ScaleEPD function

Bias-reduced scale estimator using EPD estimator

Computes the bias-reduced estimator for the scale parameter using the EPD estimator (Beirlant et al., 2016).

ScaleEPD(data, gamma, kappa, 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 EPD.
• kappa: Vector of $n-1$ estimates for $\kappa$ obtained from EPD.
• 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: $1-F(x) = A x^{-1/\gamma} ( 1+ bx^{-\beta}(1+o(1)) )$, where $A:= C^{1/\gamma}$ is the scale parameter. Using the EPD approach we replace $bx^{-\beta}$ by $\kappa/\gamma$.

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

Returns

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.

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.

Author(s)

Tom Reynkens

See Also

Scale, Scale.2o, EPD

Examples

data(secura)

# Hill estimator
H <- Hill(secura$size)
# EPD estimator
epd <- EPD(secura$size)

# Scale estimator
S <- Scale(secura$size, gamma=H$gamma, plot=FALSE)
# Bias-reduced scale estimator
Sepd <- ScaleEPD(secura$size, gamma=epd$gamma, kappa=epd$kappa, plot=FALSE)

# Plot logarithm of scale             
plot(S$k,log(S$A), xlab="k", ylab="log(Scale)", type="l")
lines(Sepd$k,log(Sepd$A), lty=2)