cQuantGPD: Estimator of large quantiles using censored GPD-MLE

In TReynkens/ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

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.

Author(s)

Tom Reynkens

References

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

See Also

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)

TReynkens/ReIns documentation built on Nov. 9, 2023, 1:29 p.m.