ReIns (version 1.0.10)

cProbMOM: Estimator of small exceedance probabilities and large return periods using censored MOM

Description

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

Usage

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

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 cProbMOM.

R

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

q

The used large 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 cMoment.

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 cProbMOM and "Estimates of large return period" for cReturnMOM.

...

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

Author

Tom Reynkens

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 MOM 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

cQuantMOM, cMoment, ProbMOM, Prob, 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)

# Moment estimator adapted for right censoring
cmom <- cMoment(Z, censored=censored, plot=TRUE)

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

# Return period
cReturnMOM(Z, censored=censored, gamma1=cmom$gamma1, q=q, plot=TRUE)