An R Package for Univariate and Bivariate Peaks Over Threshold ...

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Modified Scale−0.2 0.0 0.2 0.4 0.6 0.8●● ● ●●●●●● ●● ● ●●●● ● ●● ●●●●●●Shape−0.8 −0.6 −0.4 −0.2 0.0 0.2●● ● ● ●● ●●●●●●●● ●● ● ●● ● ●●●●●0.90 0.94 0.980.90 0.94 0.98ThresholdThresholdFigure 1: The threshold selection using the tcplot function6
Mean Residual Life PlotMean Excess0.25 0.30 0.35 0.40 0.45 0.501.0 1.5 2.0 2.5ThresholdFigure 2: The threshold selection using the mrlplot functionThe quantity E [X − µ 1 |X > µ 1 ] is linear in µ 1 . Or, E [X − µ 1 |X > µ 1 ] is simply the mean of excessesabove the threshold µ 1 which can easily be estimated using the empirical mean.A mean residual life plot consists in representing points:{(n µ) }1 ∑µ, x i,nµ − µ : µ ≤ x maxn µi=1(3.6)where n µ is the number of observations x above the threshold µ, x i,nµ is the i-th observation above thethreshold µ and x max is the maximum of the observations x.Confidence intervals can be added to this plot as the empirical mean can be supposed to be normallydistributed (Central Limit Theorem). However, normality doesn’t hold anymore for high threshold asthere are less and less excesses. Moreover, by construction, this plot always converge to the point (x max , 0).Here is another synthetic example.> x mrlplot(x, u.range = c(1, quantile(x, probs = 0.995)), col = c("green",+ "black", "green"), nt = 200)Figure 2 displays the mean residual life plot. A threshold around 2.5 should be reasonable.7
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Page 15 and 16: Pickands' Dependence FunctionA (x)0
Page 17 and 18: 3.4 Confidence IntervalsOnce a stat
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An R Package for Univariate and Bivariate Peaks Over Threshold ...

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