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

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Threshold Call: 1Number Above: 100Proportion Above: 1Estimatesscale1.684Standard Error Type: observedStandard Errorsscale0.1684Asymptotic Variance Covariancescalescale 0.02834Optimization InformationConvergence: successfulFunction Evaluations: 7Gradient Evaluations: 1> fitgpd(x, thresh = 1, scale = 2, est = "mle")Estimator: MLEDeviance: 306.4435AIC: 308.4435Varying Threshold: FALSEThreshold Call: 1Number Above: 100Proportion Above: 1Estimatesshape-0.05112Standard Error Type: observedStandard Errorsshape0.06158Asymptotic Variance Covarianceshapeshape 0.003792Optimization InformationConvergence: successfulFunction Evaluations: 28Gradient Evaluations: 6If now, we want to fit a GPD with a varying threshold, just do:> x fitgpd(x, 1:2, est = "mle")12
Estimator: MLEDeviance: -173.0829AIC: -169.0829Varying Threshold: TRUEThreshold Call: 1:2Number Above: 500Proportion Above: 1Estimatesscale shape0.29079 0.06207Standard Error Type: observedStandard Errorsscale shape0.01934 0.04929Asymptotic Variance Covariancescale shapescale 0.0003739 -0.0006681shape -0.0006681 0.0024297Optimization InformationConvergence: successfulFunction Evaluations: 43Gradient Evaluations: 12Note that the varying threshold is repeated cyclically until it matches the length of object x.3.3.2 The bivariate caseThe generic function to fit bivariate POT is fitbvgpd. There is currently 6 models for the bivariateGPD - see Annexe A. All of these models are fitted using maximum likelihood estimator. Moreover, theapproach uses censored likelihood - see (Smith et al., 1997).> x y Mlog MlogCall: fitbvgpd(data = cbind(x, y), threshold = c(0, 2), model = "log")Estimator: MLEDependence Model and Strenght:Model : Logisticlim_u Pr[ X_1 > u | X_2 > u] = 0.005Deviance: 1374.676AIC: 1384.676Marginal Threshold: 0 2Marginal Number Above: 500 500Marginal Proportion Above: 1 1Joint Number Above: 500Joint Proportion Above: 1Number of events such as {Y1 > u1} U {Y2 > u2}: 50013
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Page 11: 3.3 Fitting the GPD3.3.1 The univar
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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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