# ProbGH: Estimator of small exceedance probabilities and large return... In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

## 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

 ```1 2 3 4 5``` ```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", ...) ```

## 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.

## Details

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

## 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.

Tom Reynkens.

## 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.

`QuantGH`, `genHill`, `ProbMOM`, `Prob`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```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) ```