Hill2oQV function

Bias-reduced MLE (Quantile view)

Computes bias-reduced ML estimates of gamma based on the quantile view.

Hill.2oQV(data, start = c(1,1,1), warnings = FALSE, logk = FALSE, 
          plot = FALSE, add = FALSE, main = "Estimates of the EVI", ...)

Arguments

• data: Vector of $n$ observations.
• start: A vector of length 3 containing starting values for the first numerical optimisation (see Details). The elements are the starting values for the estimators of $\gamma$, $\mu$ and $\sigma$, respectively. Default is c(1,1,1).
• warnings: Logical indicating if possible warnings from the optimisation function are shown, default is FALSE.
• 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 of $\gamma$ should be plotted as a function of $k$, default is FALSE.
• add: Logical indicating if the estimates of $\gamma$ should be added to an existing plot, default is FALSE.
• main: Title for the plot, default is "Estimates of the EVI".
• ...: Additional arguments for the plot function, see plot for more details.

Details

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 $k$.

• gamma: Vector of the ML estimates for the EVI for each value of $k$.

• b: Vector of the ML estimates for the parameter $b$ in the regression model for each value of $k$.

• beta: Vector of the ML estimates for the parameter $\beta$ in the regression model for each value of $k$.

References

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

Beirlant J., Dierckx, G., Goegebeur Y. and Matthys, G. (1999). "Tail Index Estimation and an Exponential Regression Model." Extremes, 2, 177--200.

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

Author(s)

Tom Reynkens based on S-Plus code from Yuri Goegebeur and R code from Klaus Herrmann.

Examples

data(norwegianfire)

# Plot bias-reduced MLE (QV) as a function of k
Hill.2oQV(norwegianfire$size[norwegianfire$year==76],plot=TRUE)