ReIns (version 1.0.10)

ProbGPD: Estimator of small exceedance probabilities and large return periods using GPD-MLE

Description

Computes estimates of a small exceedance probability \(P(X>q)\) or large return period \(1/P(X>q)\) using the GPD fit for the peaks over a threshold.

Usage

ProbGPD(data, gamma, sigma, q, plot = FALSE, add = FALSE, 
        main = "Estimates of small exceedance probability", ...)
ReturnGPD(data, gamma, sigma, 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 ProbGPD.

R

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

q

The used large quantile.

Arguments

data

Vector of \(n\) observations.

gamma

Vector of \(n-1\) estimates for the EVI obtained from GPDmle.

sigma

Vector of \(n-1\) estimates for \(\sigma\) obtained from GPDmle.

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 ProbGPD and "Estimates of large return period" for ReturnGPD.

...

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

Author

Tom Reynkens.

Details

See Section 4.2.2 in Albrecher et al. (2017) for more details.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

See Also

QuantGPD, GPDmle, Prob

Examples

data(soa)

# Look at last 500 observations of SOA data
SOAdata <- sort(soa$size)[length(soa$size)-(0:499)]

# GPD-ML estimator
pot <- GPDmle(SOAdata)

# Exceedance probability
q <- 10^7
ProbGPD(SOAdata, gamma=pot$gamma, sigma=pot$sigma, q=q, plot=TRUE)

# Return period
q <- 10^7
ReturnGPD(SOAdata, gamma=pot$gamma, sigma=pot$sigma, q=q, plot=TRUE)