ReIns (version 1.0.10)

ProbMOM: Estimator of small exceedance probabilities and large return periods using 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.

Usage

ProbMOM(data, gamma, q, plot = FALSE, add = FALSE, 
        main = "Estimates of small exceedance probability", ...)
ReturnMOM(data, gamma, 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 ProbMOM.

R

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

q

The used large quantile.

Arguments

data

Vector of \(n\) observations.

gamma

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

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 ProbMOM and "Estimates of large return period" for ReturnMOM.

...

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

Author

Tom Reynkens.

Details

See Section 4.2.2 of 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.

Dekkers, A.L.M, Einmahl, J.H.J. and de Haan, L. (1989). "A Moment Estimator for the Index of an Extreme-value Distribution." Annals of Statistics, 17, 1833--1855.

See Also

QuantMOM, Moment, ProbGH, Prob

Examples

data(soa)

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

# MOM estimator
M <- Moment(SOAdata)

# Exceedance probability
q <- 10^7
ProbMOM(SOAdata, gamma=M$gamma, q=q, plot=TRUE)

# Return period
q <- 10^7
ReturnMOM(SOAdata, gamma=M$gamma, q=q, plot=TRUE)