE)

# Large quantile
p <- 10^(-4)
cQuantGPD(Z, gamma1=cpot$gamma1, sigma1=cpot$sigma1,
censored=censored, p=p, plot=TRUE)

ReIns documentation built on Nov. 3, 2023, 5:08 p.m.