ReIns (version 1.0.10)

ProbGH: Estimator of small exceedance probabilities and large return periods using generalised Hill

Description

Computes estimates of a small exceedance probability \(P(X>q)\) or large return period \(1/P(X>q)\) using the generalised Hill estimates for the EVI.

Usage

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

R

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

q

The used large quantile.

Arguments

data

Vector of \(n\) observations.

gamma

Vector of \(n-2\) estimates for the EVI obtained from genHill.

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 ProbGH and "Estimates of large return period" for ReturnGH.

...

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.

Beirlant, J., Vynckier, P. and Teugels, J.L. (1996). "Excess Function and Estimation of the Extreme-value Index". Bernoulli, 2, 293--318.

See Also

QuantGH, genHill, ProbMOM, Prob

Examples

data(soa)

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

# Hill estimator
H <- Hill(SOAdata)
# Generalised Hill estimator
gH <- genHill(SOAdata, H$gamma)

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

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