Empirical angular measure of the k largest radii

Formula: H_n(B) = (1/k) sum 1{||X_i|| >= r_n, X_i/||X_i|| in B}

Usage

Evangia(X, k)

Arguments

X

An n x 2 matrix of observations.

k

Number of upper order statistics of the radius to keep.

Value

List with H, atoms, weights, estimate (mean angle), n_used, n, method.

Details

The data are first rank-transformed to standard Frechet margins, so the radial and angular parts separate. Each of the k largest radii contributes an atom of mass 1/k at its angle w = X_1/(X_1 + X_2), which makes H a probability measure by construction; the mean constraint of a valid angular measure is mean(w) = 1/2.

References

Einmahl, de Haan & Sinha (1997), Stoch. Proc. Appl. 70(2):143-171.

Examples

set.seed(1)
Evangia(matrix(rnorm(40), 20, 2), k = 2)
#> $H
#> [1] 1
#> 
#> $atoms
#> [1] 0.3333333 0.9523810
#> 
#> $weights
#> [1] 0.5 0.5
#> 
#> $estimate
#> [1] 0.6428571
#> 
#> $n_used
#> [1] 2
#> 
#> $n
#> [1] 20
#> 
#> $method
#> [1] "empirical angular measure of the k largest radii"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"