cQuantGH: 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", ...)

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 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.

Author

Tom Reynkens

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

References

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

Examples

Run this code

# 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)

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