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", ...)
A list with following components:
Vector of the theoretical quantiles from a standard exponential distribution.
Vector of the empirical quantiles from the logarithm of the UH scores.
Vector of \(n\) observations.
Vector of \(n-1\) estimates for the EVI, typically Hill estimates are used.
Logical indicating if the quantiles should be plotted in a generalised QQ-plot, default is TRUE.
Title for the plot, default is "Generalised QQ-plot".
Additional arguments for the plot function, see plot for more details.
Tom Reynkens based on S-Plus code from Yuri Goegebeur.
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.
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.
ParetoQQ, Hill
data(soa)
# Compute Hill estimator
H <- Hill(soa$size[1:5000], plot=FALSE)$gamma
# Generalised QQ-plot
genQQ(soa$size[1:5000], gamma=H)
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