genQQ {ReIns}

Generalised quantile plot

Description

Computes the empirical quantiles of the UH scores of a data vector and the theoretical quantiles of the standard exponential distribution. These quantiles are then plotted in a generalised QQ-plot with the theoretical quantiles on the x-axis and the empirical quantiles on the y-axis.

Usage

genQQ(data, gamma, plot = TRUE, main = "Generalised QQ-plot", ...)

generalizedQQ(data, gamma, plot = TRUE, main = "Generalised QQ-plot", ...)

Arguments

data

Vector of n observations.

gamma

Vector of n-1 estimates for the EVI, typically Hill estimates are used.

plot

Logical indicating if the quantiles should be plotted in a generalised QQ-plot, default is TRUE.

main

Title for the plot, default is "Generalised QQ-plot".

...

Additional arguments for the plot function, see plot for more details.

Details

The generalizedQQ function is the same function but with a different name for compatibility with the old S-Plus code.

The UH scores are defined as UH_{j,n}=X_{n-j,n}H_{j,n} with H_{j,n} the Hill estimates, but other positive estimates for the EVI can also be used. The appropriate positive estimates for the EVI need to be specified in gamma. The generalised QQ-plot then plots

(\log((n+1)/(k+1)), \log(X_{n-k,n}H_{k,n}))

for k=1,\ldots,n-1.

See Section 4.2.2 of Albrecher et al. (2017) for more details.

Value

A list with following components:

gqq.the

Vector of the theoretical quantiles from a standard exponential distribution.

gqq.emp

Vector of the empirical quantiles from the logarithm of the UH scores.

Author(s)

Tom Reynkens based on S-Plus code from Yuri Goegebeur.

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.

Beirlant, J., Vynckier, P. and Teugels, J.L. (1996). "Excess Function and Estimation of the Extreme-value Index." Bernoulli, 2, 293–318.

See Also

ParetoQQ, Hill

Examples

data(soa)

# Compute Hill estimator
H <- Hill(soa$size[1:5000], plot=FALSE)$gamma

# Generalised QQ-plot
genQQ(soa$size[1:5000], gamma=H)