ReIns (version 1.0.7)

cProb: Estimator of small exceedance probabilities and large return periods using censored Hill

Description

Computes estimates of a small exceedance probability \(P(X>q)\) or large return period \(1/P(X>q)\) using the estimates for the EVI obtained from the Hill estimator adapted for right censoring.

Usage

cProb(data, censored, gamma1, q, plot = FALSE, add = FALSE, 
      main = "Estimates of small exceedance probability", ...)
      
cReturn(data, censored, gamma1, q, plot = FALSE, add = FALSE, 
        main = "Estimates of large return period", ...)

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.

q

The used large quantile (we estimate \(P(X>q)\) or \(1/P(X>q)\) for \(q\) large).

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 small exceedance probability" for cProb and "Estimates of large return period" for cReturn.

…

Additional arguments for the plot function, see plot for more details.

Value

A list with following components:

k

Vector of the values of the tail parameter \(k\).

P

Vector of the corresponding probability estimates, only returned for cProb.

R

Vector of the corresponding estimates for the return period, only returned for cReturn.

q

The used large quantile.

Details

The probability is estimated as $$ \hat{P}(X>q)=(1-km) \times (q/Z_{n-k,n})^{-1/H_{k,n}^c}$$ 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, cQuant, Prob, KaplanMeier

Examples

# NOT RUN {
# 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)

# Small exceedance probability
q <- 10
cProb(Z, censored=censored, gamma1=chill$gamma1, q=q, plot=TRUE)

# Return period
cReturn(Z, censored=censored, gamma1=chill$gamma1, q=q, plot=TRUE)
# }