cProb function

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

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.

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.

Details

The probability 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 $k$.

• 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.

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, cQuant, 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)

# 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)