The scale parameter \(g(w)\) in the Ledford and Tawn approach is estimated empirically for \(x\) large as $$\frac{\Pr(X_P>xw, Y_P>x(1-w))}{\Pr(X_P>x, Y_P>x)}$$ where the sample (\(X_P, Y_P\)) are observations on a common unit Pareto scale. The coefficient \(\eta\) is estimated using maximum likelihood as the shape parameter of a generalized Pareto distribution on \(\min(X_P, Y_P)\).
Usage
angextrapo(dat, qu = 0.95, w = seq(0.05, 0.95, length = 20))Arguments
- dat
an \(n\) by \(2\) matrix of multivariate observations
- qu
quantile level on uniform scale at which to threshold data. Default to 0.95
- w
vector of unique angles between 0 and 1 at which to evaluate scale empirically.
Value
a list with elements
w: angles between zero and oneg: scale function at a given value ofweta: Ledford and Tawn tail dependence coefficient
References
Ledford, A.W. and J. A. Tawn (1996), Statistics for near independence in multivariate extreme values. Biometrika, 83(1), 169--187.
Examples
angextrapo(rmev(n = 1000, model = 'log', d = 2, param = 0.5))
#> $w
#> [1] 0.05000000 0.09736842 0.14473684 0.19210526 0.23947368 0.28684211
#> [7] 0.33421053 0.38157895 0.42894737 0.47631579 0.52368421 0.57105263
#> [13] 0.61842105 0.66578947 0.71315789 0.76052632 0.80789474 0.85526316
#> [19] 0.90263158 0.95000000
#>
#> $g
#> [1] 36.275862 19.620690 13.896552 11.068966 9.448276 8.413793 7.724138
#> [8] 7.275862 7.000000 6.896552 6.896552 7.000000 7.275862 7.724138
#> [15] 8.413793 9.448276 11.068966 13.896552 19.620690 36.275862
#>
#> $eta
#> shape
#> 0.4738083
#>