genQQ function

Generalised quantile plot

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.

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

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

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

Returns

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.

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.

Author(s)

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

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)