# Scale.2o: Bias-reduced scale estimator using second order Hill... In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

## Description

Computes the bias-reduced estimator for the scale parameter using the second-order Hill estimator.

## Usage

 ```1 2``` ```Scale.2o(data, gamma, b, beta, logk = FALSE, plot = FALSE, add = FALSE, main = "Estimates of scale parameter", ...) ```

## Arguments

 `data` Vector of n observations. `gamma` Vector of n-1 estimates for the EVI obtained from `Hill.2oQV`. `b` Vector of n-1 estimates for B obtained from `Hill.2oQV`. `beta` Vector of n-1 estimates for β obtained from `Hill.2oQV`. `logk` Logical indicating if the estimates are plotted as a function of \log(k) (`logk=TRUE`) or as a function of k. Default is `FALSE`. `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 scale parameter"`. `...` Additional arguments for the `plot` function, see `plot` for more details.

## Details

The scale estimates are computed based on the following model for the CDF: 1-F(x) = A x^{-1/γ} ( 1+ bx^{-β}(1+o(1)) ), where A:= C^{1/γ} is the scale parameter.

See Section 4.2.1 of Albrecher et al. (2017) for more details.

## Value

A list with following components:

 `k` Vector of the values of the tail parameter k. `A` Vector of the corresponding scale estimates. `C` Vector of the corresponding estimates for C, see details.

Tom Reynkens

## References

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

Beirlant, J., Schoutens, W., De Spiegeleer, J., Reynkens, T. and Herrmann, K. (2016). "Hunting for Black Swans in the European Banking Sector Using Extreme Value Analysis." In: Jan Kallsen and Antonis Papapantoleon (eds.), Advanced Modelling in Mathematical Finance, Springer International Publishing, Switzerland, pp. 147–166.

`Scale`, `ScaleEPD`, `Hill.2oQV`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```data(secura) # Hill estimator H <- Hill(secura\$size) # Bias-reduced Hill estimator H2o <- Hill.2oQV(secura\$size) # Scale estimator S <- Scale(secura\$size, gamma=H\$gamma, plot=FALSE) # Bias-reduced scale estimator S2o <- Scale.2o(secura\$size, gamma=H2o\$gamma, b=H2o\$b, beta=H2o\$beta, plot=FALSE) # Plot logarithm of scale plot(S\$k,log(S\$A), xlab="k", ylab="log(Scale)", type="l") lines(S2o\$k,log(S2o\$A), lty=2) ```