cGPD function

GPD-ML estimator for right censored data

Computes ML estimates of fitting GPD to peaks over a threshold adapted for right censoring.

cGPDmle(data, censored, start = c(0.1,1), warnings = FALSE, logk = FALSE, 
        plot = FALSE, add = FALSE, main = "POT estimates of the EVI", ...)
      
cPOT(data, censored, start = c(0.1,1), warnings = FALSE, logk = FALSE, 
     plot = FALSE, add = FALSE, main = "POT estimates of the EVI", ...)

Arguments

• data: Vector of $n$ observations.
• censored: A logical vector of length $n$ indicating if an observation is censored.
• start: Vector of length 2 containing the starting values for the optimisation. The first element is the starting value for the estimator of $\gamma_1$ and the second element is the starting value for the estimator of $\sigma_1$. Default is c(0.1,1).
• warnings: Logical indicating if possible warnings from the optimisation function are shown, default is FALSE.
• logk: Logical indicating if the estimates are plotted as a function of $\log(k)$ (logk=TRUE) or as a function of $k$. Default is FALSE.
• plot: Logical indicating if the estimates of $\gamma_1$ should be plotted as a function of $k$, default is FALSE.
• add: Logical indicating if the estimates of $\gamma_1$ should be added to an existing plot, default is FALSE.
• main: Title for the plot, default is "POT estimates of the EVI".
• ...: Additional arguments for the plot function, see plot for more details.

Details

The GPD-MLE estimator for the EVI adapted for right censored data is equal to the ordinary GPD-MLE estimator for the EVI divided by the proportion of the $k$ largest observations that is non-censored. The estimates for $\sigma$ are the ordinary GPD-MLE estimates for $\sigma$.

This estimator is only suitable for right censored data.

cPOT is the same function but with a different name for compatibility with POT.

Returns

A list with following components: - k: Vector of the values of the tail parameter $k$.

• gamma1: Vector of the corresponding MLE estimates for the $\gamma_1$ parameter of the GPD.

• sigma1: Vector of the corresponding MLE estimates for the $\sigma_1$ parameter of the GPD.

References

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

Author(s)

Tom Reynkens

See Also

GPDmle, cProbGPD, cQuantGPD, cEPD

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)

# GPD-ML estimator adapted for right censoring
cpot <- cGPDmle(Z, censored=censored, plot=TRUE)