ReIns (version 1.0.7)

cProbGH: Estimator of small exceedance probabilities and large return periods using censored generalised 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 generalised Hill estimator adapted for right censoring.

Usage

cProbGH(data, censored, gamma1, q, plot = FALSE, add = FALSE, 
        main = "Estimates of small exceedance probability", ...)
cReturnGH(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 cgenHill.

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 cProbGH and "Estimates of large return period" for cReturnGH.

…

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 cProbGH.

R

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

q

The used large quantile.

Details

The probability is estimated as $$ \hat{P}(X>q)=(1-km) \times (1+ \hat{\gamma}_1/a_{k,n} \times (q-Z_{n-k,n}))^{-1/\hat{\gamma}_1}$$ 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 $$a_{k,n} = Z_{n-k,n} H_{k,n} (1-\min(\hat{\gamma}_1,0)) / \hat{p}_k$$ with \(H_{k,n}\) the ordinary Hill estimator and \(\hat{p}_k\) the proportion of the \(k\) largest observations that is non-censored.

References

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

See Also

cQuantGH, cgenHill, ProbGH, cProbMOM, 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)

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

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

# Return period
cReturnGH(Z, censored=censored, gamma1=cghill$gamma1, q=q, plot=TRUE)
# }