# ProbMOM: 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 Method of Moments estimates for the EVI.

## Usage

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

## 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.

## 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 `ProbMOM`. `R` Vector of the corresponding estimates for the return period, only returned for `ReturnMOM`. `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.

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.

`QuantMOM`, `Moment`, `ProbGH`, `Prob`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```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) ```

### Example output

```
```

ReIns documentation built on July 2, 2020, 4:03 a.m.