ReIns (version 1.0.10)

cQuant: 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 Hill estimator adapted for right censoring.

Usage

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

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.

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.

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.

Author

Tom Reynkens.

Details

The quantile is estimated as $$\hat{Q}(1-p)=Z_{n-k,n} \times ( (1-km)/p)^{H_{k,n}^c}$$ 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 \(Z_{n-k,n}\).

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.

See Also

cHill, cProb, 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)

# Hill estimator adapted for right censoring
chill <- cHill(Z, censored=censored, plot=TRUE)

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