cQuantGH: Estimator of large quantiles using censored Hill

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

Estimator of large quantiles using censored Hill

Description

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

Usage

cQuantGH(data, censored, gamma1, 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 cgenHill.
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)

with 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} = Z_{n-k,n} H_{k,n} (1-S_{Z,k,n}) / \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, and

S_{Z,k,n} = 1 - (1-M_1^2/M_2)^(-1) / 2

with

M_l = =1/k\sum_{j=1}^k (\log X_{n-j+1,n}- \log X_{n-k,n})^l.

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

cProbGH, cgenHill, QuantGH, 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)

# Generalised Hill estimator adapted for right censoring
cghill <- cgenHill(Z, censored=censored, plot=TRUE)

# Large quantile
p <- 10^(-4)
cQuantGH(Z, gamma1=cghill$gamma, censored=censored, p=p, plot=TRUE)

ReIns documentation built on April 3, 2025, 6:04 p.m.