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

Estimatesscale1 shape1 scale2 shape2 alpha1.0666 0.2287 0.5332 -0.2896 0.9961Standard Errorsscale1 shape1 scale2 shape2 alpha0.07309 0.05292 0.03009 0.03676 0.02326Asymptotic Variance Covariancescale1 shape1 scale2 shape2 alphascale1 5.342e-03 -2.387e-03 9.421e-06 -7.986e-06 6.973e-06shape1 -2.387e-03 2.801e-03 -8.893e-07 5.643e-06 -4.683e-05scale2 9.421e-06 -8.893e-07 9.055e-04 -9.400e-04 2.521e-05shape2 -7.986e-06 5.643e-06 -9.400e-04 1.351e-03 -4.686e-05alpha 6.973e-06 -4.683e-05 2.521e-05 -4.686e-05 5.409e-04Optimization InformationConvergence: successfulFunction Evaluations: 71Gradient Evaluations: 19In the summary, we can see lim_u Pr[ X_1 > u | X_2 > u] = 0.02 . This is the χ statistics of Coleset al. (1999). For the parametric model, we have:χ = 2 − V (1, 1) = 2 (1 − A(0.5))For independent variables, χ = 0 while for perfect dependence, χ = 1. In our application, the value0.02 indicates that the variables are independent – which is obvious. In this perspective, it is possible tofixed some parameters. For our purpose of independence, we can run -which is equivalent to fit x and yseparately of course:> fitbvgpd(cbind(x, y), c(0, 2), model = "log", alpha = 1)Call: fitbvgpd(data = cbind(x, y), threshold = c(0, 2), model = "log", alpha = 1)Estimator: MLEDependence Model and Strenght:Model : Logisticlim_u Pr[ X_1 > u | X_2 > u] = 0Deviance: 1374.704AIC: 1382.704Marginal 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}: 500Estimatesscale1 shape1 scale2 shape21.0667 0.2285 0.5334 -0.2899Standard Errorsscale1 shape1 scale2 shape20.07310 0.05292 0.03008 0.03672Asymptotic Variance Covariance14