ReIns (version 1.0.7)

Scale.2o: Bias-reduced scale estimator using second order Hill estimator

Description

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

Usage

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 \(\beta\) 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.

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.

Details

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

See Section 4.2.1 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., 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.

See Also

Scale, ScaleEPD, Hill.2oQV

Examples

# NOT RUN {
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)
# }