cQuant function

Estimator of large quantiles using censored Hill

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

cQuant(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 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.

Details

The quantile is estimated as

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 $Z_{n-k,n}$.

Returns

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.

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.

Author(s)

Tom Reynkens.

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)