Tests of rotational symmetry of a Bingham distribution

Tests of rotational symmetry of a Bingham distribution

Usage

watson_symmetry_test(lambda, k, n, level = 0.95)

Arguments

lambda

numeric.

k

Bingham concentrations \((0, \kappa_\text{major}, \kappa_\text{minor})\)

n

integer. Number of vectors

level

numeric. Confidence level (\(1-\alpha\)). Default is 0.95

Value

object of class "Test"

Details

Whether a Bingham distribution is a Watson distribution: $$w_b = n \sum((k_i - \text{mean}(k)) (\lambda_i - \text{mean}(\lambda)))$$ over the two smallest eigenvalues tests the bipolar form, \(w_g\) over the two largest the girdle form, each asymptotically chi-square with 2 degrees of freedom (MJ eq. 10.7.8-10.7.18). \(k_i\) are the maximum-likelihood Bingham concentrations \((0, \kappa_\text{major}, \kappa_\text{minor})\) and \(\lambda_i\) the eigenvalues of the orientation tensor, both in descending order. Mardia & Jupp's calcite example gives \(w_g = 1.33\) (girdle accepted) and \(w_b = 9.97\).

See also

Other tests: bingham_test(), fisher_ftest(), fisher_goodness_of_fit(), fisher_kent_test(), rayleigh_test3(), wellner()

Examples

r <- rbingham_eig(100, c(100, 1, 0), "Line")
e <- ot_eigen(r)$values
k <- c(0, bingham_statistics(r)$k)
watson_symmetry_test(e, k, n = nrow(r))
#> $bipolar
#> 
#> 	Watson bipolar symmetry
#> 
#> statistic = 3290.3, df = 2, p-value < 2.2e-16
#> critical value (95% confidence level) = 5.9915
#> null hypothesis rejected at the 95% confidence level: yes
#> source: Mardia & Jupp (2000), eq. 10.7.8-10.7.18 
#> 
#> 
#> $girdle
#> 
#> 	Watson girdle symmetry
#> 
#> statistic = -1046.9, df = 2, p-value = 1
#> critical value (95% confidence level) = 5.9915
#> null hypothesis rejected at the 95% confidence level: no
#> source: Mardia & Jupp (2000), eq. 10.7.8-10.7.18 
#> 
#>